Apache MADlib is an open source project that endeavors to adhere in all respects to the principles of The Apache Way.
-
MADlib grew out of discussions between database engine developers, data scientists, IT architects and academics interested in new approaches to scalable, sophisticated in-database analytics. These discussions were written up in a paper in VLDB 2009 that coined the term “MAD Skills” for data analysis. The MADlib software project began the following year as a collaboration between researchers at UC Berkeley and engineers and data scientists at EMC/Greenplum (later Pivotal).
-
In September 2015 MADlib was accepted into the Apache Software Foundation Incubator and graduated to a Top Level Project in July 2017.
-
Some of the past and present participants in this project are:
If you are interested in joining our project please consider joining our User or Developer mailing lists. Everyone is welcome.
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-
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-
-
Reporting Issues
-
-
-
We need your feedback, so if you find a bug, would like to suggest an improvement, or create a request then please follow the steps below to let us know.
-
-
Start by logging into MADlib JIRA. If you don’t have an account yet you can create one yourself.
-
To report a bug:
-
-
Create a new issue -> Bug.
-
Fill in the required (and optional) fields.
-
-
To suggest an improvement or a new feature:
-
-
Create a new issue -> New Feature.
-
Fill in the required (and optional) fields.
-
-
Submit the issue. You should receive an email confirmation. Thanks for your feedback!
Interaction and collaboration with the open source and academic communities continues to be a core foundation of our project. We have papers published in major conferences. Including:
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-
-
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-x = c(1, 2, 3)
-sum(x)
-if (sum(x) == 0){ print("yes")}
-if (sum(x) == 0){ print("yes") } else {print("no")}
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-ErrorDocument 500 http://madlib.incubator.apache.org/docs/latest
-ErrorDocument 404 http://madlib.incubator.apache.org/docs/latest
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+# API Reference
+
+Apache MADlib provides comprehensive API documentation for all algorithms and functions. The documentation is organized by functional areas and includes detailed examples, syntax, and usage patterns.
+
+## Quick Navigation
+
+### Supervised Learning
+- [Linear Regression](../versions/latest/group__grp__linreg.html) - Linear regression analysis
+- [Logistic Regression](../versions/latest/group__grp__logreg.html) - Binary and multinomial logistic regression
+- [Elastic Net](../versions/latest/group__grp__elasticnet.html) - Regularized linear and logistic regression
+- [Support Vector Machines](../versions/latest/group__grp__svm.html) - Classification and regression with SVMs
+- [Decision Trees](../versions/latest/group__grp__decision__tree.html) - Tree-based classification and regression
+- [Random Forest](../versions/latest/group__grp__random__forest.html) - Ensemble tree methods
+- [XGBoost](../versions/latest/group__grp__xgboost.html) - Gradient boosting framework
+
+### Unsupervised Learning
+- [k-Means Clustering](../versions/latest/group__grp__kmeans.html) - Partitional clustering algorithm
+- [DBSCAN](../versions/latest/group__grp__dbscan.html) - Density-based clustering
+- [Principal Component Analysis](../versions/latest/group__grp__pca.html) - Dimensionality reduction
+- [Latent Dirichlet Allocation](../versions/latest/group__grp__lda.html) - Topic modeling
+
+### Deep Learning
+- [Multi-Layer Perceptron](../versions/latest/group__grp__nn.html) - Neural networks
+- [Keras Integration](../versions/latest/group__grp__keras.html) - Deep learning with Keras/TensorFlow
+- [Model Selection](../versions/latest/group__grp__model__selection.html) - AutoML and hyperparameter tuning
+- [GPU Configuration](../versions/latest/group__grp__gpu__configuration.html) - GPU setup and management
+
+### Graph Analytics
+- [PageRank](../versions/latest/group__grp__pagerank.html) - Web page ranking algorithm
+- [Single Source Shortest Path](../versions/latest/group__grp__sssp.html) - Graph shortest paths
+- [Weakly Connected Components](../versions/latest/group__grp__wcc.html) - Graph connectivity
+- [Graph Measures](../versions/latest/group__grp__graph__measures.html) - Graph analysis metrics
+
+### Statistics and Data Analysis
+- [Descriptive Statistics](../versions/latest/group__grp__summary.html) - Summary statistics
+- [Correlation Analysis](../versions/latest/group__grp__correlation.html) - Correlation and covariance
+- [Hypothesis Testing](../versions/latest/group__grp__stats__tests.html) - Statistical tests
+- [Association Rules](../versions/latest/group__grp__assoc__rules.html) - Market basket analysis
+
+### Data Preprocessing
+- [Data Transformation](../versions/latest/group__grp__datatrans.html) - Data preparation utilities
+- [Encoding Categorical Variables](../versions/latest/group__grp__encode__categorical.html) - Categorical data handling
+- [Train/Test Split](../versions/latest/group__grp__train__test__split.html) - Data splitting utilities
+- [Sampling](../versions/latest/group__grp__sampling.html) - Data sampling methods
+
+### Model Validation
+- [Cross Validation](../versions/latest/cross__validation_8sql__in.html) - Model validation techniques
+- [Prediction Metrics](../versions/latest/pred__metrics_8sql__in.html) - Model evaluation metrics
+
+## Complete API Reference
+
+### Markdown Documentation (Developer-Friendly)
+
+For easy-to-maintain Markdown documentation of the latest versions:
+
+[**Latest Version (v2.1.0)**](latest/index.md){ .md-button .md-button--primary }
+[**Version 2.0.0**](v2.0.0/index.md){ .md-button }
+
+### HTML Documentation (Full-Featured)
+
+For the complete, searchable API documentation with full navigation and examples:
+
+[**Browse Complete HTML Documentation**](../docs/latest/index.html){ .md-button }
+
+## Version-Specific Documentation
+
+- [Latest (2.1.0)](../docs/latest/index.html)
+- [Version 2.0.0](../docs/v2.0.0/index.html)
+- [Version 1.21.0](../docs/v1.21.0/index.html)
+- [All Versions](../documentation/versions.md)
+
+## Getting Started
+
+If you're new to MADlib, we recommend starting with:
+
+1. **[Installation Guide](../documentation/installation.md)** - Set up MADlib
+2. **[Quick Start](../documentation/quick-start.md)** - Your first MADlib model
+3. **[Examples](../documentation/examples.md)** - Common use cases
+4. **API Reference** (this page) - Detailed function documentation
+
+## Search and Navigation
+
+The complete API documentation includes:
+
+- **Full-text search** across all functions and examples
+- **Interactive navigation tree** for browsing by category
+- **Cross-references** between related functions
+- **Syntax highlighting** for SQL code examples
+- **Mathematical formulas** rendered with MathJax
+
+## Contributing to Documentation
+
+The API documentation is automatically generated from source code comments. To improve the documentation:
+
+1. **Report Issues**: Use [MADlib JIRA](https://issues.apache.org/jira/browse/MADLIB) for documentation bugs
+2. **Contribute Examples**: Submit pull requests with improved examples
+3. **Improve Comments**: Enhance source code documentation
+
+For more information, see our [Contributing Guide](../community/contributing.md).
+
+---
+
+!!! tip "Pro Tip"
+ Use the search function in the complete API documentation to quickly find specific functions or examples. The search supports partial matches and function names.
\ No newline at end of file
diff --git a/docs/api/linear-regression.md b/docs/api/linear-regression.md
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+# Linear Regression
+
+Linear regression models the relationship between a scalar dependent variable and one or more explanatory variables using linear predictor functions.
+
+## Overview
+
+Linear regression is one of the most fundamental algorithms in machine learning and statistics. MADlib provides a robust implementation that works efficiently with large datasets in PostgreSQL and Greenplum Database.
+
+## Key Features
+
+- **Scalable**: Handles large datasets efficiently using parallel processing
+- **Robust**: Includes options for heteroskedasticity-consistent standard errors
+- **Flexible**: Supports various input formats and grouping variables
+- **Statistical**: Provides comprehensive statistical output including R², p-values, and confidence intervals
+
+## Basic Usage
+
+### Training a Model
+
+```sql
+SELECT madlib.linregr_train(
+ 'source_table', -- Input table
+ 'output_table', -- Output model table
+ 'dependent_var', -- Y variable
+ 'ARRAY[1, x1, x2]' -- Independent variables (include 1 for intercept)
+);
+```
+
+### Making Predictions
+
+```sql
+SELECT madlib.linregr_predict(
+ (SELECT coef FROM output_table),
+ ARRAY[1, new_x1, new_x2]
+) as prediction;
+```
+
+## Parameters
+
+| Parameter | Type | Description |
+|-----------|------|-------------|
+| `source_table` | TEXT | Name of the table containing training data |
+| `out_table` | TEXT | Name of the output table for model |
+| `dependent_varname` | TEXT | Name of the dependent variable column |
+| `independent_varname` | TEXT | Expression for independent variables |
+| `grouping_cols` | TEXT | Optional grouping columns |
+| `heteroskedasticity_option` | BOOLEAN | Enable robust standard errors |
+
+## Output
+
+The trained model includes:
+
+- **coef**: Coefficient vector
+- **r2**: R-squared value
+- **std_err**: Standard errors
+- **t_stats**: T-statistics
+- **p_values**: P-values for significance testing
+- **condition_no**: Condition number for numerical stability
+
+## Examples
+
+### Simple Linear Regression
+
+```sql
+-- Create sample data
+CREATE TABLE housing AS
+SELECT
+ random() * 1000 + 500 as sqft,
+ random() * 100000 + 200000 as price
+FROM generate_series(1, 1000);
+
+-- Train model
+SELECT madlib.linregr_train(
+ 'housing',
+ 'housing_model',
+ 'price',
+ 'ARRAY[1, sqft]'
+);
+
+-- View results
+SELECT * FROM housing_model;
+```
+
+### Multiple Linear Regression
+
+```sql
+-- Multiple predictors
+SELECT madlib.linregr_train(
+ 'housing',
+ 'housing_multi_model',
+ 'price',
+ 'ARRAY[1, sqft, bedrooms, bathrooms]'
+);
+```
+
+### Grouped Regression
+
+```sql
+-- Separate models by region
+SELECT madlib.linregr_train(
+ 'housing',
+ 'housing_by_region',
+ 'price',
+ 'ARRAY[1, sqft]',
+ 'region'
+);
+```
+
+## Advanced Features
+
+### Robust Standard Errors
+
+```sql
+-- Heteroskedasticity-consistent standard errors
+SELECT madlib.linregr_train(
+ 'housing',
+ 'housing_robust',
+ 'price',
+ 'ARRAY[1, sqft]',
+ NULL, -- no grouping
+ TRUE -- robust standard errors
+);
+```
+
+### Model Diagnostics
+
+```sql
+-- Check model quality
+SELECT
+ r2,
+ CASE WHEN r2 > 0.7 THEN 'Good fit'
+ WHEN r2 > 0.5 THEN 'Moderate fit'
+ ELSE 'Poor fit' END as fit_quality
+FROM housing_model;
+```
+
+## Performance Tips
+
+1. **Include intercept**: Always include 1 in the independent variables array for the intercept term
+2. **Scale features**: Consider scaling features if they have very different ranges
+3. **Check condition number**: High condition numbers (>1000) may indicate multicollinearity
+4. **Use appropriate data types**: FLOAT8 is recommended for numerical stability
+
+## Related Functions
+
+- [Logistic Regression](../versions/latest/group__grp__logreg.html) - For binary/categorical outcomes
+- [Elastic Net](../versions/latest/group__grp__elasticnet.html) - Regularized regression
+- [Robust Regression](../versions/latest/group__grp__robust.html) - Outlier-resistant regression
+
+## Complete Documentation
+
+For detailed syntax, additional parameters, and more examples:
+
+[**View Complete Linear Regression Documentation**](../versions/latest/group__grp__linreg.html){ .md-button .md-button--primary }
+
+## Mathematical Background
+
+Linear regression finds the best-fitting line through data points by minimizing the sum of squared residuals:
+
+$$\min_{\beta} \sum_{i=1}^{n} (y_i - X_i\beta)^2$$
+
+Where:
+- $y_i$ is the dependent variable
+- $X_i$ is the vector of independent variables
+- $\beta$ is the coefficient vector to be estimated
+
+The solution is given by the normal equation:
+$$\hat{\beta} = (X^TX)^{-1}X^Ty$$
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diff --git a/docs/asf/index.md b/docs/asf/index.md
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+---
+hide:
+ - navigation
+ - toc
+---
+# Apache Software Foundation (ASF) Links
+
+- [ASF Homepage](https://www.apache.org/)
+- [Events](https://www.apache.org/events/current-event)
+- [License](https://www.apache.org/licenses/)
+- [Code of Conduct](https://www.apache.org/foundation/policies/conduct.html)
+- [ASF Data Privacy](https://privacy.apache.org/policies/privacy-policy-public.html)
+- [ASF Security Team](https://www.apache.org/security/)
+- [Thanks to our Sponsors](https://www.apache.org/foundation/thanks.html)
+- [The Apache Software Foundation Sponsorship Program](https://www.apache.org/foundation/sponsorship.html)
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+
+
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+/* Custom styles for Apache MADlib site */
+
+/* Logo switching for dark mode */
+/* Light mode - use SVG logo for header */
+[data-md-color-scheme="default"] .md-header__button.md-logo img,
+[data-md-color-scheme="default"] .md-logo img {
+ content: url('../madlib.svg');
+}
+
+/* Dark mode - use watermark PNG logo for header */
+[data-md-color-scheme="slate"] .md-header__button.md-logo img,
+[data-md-color-scheme="slate"] .md-logo img {
+ content: url('../madlib-watermark.png');
+}
+
+/* Hero logo switching for dark mode */
+/* Light mode - use SVG logo */
+[data-md-color-scheme="default"] .hero-logo {
+ content: url('../madlib.svg');
+}
+
+/* Dark mode - use watermark PNG logo */
+[data-md-color-scheme="slate"] .hero-logo {
+ content: url('../madlib-watermark.png');
+}
+
+/* Hero section styling */
+.hero-center {
+ text-align: center;
+ margin: 2rem 0;
+}
+
+.hero-center h1 {
+ margin-bottom: 1.5rem;
+ font-size: 2.5rem;
+ font-weight: 600;
+ line-height: 1.2;
+}
+
+.hero-center img {
+ display: block;
+ margin: 0 auto;
+ max-width: 100%;
+ height: auto;
+}
+
+/* Latest news banner */
+.latest-news-banner {
+ background: linear-gradient(135deg, #667eea 0%, #764ba2 100%);
+ color: white;
+ padding: 1.5rem 2rem;
+ border-radius: 8px;
+ margin: 2rem 0;
+ box-shadow: 0 4px 6px rgba(0, 0, 0, 0.1);
+}
+
+.latest-news-banner h3 {
+ margin: 0 0 0.5rem 0;
+ font-size: 1.2rem;
+ font-weight: 600;
+ color: white;
+}
+
+.latest-news-banner p {
+ margin: 0.5rem 0;
+ font-size: 0.95rem;
+ opacity: 0.95;
+}
+
+.latest-news-banner a {
+ color: white;
+ text-decoration: underline;
+ font-weight: 500;
+}
+
+.latest-news-banner a:hover {
+ opacity: 0.8;
+}
+
+.latest-news-banner .news-date {
+ font-size: 0.85rem;
+ opacity: 0.8;
+ margin-top: 0.5rem;
+}
+
+/* Latest news section - Light mode */
+[data-md-color-scheme="default"] .latest-news-light {
+ background: #f8f9fa;
+ border-left: 4px solid #667eea;
+ color: #333;
+ padding: 1rem 2rem;
+ border-radius: 4px;
+ margin: 2rem auto;
+ text-align: center;
+ max-width: 600px;
+}
+
+[data-md-color-scheme="default"] .latest-news-light h3 {
+ margin: 0;
+ font-size: 1.1rem;
+ font-weight: 600;
+ color: #333;
+}
+
+[data-md-color-scheme="default"] .latest-news-light h3 a {
+ color: #667eea;
+ text-decoration: none;
+ font-weight: 600;
+}
+
+[data-md-color-scheme="default"] .latest-news-light h3 a:hover {
+ text-decoration: underline;
+}
+
+/* Latest news section - Dark mode */
+[data-md-color-scheme="slate"] .latest-news-light {
+ background: rgba(102, 126, 234, 0.1);
+ border-left: 4px solid #8b9ff5;
+ color: #e0e0e0;
+ padding: 1rem 2rem;
+ border-radius: 4px;
+ margin: 2rem auto;
+ text-align: center;
+ max-width: 600px;
+}
+
+[data-md-color-scheme="slate"] .latest-news-light h3 {
+ margin: 0;
+ font-size: 1.1rem;
+ font-weight: 600;
+ color: #e0e0e0;
+}
+
+[data-md-color-scheme="slate"] .latest-news-light h3 a {
+ color: #a5b4fc;
+ text-decoration: none;
+ font-weight: 600;
+}
+
+[data-md-color-scheme="slate"] .latest-news-light h3 a:hover {
+ text-decoration: underline;
+ color: #fbbf24;
+}
+
+/* Homepage specific styles - Feature cards with icons */
+.homepage-content .grid.cards img[align="left"] {
+ margin-right: 1rem;
+ margin-bottom: 0.5rem;
+ border-radius: 8px;
+ background: #f8f9fa;
+ padding: 8px;
+}
+
+/* Homepage specific styles - Center main section headings */
+.homepage-content h2 {
+ text-align: center;
+ margin: 3rem 0 2rem 0;
+}
+
+.homepage-content h3 {
+ text-align: center;
+ margin: 2rem 0 1.5rem 0;
+}
+
+/* Homepage specific styles - Center text content */
+.homepage-content .text-center {
+ text-align: center;
+ max-width: 800px;
+ margin: 0 auto;
+}
+
+.homepage-content .text-center p {
+ margin: 1rem 0;
+}
+
+/* Responsive adjustments */
+@media screen and (max-width: 768px) {
+ .hero-center h1 {
+ font-size: 2rem;
+ }
+
+ .hero-center img {
+ width: 250px;
+ }
+
+ .latest-news-banner,
+ .latest-news-light {
+ padding: 1rem 1.5rem;
+ }
+
+ .homepage-content .grid.cards img[align="left"] {
+ width: 48px !important;
+ margin-right: 0.75rem;
+ }
+}
+
+@media screen and (max-width: 480px) {
+ .hero-center h1 {
+ font-size: 1.75rem;
+ }
+
+ .hero-center img {
+ width: 200px;
+ }
+
+ .latest-news-banner,
+ .latest-news-light {
+ padding: 1rem;
+ }
+
+ .homepage-content .grid.cards img[align="left"] {
+ width: 40px !important;
+ margin-right: 0.5rem;
+ }
+}
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+authors:
+ madlib-team:
+ name: Apache MADlib
+ description: Apache MADlib
+ avatar: ../assets/madlib.svg
+ url: https://github.com/apache/madlib
\ No newline at end of file
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@@ -0,0 +1,9 @@
+---
+hide:
+ - navigation
+ - toc
+---
+
+# MADlib Blog
+
+Welcome to the Apache MADlib blog! Here you'll find the latest news, release announcements, tutorials, and insights from the MADlib community.
\ No newline at end of file
diff --git a/docs/blog/posts/madlib-1.17.0-release.md b/docs/blog/posts/madlib-1.17.0-release.md
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@@ -0,0 +1,57 @@
+---
+date: 2020-04-09
+categories:
+ - Release
+authors:
+ - madlib-team
+---
+
+# MADlib 1.17.0 Release
+
+On April 9, 2020, MADlib completed its seventh release as an Apache Software Foundation Top Level Project.
+
+
+
+## New Features
+
+### Deep Learning Advancements
+- **Model selection framework for Keras with TensorFlow backend**: Comprehensive framework for model architecture search and hyperparameter optimization with GPU acceleration
+- **Support for heterogeneous clusters**: Enhanced support for clusters where GPUs are attached to only certain segment hosts, enabling flexible deployment scenarios
+- **Support inference for imported models**: "Bring your own model" capability - perform inference on models not trained in MADlib
+- **Support transfer learning for multiple model fit function**: Advanced transfer learning capabilities for leveraging pre-trained models
+- **Generate model selection table for grid search or random search**: Automated model selection with comprehensive result tracking
+- **Helper function to get GPU type and configuration**: Utility functions for GPU discovery and configuration management
+
+### Clustering Enhancements
+- **k-Means clustering - Select optimal number of centroids**: Automated cluster number selection using elbow or silhouette methods for optimal clustering results
+
+### Platform Support
+- **PostgreSQL 12 support**: Extended compatibility to PostgreSQL 12 for broader deployment options
+
+## Improvements
+
+### Algorithm Enhancements
+- **Association rules - Add option to set number of posterior rules**: Enhanced control over association rule mining output
+- **Correlation and covariance - Improve memory usage with large number of groups**: Better memory management for statistical computations with many grouping variables
+
+### Deep Learning Performance
+- **Improve performance of mini-batch preprocessor and fit functions**: Significant performance improvements for neural network training workflows
+
+### Documentation and Infrastructure
+- **Docs - Improve installation guide on wiki**: Enhanced installation documentation for better user experience
+- **Graph - SSSP should not show vertices in output table that are unreachable**: Cleaner output for Single Source Shortest Path algorithm
+- **LDA - Add stopping criteria on perplexity**: Better convergence control for Latent Dirichlet Allocation
+
+## Download and Installation
+
+You are invited to [download the 1.17.0 release](https://dist.apache.org/repos/dist/release/madlib/1.17.0/) and [review the release notes](https://github.com/apache/madlib/blob/master/RELEASE_NOTES).
+
+For more details about the new deep learning features, please refer to the [Apache MADlib deep learning notes](https://cwiki.apache.org/confluence/display/MADLIB/Deep+Learning) and the [Jupyter notebook examples](https://github.com/apache/madlib-site/tree/asf-site/community-artifacts/Deep-learning).
+
+## What's New
+
+This release marks a significant milestone in MADlib's deep learning capabilities, introducing a comprehensive model selection framework that automates the often complex process of finding optimal neural network architectures and hyperparameters. The support for heterogeneous GPU clusters makes MADlib more flexible for real-world deployment scenarios.
+
+The "bring your own model" feature opens up new possibilities for integrating MADlib with existing machine learning workflows, while the enhanced k-means clustering with automatic cluster selection makes unsupervised learning more accessible.
+
+For technical support and questions, please visit our [community forums](../../community/index.md) or check the [documentation](../../documentation/user-guide.md).
\ No newline at end of file
diff --git a/docs/blog/posts/madlib-1.18.0-release.md b/docs/blog/posts/madlib-1.18.0-release.md
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@@ -0,0 +1,55 @@
+---
+date: 2021-04-05
+categories:
+ - Release
+authors:
+ - madlib-team
+---
+
+# MADlib 1.18.0 Release
+
+On April 5, 2021, MADlib completed its eighth release as an Apache Software Foundation Top Level Project.
+
+
+
+## New Features
+
+### Deep Learning Enhancements
+- **New grid and random search methods**: Advanced hyperparameter optimization techniques for neural networks
+- **AutoML methods Hyperband and Hyperopt**: Automated machine learning capabilities for efficient model selection and hyperparameter tuning
+- **Custom loss functions and custom metrics**: Flexibility to define domain-specific loss functions and evaluation metrics
+- **TensorBoard support**: Integration with TensorBoard for visualization and monitoring of training progress
+- **Multi-input and output support for fit and evaluate**: Enhanced neural network architecture support for complex models
+
+### Clustering Algorithms
+- **DBSCAN - Density based clustering (phase 1)**: Initial implementation of DBSCAN algorithm for density-based clustering
+
+## Improvements
+
+### Deep Learning Performance
+- **Implement cache logic to speed performance**: Intelligent caching mechanisms to reduce computation overhead
+- **Reduce GPU idle time when moving model state between workers**: Optimized GPU utilization in distributed training scenarios
+- **Use Keras version from TensorFlow**: Streamlined dependency management by using TensorFlow's built-in Keras
+- **Add top n to evaluate**: Enhanced model evaluation with top-n accuracy metrics
+
+### Graph Algorithm Enhancements
+- **Support BIGINT for all graph methods**: Extended data type support for large-scale graph processing
+
+### Infrastructure Improvements
+- **Switch to CloudBees (was Jenkins)**: Upgraded continuous integration infrastructure for better reliability and performance
+
+## Download and Installation
+
+You are invited to [download the 1.18.0 release](https://dist.apache.org/repos/dist/release/madlib/1.18.0/) and [review the release notes](https://github.com/apache/madlib/blob/master/RELEASE_NOTES).
+
+Also please refer to the [list of supported databases and OS](https://cwiki.apache.org/confluence/display/MADLIB/Database+and+OS+Support) for compatibility information.
+
+## What's New
+
+This release represents a major advancement in MADlib's deep learning capabilities, introducing AutoML features that make machine learning more accessible to users without extensive hyperparameter tuning expertise. The TensorBoard integration provides powerful visualization capabilities for monitoring and debugging neural network training.
+
+The performance improvements, particularly in GPU utilization and caching, make MADlib more suitable for production deep learning workloads at scale.
+
+For more details about the new deep learning features, please refer to the [Apache MADlib deep learning notes](https://cwiki.apache.org/confluence/display/MADLIB/Deep+Learning) and the [Jupyter notebook examples](https://github.com/apache/madlib-site/tree/asf-site/community-artifacts/Deep-learning).
+
+For technical support and questions, please visit our [community forums](../../community/index.md) or check the [documentation](../../documentation/user-guide.md).
\ No newline at end of file
diff --git a/docs/blog/posts/madlib-1.19.0-release.md b/docs/blog/posts/madlib-1.19.0-release.md
new file mode 100644
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--- /dev/null
+++ b/docs/blog/posts/madlib-1.19.0-release.md
@@ -0,0 +1,50 @@
+---
+date: 2022-03-08
+categories:
+ - Release
+authors:
+ - madlib-team
+---
+
+# MADlib 1.19.0 Release
+
+On March 8, 2022, MADlib completed its ninth release as an Apache Software Foundation Top Level Project.
+
+
+
+## New Features
+
+### Clustering Algorithms
+- **DBSCAN: Fast parallel-optimized DBSCAN**: Introduction of Density-Based Spatial Clustering of Applications with Noise (DBSCAN) algorithm with parallel optimization for large-scale clustering tasks
+
+### Neural Network Enhancements
+- **MLP: Add rmsprop and Adam optimization techniques**: Enhanced Multi-Layer Perceptron (MLP) with additional optimization algorithms for better convergence and performance
+
+## Improvements
+
+### Graph Algorithm Optimizations
+- **Improve WCC subtx count and catalog entry frequency**: Performance improvements for Weakly Connected Components algorithm, reducing transaction overhead and catalog access frequency
+
+### Neural Network Improvements
+- **MLP: Set lambda value for minibatch**: Better regularization control for mini-batch training in neural networks
+
+### Algorithm Stability
+- **GLM-multinom: Use non-temp tables in GroupIterationController**: Improved stability for multinomial GLM by using persistent tables instead of temporary ones
+
+### Infrastructure
+- **Jenkins: Add new dockerfile for PG11**: Enhanced continuous integration with PostgreSQL 11 support
+- **Build: Use dynamic_library_path for module pathname**: Improved library loading mechanism for better portability
+
+## Download and Installation
+
+You are invited to [download the 1.19.0 release](https://dist.apache.org/repos/dist/release/madlib/1.19.0/) and [review the release notes](https://github.com/apache/madlib/blob/master/RELEASE_NOTES).
+
+Also please refer to the [list of supported databases and OS](https://cwiki.apache.org/confluence/display/MADLIB/Database+and+OS+Support) for compatibility information.
+
+## What's New
+
+This release introduces DBSCAN, a powerful clustering algorithm that can find clusters of arbitrary shape and is robust to outliers. The parallel-optimized implementation makes it suitable for large-scale data analysis tasks.
+
+The neural network improvements provide more optimization options and better performance for deep learning workloads, while the various performance optimizations ensure better scalability across all algorithms.
+
+For technical support and questions, please visit our [community forums](../../community/index.md) or check the [documentation](../../documentation/user-guide.md).
\ No newline at end of file
diff --git a/docs/blog/posts/madlib-1.20.0-release.md b/docs/blog/posts/madlib-1.20.0-release.md
new file mode 100644
index 00000000..d1785f2e
--- /dev/null
+++ b/docs/blog/posts/madlib-1.20.0-release.md
@@ -0,0 +1,44 @@
+---
+date: 2022-08-03
+categories:
+ - Release
+authors:
+ - madlib-team
+---
+
+# MADlib 1.20.0 Release
+
+On August 3, 2022, MADlib completed its tenth release as an Apache Software Foundation Top Level Project.
+
+
+
+## New Features
+
+### XGBoost Integration
+- **Python based XGBoost with single and grid search executions**: Complete integration of XGBoost algorithm with both single model training and automated hyperparameter grid search capabilities
+
+### Graph Algorithm Enhancements
+- **Add multicolumn support for WCC and Pagerank**: Enhanced Weakly Connected Components (WCC) and PageRank algorithms to support composite identifiers, making them more suitable for complex real-world graph data
+
+## Improvements
+
+### Performance Optimizations
+- **Reuse update plan in GroupIterationController**: Significant performance improvement for iterative algorithms by reusing query execution plans
+- **Adjust ORCA to reduce planning time**: Optimized query planning for Elastic Net, GLM, and SVM algorithms, reducing overall execution time
+
+### Documentation
+- **Update online examples for various modules**: Refreshed and expanded examples across multiple algorithm modules for better user experience
+
+## Download and Installation
+
+You are invited to [download the 1.20.0 release](https://dist.apache.org/repos/dist/release/madlib/1.20.0/) and [review the release notes](https://github.com/apache/madlib/blob/master/RELEASE_NOTES).
+
+Also please refer to the [list of supported databases and OS](https://cwiki.apache.org/confluence/display/MADLIB/Database+and+OS+Support) for compatibility information.
+
+## What's New
+
+This release marks a significant milestone with the introduction of XGBoost, one of the most popular gradient boosting frameworks. The Python-based implementation provides seamless integration with MADlib's existing ecosystem while offering both simple training and advanced hyperparameter optimization capabilities.
+
+The graph algorithm improvements make MADlib more suitable for complex network analysis tasks, while the performance optimizations ensure better scalability for large datasets.
+
+For technical support and questions, please visit our [community forums](../../community/index.md) or check the [documentation](../../documentation/user-guide.md).
\ No newline at end of file
diff --git a/docs/blog/posts/madlib-1.21.0-release.md b/docs/blog/posts/madlib-1.21.0-release.md
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--- /dev/null
+++ b/docs/blog/posts/madlib-1.21.0-release.md
@@ -0,0 +1,40 @@
+---
+date: 2023-03-01
+categories:
+ - Release
+authors:
+ - madlib-team
+---
+
+# MADlib 1.21.0 Release
+
+On March 1, 2023, MADlib completed its eleventh release as an Apache Software Foundation Top Level Project.
+
+
+
+## New Features
+
+### Graph Algorithms
+- **Add warm start for weakly connected components**: Improved performance for iterative graph computations by allowing algorithms to resume from previous states
+- **Add multicolumn identifier support for SSSP and APSP**: Enhanced Single Source Shortest Path (SSSP) and All Pairs Shortest Path (APSP) algorithms to work with composite keys
+
+### Platform Support
+- **Add support for Photon3 OS**: Extended platform compatibility to include VMware's Photon OS 3.0
+
+## Improvements
+
+### XGBoost Enhancements
+- **Add support for bigint and varchar columns**: Expanded data type support for XGBoost algorithms, allowing for more flexible data processing
+- **Enable eval_metrics parameter**: Added evaluation metrics configuration for better model assessment during training
+
+## Download and Installation
+
+You are invited to [download the 1.21.0 release](https://dist.apache.org/repos/dist/release/madlib/1.21.0/) and [review the release notes](https://github.com/apache/madlib/blob/master/RELEASE_NOTES).
+
+Also please refer to the [list of supported databases and OS](https://cwiki.apache.org/confluence/display/MADLIB/Database+and+OS+Support) for compatibility information.
+
+## What's New
+
+This release focuses on enhancing graph algorithms and expanding XGBoost capabilities. The warm start feature for graph algorithms significantly improves performance for large-scale graph computations, while the multicolumn support makes the algorithms more flexible for real-world data scenarios.
+
+For technical support and questions, please visit our [community forums](../../community/index.md) or check the [documentation](../../documentation/user-guide.md).
\ No newline at end of file
diff --git a/docs/blog/posts/madlib-2.0.0-release.md b/docs/blog/posts/madlib-2.0.0-release.md
new file mode 100644
index 00000000..779e3ebd
--- /dev/null
+++ b/docs/blog/posts/madlib-2.0.0-release.md
@@ -0,0 +1,44 @@
+---
+date: 2023-06-20
+categories:
+ - Release
+authors:
+ - madlib-team
+---
+
+# MADlib 2.0.0 Release
+
+On June 20, 2023, MADlib completed its second major release, marking a significant milestone in the project's evolution.
+
+
+
+## New Features
+
+### Platform Support
+- **Add support for python3**: Full Python 3 compatibility for all MADlib components
+- **Add support for GP7 Beta**: Early support for Greenplum Database 7 Beta
+- **GP6 python3 extension support**: Enhanced Python 3 integration with Greenplum 6
+- **Postgres 13/14/15 support**: Extended PostgreSQL compatibility to versions 13, 14, and 15
+
+## Improvements
+
+### Machine Learning Algorithms
+- **XGBoost: Add support for version 1.7.5**: Updated to the latest XGBoost version with performance improvements and new features
+- **DL: Add support for tensorflow 2.10.1 and keras 2.10.0**: Updated deep learning framework support for better performance and compatibility
+- **DBScan: Add support for rtree 1.0.1**: Enhanced clustering algorithm with updated spatial indexing
+
+## Breaking Changes
+
+This major release includes some breaking changes. Please review the [migration guide](https://github.com/apache/madlib/blob/madlib2-master/RELEASE_NOTES) before upgrading from MADlib 1.x versions.
+
+## Download and Installation
+
+You are invited to [download the 2.0.0 release](https://dist.apache.org/repos/dist/release/madlib/2.0.0/) and [review the release notes](https://github.com/apache/madlib/blob/madlib2-master/RELEASE_NOTES).
+
+Also please refer to the [list of supported databases and OS](https://cwiki.apache.org/confluence/display/MADLIB/Installation+Guide+for+MADlib+2.X) for compatibility information.
+
+## What's New in 2.0
+
+This major release represents a significant step forward in MADlib's evolution, with comprehensive Python 3 support and expanded database platform compatibility. The updated machine learning libraries provide better performance and access to the latest algorithmic improvements.
+
+For technical support and questions, please visit our [community forums](../../community/index.md) or check the [documentation](../../documentation/user-guide.md).
\ No newline at end of file
diff --git a/docs/blog/posts/madlib-2.1.0-release.md b/docs/blog/posts/madlib-2.1.0-release.md
new file mode 100644
index 00000000..ef895168
--- /dev/null
+++ b/docs/blog/posts/madlib-2.1.0-release.md
@@ -0,0 +1,39 @@
+---
+date: 2023-09-08
+categories:
+ - Release
+authors:
+ - madlib-team
+---
+
+# MADlib 2.1.0 Release
+
+On September 8, 2023, MADlib completed its thirteenth release as an Apache Software Foundation Top Level Project.
+
+
+
+## Improvements
+
+### Build System
+- **Fix PG 15 support**: Enhanced compatibility with PostgreSQL 15
+- **Add ubuntu flag for PyXB installation**: Improved installation process on Ubuntu systems
+- **Add the actual path of $libdir to dynamic_library_path**: Better library path handling
+- **Remove PyXB as a packaged dependency**: Replaced with external pyxb-x dependency for better maintainability
+- **Use PG15 in Jenkins CI**: Updated continuous integration to test against PostgreSQL 15
+
+### Algorithm Fixes
+- **Assoc_rules: Fix SERIAL cache issue**: Resolved caching problems in association rules algorithm
+- **DL: Remove SERIAL from load_keras_model**: Improved deep learning model loading functionality
+- **CRF: Fix anyarray -> anycompatiblearray change for PG14**: Updated Conditional Random Fields for PostgreSQL 14 compatibility
+
+## Download and Installation
+
+You are invited to [download the 2.1.0 release](https://dist.apache.org/repos/dist/release/madlib/2.1.0/) and [review the release notes](https://github.com/apache/madlib/blob/madlib2-master/RELEASE_NOTES).
+
+Also please refer to the [list of supported databases and OS](https://cwiki.apache.org/confluence/display/MADLIB/Installation+Guide+for+MADlib+2.X) for compatibility information.
+
+## What's Next
+
+This release focuses on stability and compatibility improvements, ensuring MADlib works seamlessly with the latest PostgreSQL versions while maintaining backward compatibility with existing installations.
+
+For technical support and questions, please visit our [community forums](../../community/index.md) or check the [documentation](../../documentation/user-guide.md).
\ No newline at end of file
diff --git a/docs/community/contributing.md b/docs/community/contributing.md
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+++ b/docs/community/contributing.md
@@ -0,0 +1,56 @@
+# Contributing to Apache MADlib
+
+We welcome contributions from the community! This page provides information on how to contribute to the Apache MADlib project.
+
+## Getting Started
+
+### Step-by-Step Contribution Guide
+
+For detailed instructions on contributing to MADlib, please visit our [Contribution Guidelines](https://cwiki.apache.org/confluence/display/MADLIB/Contribution+Guidelines) on the MADlib Wiki.
+
+### Quick Links
+
+- **Source Code**: [GitHub Repository](https://github.com/apache/madlib)
+- **Issue Tracking**: [MADlib JIRA](https://issues.apache.org/jira/browse/MADLIB)
+- **Developer Forum**: [Subscribe](mailto:dev-subscribe@madlib.apache.org) | [Archives](https://mail-archives.apache.org/mod_mbox/madlib-dev/)
+
+## Ways to Contribute
+
+### Code Contributions
+- Bug fixes
+- New algorithms and features
+- Performance improvements
+- Documentation updates
+
+### Non-Code Contributions
+- Testing and bug reports
+- Documentation improvements
+- Community support
+- Tutorials and examples
+
+## Development Process
+
+1. **Join the Community**: Subscribe to the [developer mailing list](mailto:dev-subscribe@madlib.apache.org)
+2. **Find an Issue**: Browse [open issues](https://issues.apache.org/jira/browse/MADLIB) or propose new features
+3. **Fork and Clone**: Fork the [GitHub repository](https://github.com/apache/madlib)
+4. **Develop**: Make your changes following our coding standards
+5. **Test**: Ensure all tests pass
+6. **Submit**: Create a pull request
+
+## Developer Resources
+
+- [MADlib Architecture](https://cwiki.apache.org/confluence/display/MADLIB/Architecture)
+- [Module Anatomy](https://cwiki.apache.org/confluence/display/MADLIB/Module+Anatomy)
+- [SQL API Style Guide](https://cwiki.apache.org/confluence/display/MADLIB/SQL+API+Guide)
+- [Quick Start Guide for Developers](https://cwiki.apache.org/confluence/display/MADLIB/Quick+Start+Guide+for+Developers)
+
+## Code of Conduct
+
+Apache MADlib follows the [Apache Software Foundation Code of Conduct](https://www.apache.org/foundation/policies/conduct.html). Please be respectful and inclusive in all interactions.
+
+## Questions?
+
+If you have questions about contributing, please:
+- Ask on the [developer mailing list](mailto:dev-subscribe@madlib.apache.org)
+- Join our community discussions
+- Check the [developer documentation](https://cwiki.apache.org/confluence/display/MADLIB/)
\ No newline at end of file
diff --git a/docs/community/developers.md b/docs/community/developers.md
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+++ b/docs/community/developers.md
@@ -0,0 +1,118 @@
+# Developer Resources
+
+Resources for developers who want to contribute to Apache MADlib or integrate it into their applications.
+
+## Quick Start for Developers
+
+For developers interested in contributing to MADlib, we provide comprehensive resources to get you started quickly.
+
+[Developer Quick Start Guide](https://cwiki.apache.org/confluence/display/MADLIB/Quick+Start+Guide+for+Developers){ .md-button .md-button--primary }
+
+## Development Environment
+
+### Docker Environment
+We provide a Docker image with all necessary dependencies to compile and test MADlib. This is the fastest way to get a development environment up and running.
+
+### Build from Source
+Complete instructions for building MADlib from source on various platforms.
+
+## Architecture and Design
+
+### Core Architecture
+- [MADlib Architecture Overview](https://cwiki.apache.org/confluence/display/MADLIB/Architecture)
+- [Module Anatomy](https://cwiki.apache.org/confluence/display/MADLIB/Module+Anatomy)
+- [Technical Design Document](../design.pdf)
+
+### API Guidelines
+- [SQL API Style Guide](https://cwiki.apache.org/confluence/display/MADLIB/SQL+API+Guide)
+- Best practices for algorithm implementation
+- Performance optimization guidelines
+
+## Development Workflow
+
+### Source Code Management
+- **Repository**: [GitHub - apache/madlib](https://github.com/apache/madlib)
+- **Branching**: Feature branches and pull requests
+- **Code Review**: All changes require review
+
+### Testing
+- Unit tests for all algorithms
+- Integration tests with multiple database platforms
+- Performance benchmarks
+
+### Continuous Integration
+- Automated testing on multiple platforms
+- Code quality checks
+- Documentation generation
+
+## Contributing Guidelines
+
+### Getting Started
+1. **Join the Community**: [Developer Mailing List](mailto:dev-subscribe@madlib.apache.org)
+2. **Find Issues**: [MADlib JIRA](https://issues.apache.org/jira/browse/MADLIB)
+3. **Read Guidelines**: [Contribution Guidelines](https://cwiki.apache.org/confluence/display/MADLIB/Contribution+Guidelines)
+
+### Development Process
+- Fork and clone the repository
+- Create feature branches
+- Follow coding standards
+- Write comprehensive tests
+- Submit pull requests
+
+## Algorithm Development
+
+### Adding New Algorithms
+- Algorithm design principles
+- Implementation patterns
+- Documentation requirements
+- Testing strategies
+
+### Performance Optimization
+- Parallel processing patterns
+- Memory management
+- Database-specific optimizations
+
+## Platform Support
+
+### Supported Databases
+- PostgreSQL (11, 12, 13, 14, 15)
+- Greenplum Database (5.x, 6.x, 7.x)
+
+### Operating Systems
+- Linux (CentOS, Ubuntu, RHEL)
+- macOS
+- Docker containers
+
+## Community and Support
+
+### Communication Channels
+- [Developer Mailing List](mailto:dev-subscribe@madlib.apache.org)
+- [Mailing List Archives](https://mail-archives.apache.org/mod_mbox/madlib-dev/)
+- [GitHub Discussions](https://github.com/apache/madlib/discussions)
+
+### Regular Activities
+- Weekly developer calls
+- Quarterly planning meetings
+- Annual contributor summits
+
+## Resources and Tools
+
+### Development Tools
+- Recommended IDEs and editors
+- Debugging tools
+- Profiling utilities
+
+### Documentation Tools
+- Doxygen for API documentation
+- Sphinx for user guides
+- Wiki for design documents
+
+## Getting Help
+
+If you need help with development:
+
+1. **Check Documentation**: Start with our [developer wiki](https://cwiki.apache.org/confluence/display/MADLIB/)
+2. **Ask the Community**: Post to the [developer mailing list](mailto:dev-subscribe@madlib.apache.org)
+3. **Report Issues**: Use [JIRA](https://issues.apache.org/jira/browse/MADLIB) for bugs and feature requests
+
+We're here to help you succeed as a MADlib contributor!
\ No newline at end of file
diff --git a/docs/community/index.md b/docs/community/index.md
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+++ b/docs/community/index.md
@@ -0,0 +1,100 @@
+# Community
+
+Welcome to the Apache MADlib community! We are an open source project dedicated to bringing advanced machine learning capabilities to SQL databases.
+
+---
+
+## Reporting Issues
+
+We need your feedback, so if you find a bug, would like to suggest an improvement, or create a request then please follow the steps below to let us know.
+
+- Start by logging into [MADlib JIRA](https://issues.apache.org/jira/browse/MADLIB). If you don't have an account yet you can create one yourself.
+- To report a bug:
+ 1. Create a new issue → Bug.
+ 2. Fill in the required (and optional) fields.
+- To suggest an improvement or a new feature:
+ 1. Create a new issue → New Feature.
+ 2. Fill in the required (and optional) fields.
+- Submit the issue. You should receive an email confirmation. Thanks for your feedback!
+
+---
+
+## User Resources {#user-resources}
+
+### User Forum
+
+Our online user forum open to discuss any topics of interest to users of the product.
+
+- [Subscribe](mailto:user-subscribe@madlib.apache.org) to our user forum
+- [Browse](https://mail-archives.apache.org/mod_mbox/madlib-user/) our user forum archives
+
+---
+
+## Developer Resources {#developer-resources}
+
+### Developer Forum
+
+Our online developer forum open to discuss any topics of interest to open source contributors.
+
+- [Subscribe](mailto:dev-subscribe@madlib.apache.org) to our developer forum
+- [Browse](https://mail-archives.apache.org/mod_mbox/madlib-dev/) our developer forum archives
+
+### Contribution Guide {#contribution}
+
+Step by step instructions guiding you through the MADlib contribution model
+
+- [Contribution Guidelines](https://cwiki.apache.org/confluence/display/MADLIB/Contribution+Guidelines)
+
+### Source Code Repository
+
+- Source code available on github, contributions welcome: [https://github.com/apache/madlib](https://github.com/apache/madlib)
+
+### Design Documents
+
+There are several resources aimed at helping new developers understand how MADlib is designed
+
+- [MADlib Architecture](https://cwiki.apache.org/confluence/display/MADLIB/Architecture)
+- [Anatomy of a MADlib Module](https://cwiki.apache.org/confluence/display/MADLIB/Module+Anatomy)
+- [MADlib SQL API Style Guide](https://cwiki.apache.org/confluence/display/MADLIB/SQL+API+Guide)
+- [Technical Design Document](../design.pdf)
+
+---
+
+## Research {#research}
+
+Interaction and collaboration with the open source and academic communities continues to be a core foundation of our project. We have papers published in major conferences. Including:
+
+- [MAD Skills – VLDB 2009](https://db.cs.berkeley.edu/papers/vldb09-madskills.pdf)
+- [Hybrid In-Database Inference for Declarative Information Extraction – SIGMOD 2011](https://amplab.cs.berkeley.edu/publication/hybrid-in-database-inference-for-declarative-information-extraction/)
+- [Towards a Unified Architecture for In-Database Analytics – SIGMOD 2012](https://www.cs.stanford.edu/~chrismre/papers/bismarck-full.pdf)
+- [The MADlib Analytics Library or MAD Skills, the SQL – VLDB 2012](https://www.eecs.berkeley.edu/Pubs/TechRpts/2012/EECS-2012-38.html)
+- [Distributed Deep Learning on Data Systems: A Comparative Analysis of Approaches – 2021](https://adalabucsd.github.io/papers/TR_2021_Cerebro-DS.pdf)
+
+---
+
+## Datasets {#datasets}
+
+There is a growing set of publicly available datasets. Here are some examples:
+
+- [UCI Machine Learning Repository](https://archive.ics.uci.edu/ml/index.php)
+- [Google Dataset Search](https://datasetsearch.research.google.com/)
+- [Kaggle Datasets](https://www.kaggle.com/datasets)
+- [KDnuggets List of Datasets](https://www.kdnuggets.com/datasets/index.html)
+
+---
+
+## Related Products
+
+- [PostgreSQL](https://postgresql.org)
+- [Greenplum Database](https://greenplum.org/)
+- [PivotalR](https://cran.r-project.org/web/packages/PivotalR/)
+
+
+
+## Subscribe
+
+To stay updated with the latest MADlib news:
+
+- Subscribe to our [user mailing list](mailto:user-subscribe@madlib.apache.org)
+- Follow our [GitHub repository](https://github.com/apache/madlib)
+- Check our [community page](../community/overview.md) for more ways to get involved
\ No newline at end of file
diff --git a/docs/community/research.md b/docs/community/research.md
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index 00000000..4c73defe
--- /dev/null
+++ b/docs/community/research.md
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+# Research Papers
+
+Interaction and collaboration with the open source and academic communities continues to be a core foundation of our project. We have papers published in major conferences.
+
+## Published Papers
+
+### Foundational Papers
+
+**MAD Skills – VLDB 2009**
+The original paper that coined the term "MAD Skills" for data analysis and laid the foundation for the MADlib project.
+[Read Paper](https://db.cs.berkeley.edu/papers/vldb09-madskills.pdf)
+
+**The MADlib Analytics Library or MAD Skills, the SQL – VLDB 2012**
+Comprehensive overview of the MADlib analytics library and its SQL-based approach to machine learning.
+[Read Paper](https://www.eecs.berkeley.edu/Pubs/TechRpts/2012/EECS-2012-38.html)
+
+### Architecture and Systems
+
+**Towards a Unified Architecture for In-Database Analytics – SIGMOD 2012**
+Discusses the architectural principles behind in-database analytics systems like MADlib.
+[Read Paper](https://www.cs.stanford.edu/~chrismre/papers/bismarck-full.pdf)
+
+**Hybrid In-Database Inference for Declarative Information Extraction – SIGMOD 2011**
+Explores hybrid approaches to information extraction using in-database processing.
+[Read Paper](https://amplab.cs.berkeley.edu/publication/hybrid-in-database-inference-for-declarative-information-extraction/)
+
+### Recent Research
+
+**Distributed Deep Learning on Data Systems: A Comparative Analysis of Approaches – 2021**
+Comparative analysis of different approaches to distributed deep learning, including MADlib's approach.
+[Read Paper](https://adalabucsd.github.io/papers/TR_2021_Cerebro-DS.pdf)
+
+## Research Areas
+
+### In-Database Machine Learning
+- Scalable algorithms for large datasets
+- SQL-based machine learning interfaces
+- Integration with database query optimizers
+
+### Distributed Computing
+- Massively parallel processing (MPP) architectures
+- Shared-nothing distributed systems
+- Fault tolerance and reliability
+
+### Deep Learning Systems
+- GPU acceleration in database systems
+- Distributed neural network training
+- Model management and versioning
+
+### Graph Analytics
+- Large-scale graph processing
+- Parallel graph algorithms
+- Social network analysis
+
+## Academic Collaborations
+
+### University Partners
+
+**UC Berkeley**
+Key research and development of scalable estimators and foundational algorithms.
+
+**Stanford University**
+Research in convex optimization and statistical learning theory.
+
+**University of Florida**
+Text analytics and natural language processing research.
+
+**UC San Diego**
+Artificial neural networks on distributed systems and deep learning infrastructure.
+
+## Research Impact
+
+MADlib research has influenced:
+- Database system design for analytics workloads
+- SQL extensions for machine learning
+- Distributed machine learning frameworks
+- In-database processing paradigms
+
+## Citing MADlib
+
+If you use MADlib in your research, please cite the appropriate papers:
+
+```bibtex
+@article{hellerstein2012madlib,
+ title={The MADlib analytics library: or MAD skills, the SQL},
+ author={Hellerstein, Joseph M and R{\'e}, Christopher and Schoppmann, Florian and Wang, Daisy Zhe and Fratkin, Eugene and Gorajek, Aleksander and Ng, Kee Siong and Welton, Caleb and Feng, Xixuan and Li, Kun and others},
+ journal={Proceedings of the VLDB Endowment},
+ volume={5},
+ number={12},
+ pages={1700--1711},
+ year={2012},
+ publisher={VLDB Endowment}
+}
+```
+
+## Future Research Directions
+
+Areas of active research and development:
+- AutoML and automated model selection
+- Federated learning capabilities
+- Real-time streaming analytics
+- Advanced deep learning architectures
+- Quantum machine learning algorithms
+
+## Get Involved in Research
+
+Interested in collaborating on MADlib research?
+
+- Join our [developer community](developers.md)
+- Participate in academic conferences
+- Contribute to research publications
+- Propose new research directions
+
+For research collaborations and academic partnerships, please contact us through the [developer mailing list](mailto:dev-subscribe@madlib.apache.org).
\ No newline at end of file
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+ Contributing to MathJax - Apache MADlib
+
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+
If you're about to raise an issue because you think you've found a
+problem with MathJax, or you'd like to make a request for a new
+feature in the codebase, or any other reason… please read this first.
Please search for existing issues. Help us keep duplicate issues
+ to a minimum by checking to see if someone has already reported your
+ problem or requested your idea.
+
+
+
Please do not use the issue tracker for personal support
+ requests (use the MathJax User Group).
+
+
+
Please be civil. Keep the discussion on topic and respect the
+ opinions of others. See also our Conduct Guidelines
A bug is a demonstrable problem that is caused by the code in the repository.
+Good bug reports are extremely helpful - thank you!
+
Guidelines for bug reports:
+
+
+
Use the GitHub issue search — check if the issue has already been
+ reported.
+
+
+
Check if the issue has been fixed — look for closed issues in the
+ current milestone or try to reproduce it
+ using the latest develop branch. Please note that we only pack MathJax for releases, so on the develop branch you have to use /unpacked/MathJax.js etc. to test.
+
+
+
Share a live sample of the problem — without a live page it is usually impossible to debug problems; see also the Bug Report Template below.
+
+
+
Isolate the problem — a live sample is a starting point but if you want to speed things up create a reduced test
+ case. Be specific about your setup (browser, OS versions etc). Use services like jsbin, CodePen, JSfiddle to make collaboration on minimal test cases easier for everyone. Use the unpacked copy of MathJax ([...]/unpacked/MathJax.js etc.) for better debugging.
+
+
+
Include a screenshot/cast as a last resort — Is your issue about a layout
+ or design feature / bug but hard to reproduce or isolate? Then please provide a screenshot or screencast. Tools like LICEcap or SauceLabs allow you to quickly and easily record a screencasts. Make it an animated gif, embed it directly into your GitHub issue -- kapow!
+
+
+
Use the Bug Report template below or click this
+ link
+ to start creating a bug report with the template automatically.
+
+
+
A good bug report shouldn't leave others needing to chase you up for
+more information. Be sure to include the details of your environment.
Short and descriptive example bug report title
+
+### Issue Summary
+
+A summary of the issue and the browser/OS environment in which it occurs. If
+suitable, include the steps required to reproduce the bug.
+
+### Steps to Reproduce
+
+1. This is the first step
+2. This is the second step
+3. Further steps, etc.
+
+Any other information you want to share that is relevant to the issue
+being reported. Especially, why do you consider this to be a bug? What
+do you expect to happen instead?
+
+### Technical details:
+
+* MathJax Version: 2.3 (latest commit: f3aaf3a2a3e964df2770dc4aaaa9c87ce5f47e2c)
+* Client OS: Mac OS X 10.8.4
+* Browser: Chrome 29.0.1547.57
+
Feature requests are welcome. Before you submit one be sure to have:
+
+
Read the
+ Roadmaps,
+ use the GitHub search and check the feature hasn't already been
+ requested.
+
Take a moment to think about whether your idea fits with the scope
+ and aims of the project, or if it might better fit being a custom
+ extension.
+
Remember, it's up to you to make a strong case to convince the
+ project's leaders of the merits of this feature. Please provide as
+ much detail and context as possible, this means explaining the use
+ case and why it is likely to be common.
+
Clearly indicate whether this is a feature request for MathJax
+ core, input & output jax, or extensions.
Change requests cover both architectural and functional changes to how
+MathJax works. If you have an idea for a new or different dependency,
+a refactor, or an improvement to a feature, etc - please be sure to:
+
+
Use the GitHub search and check someone else didn't get there first
+
Take a moment to think about the best way to make a case for, and
+ explain what you're thinking. Are you sure this shouldn't really be
+ a bug report or a feature
+ request? Is it really one idea or is it many?
+ What's the context? What problem are you solving? Why is what you
+ are suggesting better than what's already there? Does it fit with
+ the Roadmap?
develop is
+ the development branch. All work on the next release happens here so
+ you should generally branch off develop. Do NOT use this branch
+ for a production site.
+
master contains the latest
+ release of MathJax. This branch may be used in production. Do
+ NOT use this branch to work on MathJax's source.
Pull requests are awesome. If you're looking to raise a PR for
+something which doesn't have an open issue, please think carefully
+about raising an issue which your PR can close,
+especially if you're fixing a bug. This makes it more likely that
+there will be enough information available for your PR to be properly
+tested and merged.
If you're not completely clear on how to submit / update / do Pull
+Requests, please check out our source control
+policies. For
+more insights, chech the excellent in depth Git Workflow
+guide from
+Ghost, in particular
Never underestimate just how useful quality assurance is. If you're
+looking to get involved with the code base and don't know where to
+start, checking out and testing a pull request is one of the most
+useful things you could do.
+
If you want to get involved with testing MathJax, there is a set of QA
+Documentation in our testing
+framework.
MathJax's main documentation can be found at docs.mathjax.org.
+The source of the docs is hosted in the
+mathjax/mathjax-docs repo here on GitHub.
+
The documentation is generated using Sphinx-doc and hosted on
+Read the docs.
+You can clone the repo and submit pull requests following the
+pull-request guidelines.
If you wish to add or update translations of MathJax, please do it on
+TranslateWiki.net
+(and while you're there you can help other open source projects,
+too, because you're awesome!).
+
For bug reports and other questions that don't fit on
+TranslateWiki.net, head over to the
+mathjax/mathjax-i18n
+repository.
We are committed to providing a friendly, safe and welcoming environment for
+all, regardless of gender, sexual orientation, disability, ethnicity, religion,
+or similar personal characteristic.
+
Please be kind and courteous. There's no need to be mean or rude.
+Respect that people have differences of opinion and that every design or
+implementation choice carries a trade-off and numerous costs. There is seldom
+a right answer, merely an optimal answer given a set of values and
+circumstances.
+
Please keep unstructured critique to a minimum. If you have solid ideas you
+want to experiment with, make a fork and see how it works.
+
We will exclude you from interaction if you insult, demean or harass anyone.
+That is not welcome behaviour. We interpret the term "harassment" as
+including the definition in the
+Citizen Code of Conduct;
+if you have any lack of clarity about what might be included in that concept,
+please read their definition. In particular, we don't tolerate behavior that
+excludes people in socially marginalized groups.
+
Private harassment is also unacceptable. No matter who you are, if you feel
+you have been or are being harassed or made uncomfortable by a community
+member, please contact one of the channel ops or any of the
+MathJax core team
+immediately. Whether you're a regular contributor or a newcomer, we care about
+making this community a safe place for you and we've got your back.
+
Likewise any spamming, trolling, flaming, baiting or other attention-stealing
+behaviour is not welcome.
+
+
+
+
+
+
+
+
+
+
+
+
\ No newline at end of file
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+ MathJax - Apache MADlib
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MathJax is an open-source JavaScript display engine for LaTeX, MathML, and
+AsciiMath notation that works in all modern browsers. It was designed with
+the goal of consolidating the recent advances in web technologies into a
+single, definitive, math-on-the-web platform supporting the major browsers
+and operating systems. It requires no setup on the part of the user (no
+plugins to download or software to install), so the page author can write
+web documents that include mathematics and be confident that users will be
+able to view it naturally and easily. Simply include MathJax and some
+mathematics in a web page, and MathJax does the rest.
+
Some of the main features of MathJax include:
+
+
+
High-quality display of LaTeX, MathML, and AsciiMath notation in HTML pages
+
+
+
Supported in most browsers with no plug-ins, extra fonts, or special
+ setup for the reader
+
+
+
Easy for authors, flexible for publishers, extensible for developers
+
+
+
Supports math accessibility, cut-and-paste interoperability, and other
+ advanced functionality
+
+
+
Powerful API for integration with other web applications
The main MathJax website is http://www.mathjax.org, and it includes
+announcements and other important information. MathJax is maintained and
+distributed on GitHub at http://github.com/mathjax/MathJax. A user forum
+for asking questions and getting assistance is hosted at Google, and the
+bug tracker is hosted at GitHub:
Before reporting a bug, please check that it has not already been reported.
+Also, please use the bug tracker for reporting bugs rather than the help forum.
+
+
+
+
+
+
+
+
+
+
+
+
\ No newline at end of file
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+MADlib: apsp.sql_in File Reference
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Licensed to the Apache Software Foundation (ASF) under one or more contributor license agreements. See the NOTICE file distributed with this work for additional information regarding copyright ownership. The ASF licenses this file to you under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at
Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License.
Arima function for forecasting of timeseries data.
+More...
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+Functions
+
void
arima_train (text input_table, text output_table, text timestamp_column, text timeseries_column, text grouping_columns, boolean include_mean, integer[] non_seasonal_orders, text optimizer_params)
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void
arima_train (text input_table, text output_table, text timestamp_column, text timeseries_column, text grouping_columns, boolean include_mean, integer[] non_seasonal_orders)
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+
void
arima_train (text input_table, text output_table, text timestamp_column, text timeseries_column, text grouping_columns, boolean include_mean)
+
+
void
arima_train (text input_table, text output_table, text timestamp_column, text timeseries_column, text grouping_columns)
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void
arima_train (text input_table, text output_table, text timestamp_column, text timeseries_column)
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void
arima_forecast (text model_table, text output_table, integer steps_ahead)
Checks whether one array contains the other. This function returns TRUE if each non-zero element in the right array equals to the element with the same index in the left array. More...
This function finds the sum of the values in the array. NULLs are ignored. Return type is always FLOAT8 regardless of input. This function is meant to replace array_sum() in the cases when sum may overflow the element type. More...
This function takes an array as the input and finds absolute value of each element in the array, returning the resulting array. It requires that all the values are NON-NULL. More...
This function takes an array as the input and apply cos function element-wise, returning the resulting array. It requires that all the values are NON-NULL. Return type is the same as the input type. More...
This function takes an array as the input and executes element-wise multiplication by the scalar provided as the second argument, returning the resulting array. It requires that all the values are NON-NULL. Return type is the same as the input type. More...
This function takes an array as the input and executes element-wise addition by the scalar provided as the second argument, returning the resulting array. It requires that all the values are NON-NULL. Return type is the same as the input type. More...
This function takes an array as the input and keep only elements that satisfy the operator on specified scalar. It requires that all the values are NON-NULL. Return type is the same as the input type. More...
This function takes an array as the input and removes elements that equal to specified scalar. It requires that all the values are NON-NULL. Return type is the same as the input type. More...
This function takes an array as the input and removes elements that equal to 0. It requires that all the values are NON-NULL. Return type is the same as the input type. More...
This function takes an array as the input and finds square root of each element in the array, returning the resulting array. It requires that all the values are NON-NULL. More...
This function takes an array and a float8 as the input and finds power of each element in the array, returning the resulting array. It requires that all the values are NON-NULL. More...
This function takes an array as the input and finds square of each element in the array, returning the resulting array. It requires that all the values are NON-NULL. More...
This function takes an array as the input and computes the cumulative sum with the first element being the same. It requires that all the values are NON-NULL. More...
This function takes an array as the input and computes the cumulative product with the first element being the same. It requires that all the values are NON-NULL. More...
This function takes a 2-D array as the input and unnests it by one level. It returns a set of 1-D arrays that correspond to rows of the input array as well as an ID column containing row positions occupied by those 1-D arrays within the 2-D array (the ID column values start with 1 and not 0) More...
The assoc_rules function computes association rules for a given set of data. The data is assumed to have two dimensions; items (between which we are trying to discover associations), and a transaction id. This tranaction id groups the items by event and could also be a user id, date, etc. depending on the context of the data. This function assumes the data is stored in two columns with one transaction id and one item per row.
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assoc_rules (float8 support, float8 confidence, text tid_col, text item_col, text input_table, text output_schema, boolean verbose, integer max_itemset_size, integer max_lhs_size, integer max_rhs_size)
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assoc_rules_results
assoc_rules (float8 support, float8 confidence, text tid_col, text item_col, text input_table, text output_schema, boolean verbose, integer max_itemset_size, integer max_lhs_size)
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assoc_rules_results
assoc_rules (float8 support, float8 confidence, text tid_col, text item_col, text input_table, text output_schema, boolean verbose, integer max_itemset_size)
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assoc_rules_results
assoc_rules (float8 support, float8 confidence, text tid_col, text item_col, text input_table, text output_schema)
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The short form of the above function with vobose removed. More...
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assoc_rules_results
assoc_rules (float8 support, float8 confidence, text tid_col, text item_col, text input_table, text output_schema, boolean verbose)
minimum level of support needed for each itemset to be included in result
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confidence
minimum level of confidence needed for each rule to be included in result
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tid_col
name of the column storing the transaction ids
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item_col
name of the column storing the products
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input_table
name of the table where the data is stored
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output_schema
name of the schema where the final results will be stored
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verbose
determining if output contains comments
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Returns
The schema and table name containing association rules, and total number of rules found.
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This function computes the association rules between products in a data set. It reads the name of the table, the column names of the product and ids, and computes ssociation rules using the Apriori algorithm, and subject to the support and confidence constraints as input by the user. This version of association rules has verbose functionality. When verbose is true, output of function includes iteration steps and comments on Apriori algorithm steps.
SQL functions for balanced data sets sampling.
+More...
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+Functions
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void
balance_sample (text source_table, text output_table, text class_col, varchar class_sizes, integer output_table_size, text grouping_cols, boolean with_replacement, boolean keep_null)
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void
balance_sample (text source_table, text output_table, text class_col, varchar class_sizes, integer output_table_size, text grouping_cols, boolean with_replacement)
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void
balance_sample (text source_table, text output_table, text class_col, varchar class_sizes, integer output_table_size, text grouping_cols)
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void
balance_sample (text source_table, text output_table, text class_col, varchar class_sizes, integer output_table_size)
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void
balance_sample (text source_table, text output_table, text class_col, varchar class_sizes)
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void
balance_sample (text source_table, text output_table, text class_col)
Licensed to the Apache Software Foundation (ASF) under one or more contributor license agreements. See the NOTICE file distributed with this work for additional information regarding copyright ownership. The ASF licenses this file to you under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at
Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License.
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Date
12/14/2017
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See also
Given a table, balanced sampling returns a sampled data set with specified proportions for each class (defaults to uniform sampling).
graph_bfs (text vertex_table, text vertex_id, text edge_table, text edge_args, bigint source_vertex, text out_table, int max_distance, boolean directed, text grouping_cols)
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void
graph_bfs (text vertex_table, text vertex_id, text edge_table, text edge_args, bigint source_vertex, text out_table, int max_distance, boolean directed)
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void
graph_bfs (text vertex_table, text vertex_id, text edge_table, text edge_args, bigint source_vertex, text out_table, int max_distance)
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void
graph_bfs (text vertex_table, text vertex_id, text edge_table, text edge_args, bigint source_vertex, text out_table)
Licensed to the Apache Software Foundation (ASF) under one or more contributor license agreements. See the NOTICE file distributed with this work for additional information regarding copyright ownership. The ASF licenses this file to you under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at
Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License.
clustered_variance_logregr (text source_table, text out_table, text depvar, text indvar, text clustervar, text grouping_col, integer max_iter, text optimizer, float8 tolerance, boolean verbose_mode)
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void
clustered_variance_logregr (text source_table, text out_table, text depvar, text indvar, text clustervar)
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void
clustered_variance_logregr (text source_table, text out_table, text depvar, text indvar, text clustervar, text grouping_col)
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void
clustered_variance_logregr (text source_table, text out_table, text depvar, text indvar, text clustervar, text grouping_col, integer max_iter)
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void
clustered_variance_logregr (text source_table, text out_table, text depvar, text indvar, text clustervar, text grouping_col, integer max_iter, text optimizer)
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void
clustered_variance_logregr (text source_table, text out_table, text depvar, text indvar, text clustervar, text grouping_col, integer max_iter, text optimizer, float8 tolerance)
clustered_variance_mlogregr (text source_table, text out_table, text dependent_varname, text independent_varname, text cluster_varname, integer ref_category, text grouping_cols, text optimizer_params, boolean verbose_mode)
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void
clustered_variance_mlogregr (text source_table, text out_table, text dependent_varname, text independent_varname, text cluster_varname, integer ref_category, text grouping_cols, text optimizer_params)
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void
clustered_variance_mlogregr (text source_table, text out_table, text dependent_varname, text independent_varname, text cluster_varname, integer ref_category, text grouping_cols)
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void
clustered_variance_mlogregr (text source_table, text out_table, text dependent_varname, text independent_varname, text cluster_varname, integer ref_category)
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void
clustered_variance_mlogregr (text source_table, text out_table, text dependent_varname, text independent_varname, text cluster_varname)
clustered_variance_mlogregr (text source_table, text out_table, text dependent_varname, text independent_varname, text cluster_varname, integer ref_category, text grouping_cols, integer max_iter, varchar optimizer, float8 tolerance, boolean verbose_mode)
+
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void
clustered_variance_mlogregr (text source_table, text out_table, text dependent_varname, text independent_varname, text cluster_varname, integer ref_category, text grouping_cols, integer max_iter, varchar optimizer, float8 tolerance)
+
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void
clustered_variance_mlogregr (text source_table, text out_table, text dependent_varname, text independent_varname, text cluster_varname, integer ref_category, text grouping_cols, integer max_iter, varchar optimizer)
+
+
void
clustered_variance_mlogregr (text source_table, text out_table, text dependent_varname, text independent_varname, text cluster_varname, integer ref_category, text grouping_cols, integer max_iter)
Licensed to the Apache Software Foundation (ASF) under one or more contributor license agreements. See the NOTICE file distributed with this work for additional information regarding copyright ownership. The ASF licenses this file to you under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at
Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License.
Name of the table that contains the Cox Proportional Hazards model
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output_table
Name of the output table to contain the statistics to test the proportional hazards assumption
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Returns
An output table with following columns: Columns of the matrix contain the correlation coefficient between transformed , a chi-square, and the two-sided p-value.
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rho FLOAT8[] - Array of correlation coefficient between survival time and the scaled Schoenfeld residuals
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chi_square FLOAT8[] - Chi-square test statistic for the
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p_value FLOAT8[] - Two-side p-value for the chi-square test
Licensed to the Apache Software Foundation (ASF) under one or more contributor license agreements. See the NOTICE file distributed with this work for additional information regarding copyright ownership. The ASF licenses this file to you under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at
Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License.
+
Date
June 2014
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See also
Calculates dummy-coded indicator variables for categorical variables
Name of the source relation containing the training data
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sparse_R
Name of the sparse single state feature column (of type DOUBLE PRECISION[])
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dense_M
Name of the dense two state feature column (of type DOUBLE PRECISION[])
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sparse_M
Name of the sparse two state feature column (of type DOUBLE PRECISION[])
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featureSize
Name of feature size column (of type DOUBLE PRECISION)
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tagSize
The number of tags in the tag set
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featureset
The unique feature set
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crf_feature
The Name of output feature table
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maxNumIterations
The maximum number of iterations
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Returns
a composite value:
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coef FLOAT8[] - Array of coefficients, \( \boldsymbol c \)
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log_likelihood FLOAT8 - Log-likelihood \( l(\boldsymbol c) \)
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num_iterations INTEGER - The number of iterations before the algorithm terminated
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+A 'crf_feature' table is used to store all the features and corresponding weights
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Note
This function starts an iterative algorithm. It is not an aggregate function. Source and column names have to be passed as strings (due to limitations of the SQL syntax).
This feature extraction function will produce two factor tables, "m table" (viterbi_mtbl) and "r table" (viterbi_rtbl). The viterbi_mtbl table and viterbi_rtbl table are used to calculate the best label sequence for each sentence.
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viterbi_mtbl table encodes the edge features which are solely dependent on upon current label and previous y value. The m table has three columns which are prev_label, label, and value respectively. If the number of labels in \( n \), then the m factor table will \( n^2 \) rows. Each row encodes the transition feature weight value from the previous label to the current label.
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startFeature is considered as a special edge feature which is from the beginning to the first token. Likewise, endFeature can be considered as a special edge feature which is from the last token to the very end. So m table encodes the edgeFeature, startFeature, and endFeature. If the total number of labels in the label space is 45 from 0 to 44, then the m factor array is as follows:
+ 0 1 2 3 4 5...44
+startFeature -1 a a a a a a...a
+edgeFeature 0 a a a a a a...a
+edgeFeature 1 a a a a a a...a
+...
+edgeFeature 44 a a a a a a...a
+endFeature 45 a a a a a a...a
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viterbi_r table is related to specific tokens. It encodes the single state features, e.g., wordFeature, RegexFeature for all tokens. The r table is represented in the following way.
+ 0 1 2 3 4...44
+token1 a a a a a...a
+token2 a a a a a...a
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Parameters
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test_segment_tbl
Name of table containing all the tokenized testing sentences.
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dictionary_tbl
Name of table containing the dictionary_tbl.
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label_tbl
Name of table containing the the label space used in POS or other NLP tasks.
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regex_tbl
Name of table containing all the regular expressions to capture regex features.
__dbscan_leaf (__dbscan_record db_rec, real eps, int min_samples, text metric, bigint num_internal_points, bigint num_external_points)
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Detailed Description
+
Licensed to the Apache Software Foundation (ASF) under one or more contributor license agreements. See the NOTICE file distributed with this work for additional information regarding copyright ownership. The ASF licenses this file to you under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at
Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License.
tree_train (text training_table_name, text output_table_name, text id_col_name, text dependent_variable, text list_of_features, text list_of_features_to_exclude, text split_criterion, text grouping_cols, text weights, integer max_depth, integer min_split, integer min_bucket, integer n_bins, text pruning_params, text null_handling_params, boolean verbose_mode)
__build_tree (boolean is_classification, text split_criterion, text training_table_name, text output_table_name, text id_col_name, text dependent_variable, boolean dep_is_bool, text list_of_features, varchar[] cat_features, varchar[] ordered_cat_features, varchar[] boolean_cats, varchar[] con_features, text grouping_cols, text weights, integer max_depth, integer min_split, integer min_bucket, integer n_bins, text cp_table, smallint max_n_surr, text msg_level, text null_proxy, integer n_folds)
tree_train (text training_table_name, text output_table_name, text id_col_name, text dependent_variable, text list_of_features, text list_of_features_to_exclude, text split_criterion, text grouping_cols, text weights, integer max_depth, integer min_split, integer min_bucket, integer n_bins, text pruning_params, text null_handling_params)
+
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void
tree_train (text training_table_name, text output_table_name, text id_col_name, text dependent_variable, text list_of_features, text list_of_features_to_exclude, text split_criterion, text grouping_cols, text weights, integer max_depth, integer min_split, integer min_bucket, integer n_bins, text pruning_params)
+
+
void
tree_train (text training_table_name, text output_table_name, text id_col_name, text dependent_variable, text list_of_features, text list_of_features_to_exclude, text split_criterion, text grouping_cols, text weights, integer max_depth, integer min_split, integer min_bucket, integer n_bins)
+
+
void
tree_train (text training_table_name, text output_table_name, text id_col_name, text dependent_variable, text list_of_features, text list_of_features_to_exclude, text split_criterion, text grouping_cols, text weights, integer max_depth, integer min_split, integer min_bucket)
+
+
void
tree_train (text training_table_name, text output_table_name, text id_col_name, text dependent_variable, text list_of_features, text list_of_features_to_exclude, text split_criterion, text grouping_cols, text weights, integer max_depth, integer min_split)
+
+
void
tree_train (text training_table_name, text output_table_name, text id_col_name, text dependent_variable, text list_of_features, text list_of_features_to_exclude, text split_criterion, text grouping_cols, text weights, integer max_depth)
+
+
void
tree_train (text training_table_name, text output_table_name, text id_col_name, text dependent_variable, text list_of_features, text list_of_features_to_exclude, text split_criterion, text grouping_cols, text weights)
+
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void
tree_train (text training_table_name, text output_table_name, text id_col_name, text dependent_variable, text list_of_features, text list_of_features_to_exclude, text split_criterion, text grouping_cols)
+
+
void
tree_train (text training_table_name, text output_table_name, text id_col_name, text dependent_variable, text list_of_features, text list_of_features_to_exclude, text split_criterion)
+
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void
tree_train (text training_table_name, text output_table_name, text id_col_name, text dependent_variable, text list_of_features, text list_of_features_to_exclude)
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void
tree_train (text training_table_name, text output_table_name, text id_col_name, text dependent_variable, text list_of_features)
Name of the table containing the decision tree model
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source
Name of table containing prediction data
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output
Name of table to output prediction results
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pred_type
OPTIONAL (Default = 'response'). For regression trees, 'response', implies output is the predicted value. For classification trees, this can be 'response', giving the classification prediction as output, or ‘prob’, giving the class probabilities as output (for two classes, only a single probability value is output that corresponds to the first class when the two classes are sorted by name; in case of more than two classes, an array of class probabilities (a probability of each class) is output).
Various options to compute the feature to split a node. Available options are 'gini', 'cross-entropy', and 'misclassification'. The "cart" algorithm provides an additional option of 'mse'.
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training_table_name
Name of the table containing data.
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output_table_name
Name of the table to output the model.
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id_col_name
Name of column containing the id information in training data.
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dependent_variable
Name of the column that contains the output for training. Boolean, integer and text are considered classification outputs, while float values are considered regression outputs.
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list_of_features
List of column names (comma-separated string) to use as predictors. Can also be a ‘*’ implying all columns are to be used as predictors (except the ones included in the next argument). Boolean, integer, and text columns are considered categorical columns.
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list_of_features_to_exclude
OPTIONAL. List of column names (comma-separated string) to exlude from the predictors list.
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grouping_cols
OPTIONAL. List of column names (comma-separated string) to group the data by. This will lead to creating multiple decision trees, one for each group.
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weights
OPTIONAL. Column name containing weights for each observation.
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max_depth
OPTIONAL (Default = 7). Set the maximum depth of any node of the final tree, with the root node counted as depth 0. A deeper tree can lead to better prediction but will also result in longer processing time and higher memory usage.
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min_split
OPTIONAL (Default = 20). Minimum number of observations that must exist in a node for a split to be attempted.
+
min_bucket
OPTIONAL (Default = minsplit/3). Minimum number of observations in any terminal node. If only one of minbucket or minsplit is specified, minsplit is set to minbucket*3 or minbucket to minsplit/3, as appropriate.
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n_bins
optional (default = 20) number of bins to use during binning. continuous-valued features are binned into discrete bins (per the quartile values) to compute split bound- aries. this global parameter is used to compute the resolution of the bins. higher number of bins will lead to higher processing time.
+
pruning_params
(default: cp=0) pruning parameter string containing key-value pairs. the keys can be: cp (default = 0.01) a complexity parameter that determines that a split is attempted only if it decreases the overall lack of fit by a factor of ‘cp’. n_folds (default = 0) number of cross-validation folds
+
verbose_mode
optional (default = false) prints status information on the splits performed and any other information useful for debugging.
Column containing the left hand side of the system
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right_hand_side
Column containing the right hand side of the system
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grouping_cols
Columns to group by
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optimizer
Optimizer to be used
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optimizer_options
Optimal parameters for the algorithms
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+
+
+
Returns
void
+
Usage
For function summary information. Run sql> select linear_solver_dense('help'); OR sql> select linear_solver_dense(); OR sql> select linear_solver_dense('?'); For function usage information. Run sql> select linear_solver_dense('usage');
The assoc_rules function computes association rules for a given set of data. The data is assumed to have two dimensions; items (between which we are trying to discover associations), and a transaction id. This tranaction id groups the items by event and could also be a user id, date, etc. depending on the context of the data. This function assumes the data is stored in two columns with one transaction id and one item per row.
SQL functions for elastic net regularization.
+More...
+
+
+Functions
+
void
elastic_net_train (text tbl_source, text tbl_result, text col_dep_var, text col_ind_var, text regress_family, float8 alpha, float8 lambda_value, boolean standardize, text grouping_col, text optimizer, text optimizer_params, text excluded, integer max_iter, float8 tolerance)
elastic_net_train (text tbl_source, text tbl_result, text col_ind_var, text col_dep_var, text regress_family, float8 alpha, float8 lambda_value, boolean standardization, text grouping_columns, text optimizer, text optimizer_params, text excluded, integer max_iter)
+
+
void
elastic_net_train (text tbl_source, text tbl_result, text col_ind_var, text col_dep_var, text regress_family, float8 alpha, float8 lambda_value, boolean standardization, text grouping_columns, text optimizer, text optimizer_params, text excluded)
+
+
void
elastic_net_train (text tbl_source, text tbl_result, text col_ind_var, text col_dep_var, text regress_family, float8 alpha, float8 lambda_value, boolean standardization, text grouping_columns, text optimizer, text optimizer_params)
+
+
void
elastic_net_train (text tbl_source, text tbl_result, text col_ind_var, text col_dep_var, text regress_family, float8 alpha, float8 lambda_value, boolean standardization, text grouping_columns, text optimizer)
+
+
void
elastic_net_train (text tbl_source, text tbl_result, text col_ind_var, text col_dep_var, text regress_family, float8 alpha, float8 lambda_value, boolean standardization, text grouping_columns)
+
+
void
elastic_net_train (text tbl_source, text tbl_result, text col_ind_var, text col_dep_var, text regress_family, float8 alpha, float8 lambda_value, boolean standardization)
+
+
void
elastic_net_train (text tbl_source, text tbl_result, text col_ind_var, text col_dep_var, text regress_family, float8 alpha, float8 lambda_value)
Name of independent variable column, independent variable is an array
+
col_dep_var
Name of dependent variable column
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regress_family
Response type (gaussian or binomial)
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alpha
The elastic net parameter, [0, 1]
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lambda_value
The regularization parameter
+
standardize
Whether to normalize the variables (default True)
+
grouping_col
List of columns on which to apply grouping (currently only a placeholder)
+
optimizer
The optimization algorithm, 'fista' or 'igd'. Default is 'fista'
+
optimizer_params
Parameters of the above optimizer, the format is 'arg = value, ...'. Default is NULL
+
excluded
Which columns to exclude? Default is NULL (applicable only if col_ind_var is set as * or a column of array, column names as 'col1, col2, ...' if col_ind_var is '*'; element indices as '1,2,3, ...' if col_ind_var is a column of array)
+
max_iter
Maximum number of iterations to run the algorithm (default value of 1000)
Licensed to the Apache Software Foundation (ASF) under one or more contributor license agreements. See the NOTICE file distributed with this work for additional information regarding copyright ownership. The ASF licenses this file to you under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at
Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License.
This page explains how to interpret the graphs that are generated by doxygen.
+
Consider the following example:
/*! Invisible class because of truncation */
class Invisible { };
/*! Truncated class, inheritance relation is hidden */
class Truncated : public Invisible { };
/* Class not documented with doxygen comments */
class Undocumented { };
/*! Class that is inherited using public inheritance */
class PublicBase : public Truncated { };
/*! A template class */
template<class T> class Templ { };
/*! Class that is inherited using protected inheritance */
class ProtectedBase { };
/*! Class that is inherited using private inheritance */
class PrivateBase { };
/*! Class that is used by the Inherited class */
class Used { };
/*! Super class that inherits a number of other classes */
class Inherited : public PublicBase,
protected ProtectedBase,
private PrivateBase,
public Undocumented,
public Templ<int>
{
private:
Used *m_usedClass;
};
This will result in the following graph:
+
The boxes in the above graph have the following meaning:
+
+
+A filled gray box represents the struct or class for which the graph is generated.
+
+A box with a black border denotes a documented struct or class.
+
+A box with a gray border denotes an undocumented struct or class.
+
+A box with a red border denotes a documented struct or class forwhich not all inheritance/containment relations are shown. A graph is truncated if it does not fit within the specified boundaries.
+
+
The arrows have the following meaning:
+
+
+A dark blue arrow is used to visualize a public inheritance relation between two classes.
+
+A dark green arrow is used for protected inheritance.
+
+A dark red arrow is used for private inheritance.
+
+A purple dashed arrow is used if a class is contained or used by another class. The arrow is labelled with the variable(s) through which the pointed class or struct is accessible.
+
+A yellow dashed arrow denotes a relation between a template instance and the template class it was instantiated from. The arrow is labelled with the template parameters of the instance.
+
+
+
+
+
+
+
Generated on Mon Aug 28 2023 08:03:58 for MADlib by
+
+ 1.8.13
+
+
+
+
diff --git a/docs/docs/v2.1.0/graph_legend.md5 b/docs/docs/v2.1.0/graph_legend.md5
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diff --git a/docs/docs/v2.1.0/group__grp__apsp.html b/docs/docs/v2.1.0/group__grp__apsp.html
new file mode 100644
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@@ -0,0 +1,388 @@
+
+
+
+
+
+
+
+
+MADlib: All Pairs Shortest Path
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
The all pairs shortest paths (APSP) algorithm finds the length (summed weights) of the shortest paths between all pairs of vertices, such that the sum of the weights of the path edges is minimized.
+
Warning
APSP is an expensive algorithm for run-time because it finds the shortest path between all nodes in the graph. It is recommended that you start with a small graph to get a sense of run-time for your use case, then increase size carefully from there. The worst case run-time for this implementation is O(V^2 * E) where V is the number of vertices and E is the number of edges. In practice, run-time will be generally be much less than this, but it depends on the graph. On a Greenplum cluster, the edge table should be distributed by the source vertex id column for better performance.
TEXT. Name of the table containing the vertex data for the graph. Must contain the column specified in the 'vertex_id' parameter below.
+
+
+
vertex_id
+
TEXT, default = 'id'. Name of the column in 'vertex_table' containing vertex ids. The vertex ids can be of type INTEGER or BIGINT with no duplicates. They do not need to be contiguous.
+
+
+
edge_table
+
TEXT. Name of the table containing the edge data. The edge table must contain columns for source vertex, destination vertex and edge weight. Column naming convention is described below in the 'edge_args' parameter.
+
+
+
edge_args
+
TEXT. A comma-delimited string containing multiple named arguments of the form "name=value". The following parameters are supported for this string argument:
+
src (INTEGER or BIGINT): Name of the column containing the source vertex ids in the edge table. Default column name is 'src'.
+
dest (INTEGER or BIGINT): Name of the column containing the destination vertex ids in the edge table. Default column name is 'dest'.
+
weight (FLOAT8): Name of the column containing the edge weights in the edge table. Default column name is 'weight'.
+
+
+
+
out_table
+
TEXT. Name of the table to store the result of APSP. It contains a row for every vertex of every group and have the following columns (in addition to the grouping columns):
+
source_vertex: The id for the source vertex. Will use the input edge column 'src' for column naming.
+
dest_vertex: The id for the destination vertex. Will use the input edge column 'dest' for column naming.
+
weight: The total weight of the shortest path from the source vertex to the destination vertex. Will use the input parameter 'weight' for column naming.
+
parent: The parent of the destination vertex in the shortest path from source. Parent will equal dest_vertex if there are no intermediate vertices. Will use 'parent' for column naming.
+
+
A summary table named <out_table>_summary is also created. This is an internal table that keeps a record of the input parameters and is used by the path retrieval function described below.
+
+
+
grouping_cols (optional)
+
TEXT, default = NULL. List of columns used to group the input into discrete subgraphs. These columns must exist in the edge table. When this value is null, no grouping is used and a single APSP result is generated.
+
+
Path Retrieval
+
The path retrieval function returns the shortest path from the source vertex to a specified desination vertex.
TEXT. Name of the table that contains the APSP output.
+
+
+
source_vertex
+
INTEGER or BIGINT. The vertex that will be the source of the desired path.
+
+
+
dest_vertex
+
INTEGER or BIGINT. The vertex that will be the destination of the desired path.
+
+
+
path_table
+
TEXT. Name of the output table that contains the path. It contains a row for every group and has the following columns:
+
grouping_cols: The grouping columns given in the creation of the APSP table. If there are no grouping columns, these columns will not exist and the table will have a single row.
+
path (ARRAY): The shortest path from the source vertex to the destination vertex.
+
+
+
+
+
Examples
+
+
Create vertex and edge tables to represent the graph:
+DROP TABLE IF EXISTS out, out_summary;
+SELECT madlib.graph_apsp(
+ 'vertex', -- Vertex table
+ NULL, -- Vertix id column (NULL means use default naming)
+ 'edge', -- Edge table
+ NULL, -- Edge arguments (NULL means use default naming)
+ 'out'); -- Output table of shortest paths
+SELECT * FROM out ORDER BY src,dest;
+
+DROP TABLE IF EXISTS out_path;
+SELECT madlib.graph_apsp_get_path('out',0,5,'out_path');
+SELECT * FROM out_path;
+
+ path
+---------
+ {0,2,5}
+
+
Now let's do a similar example except using different column names in the tables (i.e., not the defaults). Create the vertex and edge tables:
+DROP TABLE IF EXISTS vertex_alt, edge_alt;
+CREATE TABLE vertex_alt AS SELECT id AS v_id FROM vertex;
+CREATE TABLE edge_alt AS SELECT src AS e_src, dest, weight AS e_weight FROM edge;
+
+
Calculate the shortest paths:
+DROP TABLE IF EXISTS out_alt, out_alt_summary;
+SELECT madlib.graph_apsp(
+ 'vertex_alt', -- Vertex table
+ 'v_id', -- Vertex id column
+ 'edge_alt', -- Edge table
+ 'src=e_src, weight=e_weight', -- Edge arguments
+ 'out_alt'); -- Output table of shortest paths
+SELECT * FROM out_alt ORDER BY e_src, dest;
+
Create a graph with 2 groups and find APSP for each group:
+DROP TABLE IF EXISTS edge_gr;
+CREATE TABLE edge_gr AS
+(
+ SELECT *, 0 AS grp FROM edge
+ UNION
+ SELECT *, 1 AS grp FROM edge WHERE src < 6 AND dest < 6
+);
+INSERT INTO edge_gr VALUES
+(4,5,-20,1);
+
+
Find APSP for all groups:
+DROP TABLE IF EXISTS out_gr, out_gr_summary;
+SELECT madlib.graph_apsp(
+ 'vertex', -- Vertex table
+ NULL, -- Vertex id column (NULL means use default naming)
+ 'edge_gr', -- Edge table
+ NULL, -- Edge arguments (NULL means use default naming)
+ 'out_gr', -- Output table of shortest paths
+ 'grp' -- Grouping columns
+);
+SELECT * FROM out_gr WHERE src < 2 ORDER BY grp,src,dest;
+
Graphs with negative edges are supported but graphs with negative cycles are not.
+
The implementation for APSP is analogous to a matrix multiplication operation. Please refer to the MADlib design document and references [1] and [2] for more details.
+
Also see the Grail project [3] for more background on graph analytics processing in relational databases.
Given a time series of data X, the Autoregressive Integrated Moving Average (ARIMA) model is a tool for understanding and, perhaps, predicting future values in the series. The model consists of three parts, an autoregressive (AR) part, a moving average (MA) part, and an integrated (I) part where an initial differencing step can be applied to remove any non-stationarity in the signal. The model is generally referred to as an ARIMA(p, d, q) model where parameters p, d, and q are non-negative integers that refer to the order of the autoregressive, integrated, and moving average parts of the model respectively.
+
Training Function
+
The ARIMA training function has the following syntax.
TEXT. The name of the table containing time series data.
+
+
+
output_table
+
TEXT. The name of the table to store the ARIMA model. Three tables are created, with names based on the value of the output_table argument in the training function:
+
+
output_table: Table containing the ARIMA model. Contains the following columns:
+
+
mean
Model mean (only if 'include_mean' is TRUE)
+
+
mean_std_error
Standard errors for mean
+
+
ar_params
Auto-regressions parameters of the ARIMA model
+
+
ar_std_errors
Standard errors for AR parameters
+
+
ma_params
Moving average parameters of the ARIMA model
+
+
ma_std_errors
Standard errors for MA parameters
+
+
+
output_table_summary: Table containing descriptive statistics of the ARIMA model. Contains the following columns:
+
+
input_table
Table name with the source data
+
+
timestamp_col
Column name in the source table that contains the timestamp index to data
+
+
timeseries_col
Column name in the source table that contains the data values
+
+
non_seasonal_orders
Orders of the non-seasonal ARIMA model
+
+
include_mean
TRUE if intercept was included in ARIMA model
+
+
residual_variance
Variance of the residuals
+
+
log_likelihood
Log likelihood value (when using MLE)
+
+
iter_num
The number of iterations executed
+
+
exec_time
Total time taken to train the model
+
+
+
output_table_residual: Table containing the residuals for each data point in 'input_table'. Contains the following columns:
+
+
timestamp_col
Same as the 'timestamp_col' parameter (all indices from source table included except the first d elements, where d is the differencing order value from 'non_seasonal_orders')
+
+
residual
Residual value for each data point
+
+
+
+
+
+
+
timestamp_column
+
TEXT. The name of the column containing the timestamp (index) data. This could be a serial index (INTEGER) or date/time value (TIMESTAMP).
+
+
+
timeseries_column
+
TEXT. The name of the column containing the time series data. This data is currently restricted to DOUBLE PRECISION.
+
+
+
grouping_columns (not currently implemented)
+
TEXT, default: NULL.
+
A comma-separated list of column names used to group the input dataset into discrete groups, training one ARIMA model per group. It is similar to the SQL GROUP BY clause. When this value is null, no grouping is used and a single result model is generated.
+
Note
Grouping is not currently implemented for ARIMA, but will be added in the future. Any non-NULL value for this parameter is ignored.
+
+
include_mean (optional)
+
BOOLEAN, default: FALSE. Mean value of the data series is added in the ARIMA model if this variable is True.
+
+
+
non_seasonal_orders (optional)
+
INTEGER[], default: 'ARRAY[1,1,1]'. Orders of the ARIMA model. The orders are [p, d, q], where parameters p, d, and q are non-negative integers that refer to the order of the autoregressive, integrated, and moving average parts of the model respectively.
+
+
+
optimizer_params (optional)
+
TEXT. Comma-separated list of optimizer-specific parameters of the form ‘name=value'. The order of the parameters does not matter. The following parameters are recognized:
+
max_iter: Maximum number of iterations to run learning algorithm (Default = 100)
+
tau: Computes the initial step size for gradient algorithm (Default = 0.001)
+
e1: Algorithm-specific threshold for convergence (Default = 1e-15)
+
e2: Algorithm-specific threshold for convergence (Default = 1e-15)
+
e3: Algorithm-specific threshold for convergence (Default = 1e-15)
+
hessian_delta: Delta parameter to compute a numerical approximation of the Hessian matrix (Default = 1e-6)
+
+
+
+
Forecasting Function
+
The ARIMA forecast function has the following syntax.
An ARIMA model is an auto-regressive integrated moving average model. An ARIMA model is typically expressed in the form
+\[ (1 - \phi(B)) Y_t = (1 + \theta(B)) Z_t, \]
+
+
+
where \(B\) is the backshift operator. The time \( t \) is from \( 1 \) to \( N \).
+
ARIMA models involve the following variables:
+
The values of the time series: \( X_t \).
+
Parameters of the model: \( p \), \( q \), and \( d \); \( d \) is the differencing order, \( p \) is the order of the AR operator, and \( q \) is the order of the MA operator.
+
The AR operator: \( \phi(B) \).
+
The MA operator: \( \theta(B) \).
+
The lag difference: \( Y_{t} \), where \( Y_{t} = (1-B)^{d}(X_{t} - \mu) \).
+
The mean value: \( \mu \), which is set to be zero for \( d>0 \) and estimated from the data when d=0.
+
The error terms: \( Z_t \).
+
+
The auto regression operator models the prediction for the next observation as some linear combination of the previous observations. More formally, an AR operator of order \( p \) is defined as
The moving average operator is similar, and it models the prediction for the next observation as a linear combination of the errors in the previous prediction errors. More formally, the MA operator of order \( q \) is defined as
We estimate the parameters using the Levenberg-Marquardt Algorithm. In mathematics and computing, the Levenberg-Marquardt algorithm (LMA), also known as the damped least-squares (DLS) method, provides a numerical solution to the problem of minimizing a function, generally nonlinear, over a space of parameters of the function.
+
Like other numeric minimization algorithms, LMA is an iterative procedure. To start a minimization, the user has to provide an initial guess for the parameter vector, $p$, as well as some tuning parameters \(\tau, \epsilon_1, \epsilon_2, \epsilon_3,\).
+
Literature
+
[1] Rob J Hyndman and George Athanasopoulos: Forecasting: principles and practice, http://otexts.com/fpp/
+
[2] Robert H. Shumway, David S. Stoffer: Time Series Analysis and Its Applications With R Examples, Third edition Springer Texts in Statistics, 2010
+
[3] Henri Gavin: The Levenberg-Marquardt method for nonlinear least squares curve-fitting problems, 2011
This module provides a set of basic array operations implemented in C and SQL. It is a support module for several machine learning algorithms that require fast array operations.
+
Implementation Notes
+
All functions (except normalize() and array_filter()) described in this module work with 2-D arrays.
+
These functions support several numeric types:
+
SMALLINT
+
INTEGER
+
BIGINT
+
REAL
+
DOUBLE PRECISION (FLOAT8)
+
NUMERIC (internally casted into FLOAT8, loss of precisions can happen)
Checks whether one array contains the other. This function returns TRUE if each non-zero element in the right array equals to the element with the same index in the left array.
This function finds the maximum value and corresponding index in the array. NULLs are ignored. Return type is array in format [max, index], and its element type is the same as the input type.
This function finds the minimum value and corresponding index in the array. NULLs are ignored. Return type is array in format [min, index], and its element type is the same as the input type.
This function finds the sum of the values in the array. NULLs are ignored. Return type is always FLOAT8 regardless of input. This function is meant to replace array_sum() in cases when a sum may overflow the element type.
This function takes an array as the input and finds abs of each element in the array, returning the resulting array. It requires that all the values are NON-NULL.
This function takes an array as the input and keep only elements that satisfy the operator on specified scalar. It requires that the array is 1-D and all the values are NON-NULL. Return type is the same as the input type. By default, this function removes all zeros.
This function takes an array as the input and executes element-wise multiplication by the scalar provided as the second argument, returning the resulting array. It requires that all the values are NON-NULL. Return type is the same as the input type.
This function takes an array as the input and executes element-wise addition of the scalar provided as the second argument, returning the resulting array. It requires that all the values are NON-NULL. Return type is the same as the input type.
This function takes an array as the input and finds square root of each element in the array, returning the resulting array. It requires that all the values are NON-NULL.
This function takes an array and a float8 as the input and finds power of each element in the array, returning the resulting array. It requires that all the values are NON-NULL.
This function takes an array as the input and finds square of each element in the array, returning the resulting array. It requires that all the values are NON-NULL.
This function takes a 2-D array as the input and unnests it by one level. It returns a set of 1-D arrays that correspond to rows of the input array as well as an ID column with values corresponding to row positions occupied by those 1-D arrays within the 2-D array.
+
+
+
+
+
Examples
+
+
Create a database table with two integer array columns and add some data.
Perform element-wise array multiplication and division. The row with id=2 is excluded because the divisor array contains zero, which would cause a divide-by-zero error.
Unnest a column of 2-D arrays into sets of 1-D arrays.
+SELECT id, (madlib.array_unnest_2d_to_1d(val)).*
+FROM (
+ SELECT 1::INT AS id, ARRAY[[1.3,2.0,3.2],[10.3,20.0,32.2]]::FLOAT8[][] AS val
+ UNION ALL
+ SELECT 2, ARRAY[[pi(),pi()/2],[2*pi(),pi()],[pi()/4,4*pi()]]::FLOAT8[][]
+) t
+ORDER BY 1,2;
+
If the function is called without the .* notation then it will return a composite record type with two attributes: the row ID and corresponding unnested array result.
These modules provide basic mathematical operations to be run on array and matrices.
+
For a distributed system, a matrix cannot simply be represented as a 2D array of numbers in memory. We provide two forms of distributed representation of a matrix:
+
+
Dense: The matrix is represented as a distributed collection of 1-D arrays. An example 3x10 matrix would be the below table:
All matrix operations work with either form of representation.
+
+
In many cases, a matrix function can be decomposed to vector operations applied independently on each row of a matrix (or corresponding rows of two matrices). We have also provided access to these internal vector operations (Array Operations) for greater flexibility. Matrix operations like matrix_add use the corresponding vector operation (array_add) and also include additional validation and formating. Other functions like matrix_mult are complex and use a combination of such vector operations and other SQL operations.
+
It's important to note that these array functions are only available for the dense format representation of the matrix. In general, the scope of a single array function invocation is limited to only an array (1-dimensional or 2-dimensional) that fits in memory. When such function is executed on a table of arrays, the function is called multiple times - once for each array (or pair of arrays). On contrary, scope of a single matrix function invocation is the complete matrix stored as a distributed table.
This module implements the association rules data mining technique on a transactional data set. Given the names of a table and the columns, minimum support and confidence values, this function generates all single and multidimensional association rules that meet the minimum thresholds.
+
Association rule mining is a widely used technique for discovering relationships between variables in a large data set (e.g., items in a store that are commonly purchased together). The classic market basket analysis example using association rules is the "beer and diapers" rule. According to data mining urban legend, a study of customer purchase behavior in a supermarket found that men often purchased beer and diapers together. After making this discovery, the managers strategically placed beer and diapers closer together on the shelves and saw a dramatic increase in sales. In addition to market basket analysis, association rules are also used in bioinformatics, web analytics, and several other fields.
+
This type of data mining algorithm uses transactional data. Every transaction event has a unique identifier, and each transaction consists of a set of items (or itemset). Purchases are considered binary (either it was purchased or not), and this implementation does not take into consideration the quantity of each item. For the MADlib association rules function, it is assumed that the data is stored in two columns with one item and transaction id per row. Transactions with multiple items will span multiple rows with one row per item.
Association rules take the form "If X, then Y", where X and Y are non-empty itemsets. X and Y are called the antecedent and consequent, or the left-hand-side and right-hand-side, of the rule respectively. Using our previous example, the association rule may state "If {diapers}, then {beer}" with .2 support and .85 confidence.
+
The following metrics are defined for any given itemset "X".
+
Count: The number of transactions that contain X
+
Support: The ratio of transactions that contain X to all transactions, T
+\[ S (X) = \frac{Total X}{Total transactions} \]
+
+
+
+
Given any association rule "If X, then Y", the association rules function will also calculate the following metrics:
+
Count: The number of transactions that contain X,Y
+
Support: The ratio of transactions that contain X,Y to all transactions, T
Confidence: The ratio of transactions that contain \( X,Y \) to transactions that contain \( X \). One could view this metric as the conditional probability of \( Y \) , given \( X \) . \( P(Y|X) \)
+\[ C (X \Rightarrow Y) = \frac{s(X \cap Y )}{s(X)} \]
+
+
+
Lift: The ratio of observed support of \( X,Y \) to the expected support of \( X,Y \) , assuming \( X \) and \( Y \) are independent.
+\[ L (X \Rightarrow Y) = \frac{s(X \cap Y )}{s(X) \cdot s(Y)} \]
+
+
+
Conviction: The ratio of expected support of \( X \) occurring without \( Y \) assuming \( X \) and \( \neg Y \) are independent, to the observed support of \( X \) occuring without \( Y \). If conviction is greater than 1, then this metric shows that incorrect predictions ( \( X \Rightarrow Y \) ) occur less often than if these two actions were independent. This metric can be viewed as the ratio that the association rule would be incorrect if the actions were independent (i.e. a conviction of 1.5 indicates that if the variables were independent, this rule would be incorrect 50% more often.)
Although there are many algorithms that generate association rules, the classic algorithm is called Apriori [1] which we have implemented in this module. It is a breadth-first search, as opposed to depth-first searches like Eclat. Frequent itemsets of order \( n \) are generated from sets of order \( n - 1 \). Using the downward closure property, all sets must have frequent subsets. There are two steps in this algorithm; generating frequent itemsets, and using these itemsets to construct the association rules. A simplified version of the algorithm is as follows, and assumes a minimum level of support and confidence is provided:
+
Initialstep
+
Generate all itemsets of order 1.
+
Eliminate itemsets that have support less than minimum support.
+
+
Mainalgorithm
+
For \( n \ge 2 \), generate itemsets of order \( n \) by combining the itemsets of order \( n - 1 \). This is done by doing the union of two itemsets that have identical items except one.
+
Eliminate itemsets that have (n-1) order subsets with insufficient support.
+
Eliminate itemsets with insufficient support.
+
Repeat until itemsets cannot be generated, or maximum itemset size is exceeded.
+
+
Associationrulegeneration
+
Given a frequent itemset \( A \) generated from the Apriori algorithm, and all subsets \( B \) , we generate rules such that \( B \Rightarrow (A - B) \) meets minimum confidence requirements.
+
Note
Beware of combinatorial explosion. The Apriori algorithm can potentially generate a huge number of rules, even for fairly simple data sets, resulting in run times that are unreasonably long. To avoid this, it is recommended to cap the maximum itemset size to a small number to start with, then increase it gradually. Similarly, max_LHS_size and max_RHS_size limit the number of items on the LHS and RHS of the rules and can significantly reduce run times. Support and confidence values are parameters that can also be used to control rule generation.
This generates all association rules that satisfy the specified minimum support and confidence.
+
Arguments
+
support
+
Minimum level of support needed for each itemset to be included in result.
+
+
+
confidence
+
Minimum level of confidence needed for each rule to be included in result.
+
+
+
tid_col
+
Name of the column storing the transaction ids.
+
+
+
item_col
+
Name of the column storing the products.
+
+
+
input_table
+
Name of the table containing the input data.
+
The input data is expected to be of the following form:
{TABLE|VIEW} input_table (
+ trans_id INTEGER,
+ product TEXT
+)
The algorithm maps the product names to consecutive integer ids starting at 1. If they are already structured this way, then the ids will not change.
+
+
+
output_schema
+
The name of the schema where the final results will be stored. The schema must be created before calling the function. Alternatively, use NULL to output to the current schema.
+
The results containing the rules, support, count, confidence, lift, and conviction are stored in the table assoc_rules in the schema specified by output_schema.
+
The table has the following columns.
+
+
ruleid
integer
+
+
pre
text
+
+
post
text
+
+
count
integer
+
+
support
double
+
+
confidence
double
+
+
lift
double
+
+
conviction
double
+
+
On Greenplum Database, the table is distributed by the ruleid column.
+
The pre and post columns are the itemsets of left and right hand sides of the association rule respectively. The support, confidence, lift, and conviction columns are calculated as described earlier.
+
+
+
verbose (optional)
+
BOOLEAN, default: FALSE. Determines if details are printed for each iteration as the algorithm progresses.
+
+
+
max_itemset_size (optional)
+
INTEGER, default: 10. Determines the maximum size of frequent itemsets that are used for generating association rules. Must be 2 or more. This parameter can be used to reduce run time for data sets where itemset size is large, which is a common situation. If your query is not returning or is running too long, try using a lower value for this parameter.
+
+
+
max_LHS_size (optional)
+
INTEGER, default: NULL. Determines the maximum size of the left hand side of the rule. Must be 1 or more. This parameter can be used to reduce run time.
+
+
+
max_RHS_size (optional)
+
INTEGER, default: NULL. Determines the maximum size of the right hand side of the rule. Must be 1 or more. This parameter can be used to reduce run time. For example, setting to 1 can significantly reduce run time if this makes sense for your use case. (The apriori algorithm in the R package arules [2] only supports a RHS of 1.)
+
+
Examples
+
Let's look at some sample transactional data and generate association rules.
Let \( min(support) = .25 \) and \( min(confidence) = .5 \), and the output schema is set to NULL indicating output to the current schema. In this example we set verbose to TRUE so that we have some insight into progress of the function. We can now generate association rules as follows:
+DROP TABLE IF EXISTS assoc_rules;
+SELECT * FROM madlib.assoc_rules( .25, -- Support
+ .5, -- Confidence
+ 'trans_id', -- Transaction id col
+ 'product', -- Product col
+ 'test_data', -- Input data
+ NULL, -- Output schema
+ TRUE -- Verbose output
+ );
+
Post-processing can now be done on the output table in the case that you want to filter the results. For example, if you want any single item on the left hand side and a particular item on the right hand side:
+SELECT * FROM assoc_rules WHERE array_upper(pre,1) = 1 AND post = array['beer'];
+
The association rules function always creates a table named assoc_rules. Make a copy of this table before running the function again if you would like to keep multiple association rule tables. This behavior will be improved in a later release.
This module contains automated machine learning (autoML) methods for model architecture search and hyperparameter tuning.
+
The goal of autoML when training deep nets is to reduce the amount of hand-tuning by data scientists to produce a model of acceptable accuracy, compared to manual methods like grid or random search. The two autoML methods implemented here are Hyperband and Hyperopt. If you want to use grid or random search, please refer to Define Model Configurations.
+
Hyperband is an effective model selection algorithm that utilizes the idea of successive halving. It accelerates random search through adaptive resource allocation and early stopping [1]. The implementation here is designed to keep MPP database cluster resources as busy as possible when executing the Hyperband schedule.
+
There is also a utility function for printing out the Hyperband schedule for a given set of input parameters, to give you a sense of how long a run might take before starting.
+
Hyperopt is meta-modeling approach for automated hyperparameter optimization [2]. It intelligently explores the search space while narrowing down to the best estimated parameters. Within Hyperopt we support random search and Tree of Parzen Estimators (TPE) approach.
+
Note
AutoML methods do not currently support multi-input or multi-output neural networks.
TEXT. Name of the table containing the training data. This is the name of the output table from the image preprocessor. Independent and dependent variables are specified in the preprocessor step which is why you do not need to explictly state them here. Configurations will be evaluated by the autoML methods on the basis of training loss, unless a validation table is specified below, in which case validation loss will be used.
+
+
+
model_output_table
+
TEXT. Name of the output table containing the multiple models created.
Note
'pg_temp' is not allowed as an output table schema. Details of output tables are shown below.
+
+
model_arch_table
+
VARCHAR. Table containing model architectures and weights. For more information on this table refer to Define Model Architectures.
+
+
+
model_selection_table
+
VARCHAR. Model selection table created by this method. A summary table named <model_selection_table>_summary is also created. Contents of both of these tables are described below.
+
+
+
model_id_list
+
INTEGER[]. Array of model IDs from the 'model_arch_table' to be included in the run combinations. For hyperparameter search, this will typically be one model ID. For model architecture search, this will be the different model IDs that you want to try.
+
+
+
compile_params_grid
+
VARCHAR. String representation of a Python dictionary of compile parameters to be tested. Each entry of the dictionary should consist of keys as compile parameter names, and values as a Python list of compile parameter values to be passed to Keras. Also, optimizer parameters are a nested dictionary to allow different optimizer types to have different parameters or ranges of parameters. Here is an example:
The following types of sampling are supported: 'linear', 'log' and 'log_near_one'. The 'log_near_one' sampling is useful for exponentially weighted average types of parameters like momentum, which are very sensitive to changes near 1. It has the effect of producing more values near 1 than regular log-based sampling. However, 'log_near_one' is only supported for Hyperband, not for Hyperopt.
+
Note
+
Custom loss functions and custom metrics can be used as defined in Define Custom Functions. List the custom function name and provide the name of the table where the serialized Python objects reside using the parameter 'object_table' below.
+
The following loss function is not supported: sparse_categorical_crossentropy. The following metrics are not supported: sparse_categorical_accuracy, sparse_top_k_categorical_accuracy.
+
The Keras accuracy parameter top_k_categorical_accuracy returns top 5 accuracy by default. If you want a different top k value, use the helper function Top k Accuracy Function to create a custom Python function to compute the top k accuracy that you want.
+
+
+
+
fit_params_grid
+
VARCHAR. String representation of a Python dictionary of fit parameters to be tested. Each entry of the dictionary should consist of keys as fit parameter names, and values as a Python list of fit parameter values to be passed to Keras. Here is an example:
Callbacks are not currently supported except for TensorBoard which you can specify in the usual way, e.g., 'callbacks': ['[TensorBoard(log_dir="/tmp/logs/fit")]']
+
+
automl_method (optional)
+
VARCHAR, default 'hyperband'. Name of the autoML algorithm to run. Can be either 'hyperband' or 'hyperopt' (case insensitive).
+
Note
If you select 'hyperopt', then the Hyperopt package must be installed on the main node of the database cluster [3]. Hyperband does not need any separate package installation.
+
+
+
automl_params (optional)
+
VARCHAR, default depends on the method. Parameters for the chosen autoML method in a comma-separated string of key-value pairs. Please refer to references [1] and [2] for more details on the definition of these parameters.
+
+
Hyperband params:
+
+
R
+
Default: 6. Maximum amount of resources (i.e., iterations) to allocate to a single configuration in a round of Hyperband.
+
eta
+
Default: 3. Controls the proportion of configurations discarded in each round of successive halving. For example, for eta=3 will keep the best 1/3 the configurations for the next round.
+
skip_last
+
Default: 0. The number of last rounds to skip. For example, 'skip_last=1' will skip the last round (i.e., last entry in each bracket), which is standard random search and can be expensive when run for the total R iterations.
+
+
+
Hyperopt params:
+
+
num_configs
+
Default: 20. Number of trials to evaluate.
+
num_iterations
+
Default: 5. Number of iterations to run for each trial.
+
algorithm
+
Default: 'tpe'. Name of the algorithm to explore the search space in Hyperopt ('rand' or 'tpe').
+
+
+
+
random_state (optional)
+
INTEGER, default NULL. Pseudo random number generator state used for random uniform sampling from lists of possible values. Pass an integer to evaluate a fixed set of configurations.
Note
Specifying a random state does not guarantee result reproducibility of the best configuration or the best train/validation accuracy/loss. It only guarantees that the same set of configurations will be chosen for evaluation.
+
+
object_table (optional)
+
VARCHAR, default: NULL. Name of the table containing Python objects in the case that custom loss functions, metrics or top k categorical accuracy are specified in the 'compile_params_grid'.
+
+
+
validation_table (optional)
+
TEXT, default: none. Name of the table containing the validation dataset. Note that the validation dataset must be preprocessed in the same way as the training dataset, so this is the name of the output table from running the image preprocessor on the validation dataset. Using a validation dataset can mean a longer training time depending on its size, and the configurations in autoML will be evaluated on the basis of validation loss instead of training loss.
+
+
+
metrics_compute_frequency (optional)
+
INTEGER, default: once at the end of training. Frequency to compute per-iteration metrics for the training dataset and validation dataset (if specified). There can be considerable cost to computing metrics every iteration, especially if the training dataset is large. This parameter is a way of controlling the frequency of those computations. For example, if you specify 5, then metrics will be computed every 5 iterations as well as at the end of training. If you use the default, metrics will be computed only once after training has completed.
+
+
+
name (optional)
+
TEXT, default: NULL. Free text string to provide a name, if desired.
+
+
+
description (optional)
+
TEXT, default: NULL. Free text string to provide a description, if desired.
+
+
+
use_caching (optional)
+
BOOLEAN, default: FALSE. Use caching of images in memory on the segment in order to speed up processing.
+
Note
When set to TRUE, image byte arrays on each segment are maintained in cache (GD). This can speed up training significantly, however the memory usage per segment increases. In effect, it requires enough available memory on a segment so that all images residing on that segment can be read into memory.
+
+
+
Output tables
+ The model selection output table <model_selection_table> has only one row containing the best model configuration from autoML, based on the training/validation loss. It contains the following columns:
+
+
mst_key
INTEGER. ID that defines a unique tuple for model architecture-compile parameters-fit parameters.
+
+
model_id
VARCHAR. Model architecture ID from the 'model_arch_table'.
+
+
compile_params
VARCHAR. Keras compile parameters.
+
+
fit_params
VARCHAR. Keras fit parameters.
+
+
A summary table named <model_selection_table>_summary is also created, which contains the following columns:
+
+
model_arch_table
VARCHAR. Name of the model architecture table containing the model architecture IDs.
+
+
object_table
VARCHAR. Name of the object table containing the serialized Python objects for custom loss functions, custom metrics and top k categorical accuracy. If there are none, this field will be blank.
+
+
The model output table produced by autoML contains columns below. There is one row per model configuration generated:
+
+
mst_key
INTEGER. ID that defines a unique tuple for model architecture-compile parameters-fit parameters, as defined in the 'model_selection_table'.
+
+
model_weights
BYTEA8. Byte array containing the weights of the neural net.
+
+
model_arch
TEXT. A JSON representation of the model architecture used in training.
+
+
An info table named <model_output_table>_info is also created, which has the columns below. There is one row per model:
+
+
mst_key
INTEGER. ID that defines a unique tuple for model architecture-compile parameters-fit parameters, for each model configuration generated.
+
+
model_id
INTEGER. ID that defines model in the 'model_arch_table'.
+
+
compile_params
Compile parameters passed to Keras.
+
+
fit_params
Fit parameters passed to Keras.
+
+
model_type
General identifier for type of model trained. Currently says 'madlib_keras'.
+
+
model_size
Size of the model in KB. Models are stored in 'bytea' data format which is used for binary strings in PostgreSQL type databases.
+
+
metrics_elapsed_time
Array of elapsed time for metric computations as per the 'metrics_compute_frequency' parameter. Useful for drawing a curve showing loss, accuracy or other metrics as a function of time. For example, if 'metrics_compute_frequency=5' this would be an array of elapsed time for every 5th iteration, plus the last iteration.
+
+
metrics_type
Metric specified in the 'compile_params'.
+
+
training_metrics_final
Final value of the training metric after all iterations have completed. The metric reported is the one specified in the 'metrics_type' parameter.
+
+
training_loss_final
Final value of the training loss after all iterations have completed.
+
+
training_metrics
Array of training metrics as per the 'metrics_compute_frequency' parameter. For example, if 'metrics_compute_frequency=5' this would be an array of metrics for every 5th iteration, plus the last iteration.
+
+
training_loss
Array of training losses as per the 'metrics_compute_frequency' parameter. For example, if 'metrics_compute_frequency=5' this would be an array of losses for every 5th iteration, plus the last iteration.
+
+
validation_metrics_final
Final value of the validation metric after all iterations have completed. The metric reported is the one specified in the 'metrics_type' parameter.
+
+
validation_loss_final
Final value of the validation loss after all iterations have completed.
+
+
validation_metrics
Array of validation metrics as per the 'metrics_compute_frequency' parameter. For example, if 'metrics_compute_frequency=5' this would be an array of metrics for every 5th iteration, plus the last iteration.
+
+
validation_loss
Array of validation losses as per the 'metrics_compute_frequency' parameter. For example, if 'metrics_compute_frequency=5' this would be an array of losses for every 5th iteration, plus the last iteration.
+
+
metrics_iters
Array indicating the iterations for which metrics are calculated, as derived from the parameters 'metrics_compute_frequency' and iterations decided by the autoML algorithm. For example, if 'num_iterations=5' and 'metrics_compute_frequency=2', then 'metrics_iters' value would be {2,4,5} indicating that metrics were computed at iterations 2, 4 and 5 (at the end). If 'num_iterations=5' and 'metrics_compute_frequency=1', then 'metrics_iters' value would be {1,2,3,4,5} indicating that metrics were computed at every iteration.
+
Note that 'metrics_iters' values are for the overall iterations. For some models, the count might start at a later iteration based on the schedule. This representation is selected to simplify representing the results in iteration-metric graphs.
+
+
+
s
Bracket number from Hyperband schedule. This column is not present for Hyperopt.
+
+
i
Latest evaluated round number from Hyperband schedule. This column is not present for Hyperopt.
+
+
A summary table named <model_output_table>_summary is also created, which has the following columns:
+
+
source_table
Source table used for training.
+
+
validation_table
Name of the table containing the validation dataset (if specified).
+
+
model
Name of the output table containing the model for each model selection tuple.
+
+
model_info
Name of the output table containing the model performance and other info for each model selection tuple.
+
+
dependent_varname
Dependent variable column from the original source table in the image preprocessing step.
+
+
independent_varname
Independent variables column from the original source table in the image preprocessing step.
+
+
model_arch_table
Name of the table containing the model architecture and (optionally) the initial model weights.
+
+
model selection table
Name of the mst table containing the best configuration.
+
+
automl_method
Name of the autoML method used.
+
+
automl_params
AutoML parameter values.
+
+
random_state
Chosen random seed.
+
+
metrics_compute_frequency
Frequency that per-iteration metrics are computed for the training dataset and validation datasets.
+
+
name
Name of the training run (free text).
+
+
description
Description of the training run (free text).
+
+
start_training_time
Timestamp for start of training.
+
+
end_training_time
Timestamp for end of training.
+
+
madlib_version
Version of MADlib used.
+
+
num_classes
Count of distinct classes values used.
+
+
<dependent_varname>_class_values
Array of actual class values used for a particular dependent variable. A column will be generated for each dependent variable.
+
+
dependent_vartype
Data type of the dependent variable.
+
+
normalizing_constant
Normalizing constant used from the image preprocessing step.
+
+
Print Hyperband Schedule
+
This utility prints out the schedule for a set of input parameters. It does not run the Hyperband method, rather it just prints out the schedule so you can see what the brackets look like. Refer to [1] for information on Hyperband schedules.
VARCHAR. Name of output table containing hyperband schedule.
+
+
+
R
+
INTEGER. Maximum number of resources (i.e., iterations) to allocate to a single configuration in a round of Hyperband.
+
+
+
eta
+
INTEGER. Controls the proportion of configurations discarded in each round of successive halving. For example, for eta=3 will keep the best 1/3 the configurations for the next round.
+
+
+
skip_last
+
INTEGER. The number of last rounds to skip. For example, 'skip_last=1' will skip the last round (i.e., last entry in each bracket), which is standard random search and can be expensive when run for the total R iterations.
+
+
+
+
Output table
+ The hyperband schedule output table contains the following columns:
Hyperopt must be installed on the main node of the database cluster if you want to use the Hyperopt method of autoML. You can pip install it in the usual way [3]. Hyperband does not require any separate package installation.
+
In practice you may need to do more than one run of an autoML method to arrive at a model with adequate accuracy. One approach is to set the search space to be quite broad initially, then observe which hyperparameter ranges and model architectures seem to be doing the best. Subesquent runs can then zoom in on those good ones in order to fine tune the model.
+
+
Literature
+
[1] Li et al., "Hyperband: A Novel Bandit-Based Approach to
+Hyperparameter Optimization", Journal of Machine Learning Research 18 (2018) 1-52.
+
[2] J. Bergstra, D. Yamins, D. D. Cox, "Making a Science of Model Search:
+Hyperparameter Optimization in Hundreds of Dimensions for Vision Architectures," Proceedings of the 30th International Conference on Machine Learning, Atlanta, Georgia, USA, 2013. JMLR: W&CP volume 28.
Some classification algorithms only perform optimally when the number of samples in each class is roughly the same. Highly skewed datasets are common in many domains (e.g., fraud detection), so resampling to offset this imbalance can produce a better decision boundary.
+
This module offers a number of resampling techniques including undersampling majority classes, oversampling minority classes, and combinations of the two.
TEXT. Name of the table containing the input data.
+
+
+
output_table
+
TEXT. Name of output table that contains the sampled data. The output table contains all columns present in the source table, plus a new generated id called "__madlib_id__" added as the first column.
+
+
+
class_col
+
TEXT, Name of the column containing the class to be balanced.
+
+
+
class_sizes (optional)
+
VARCHAR, default ‘uniform’. Parameter to define the size of the different class values. (Class values are sometimes also called levels). Can be set to the following:
+
+
+‘uniform’: All class values will be resampled to have the same number of rows.
+
+'undersample': Undersample such that all class values end up with the same number of observations as the minority class. Done without replacement by default unless the parameter ‘with_replacement’ is set to TRUE.
+
+'oversample': Oversample with replacement such that all class values end up with the same number of observations as the majority class. Not affected by the parameter ‘with_replacement’ since oversampling is always done with replacement. Short forms of the above will work too, e.g., 'uni' works the same as 'uniform'.
+
+
Alternatively, you can also explicitly set class size in a string containing a comma-delimited list. Order does not matter and all class values do not need to be specified. Use the format “class_value_1=x, class_value_2=y, …” where 'class_value' in the list must exist in the column 'class_col'. Set to an integer representing the desired number of observations. E.g., ‘red=3000, blue=4000’ means you want to resample the dataset to result in exactly 3000 red and 4000 blue rows in the ‘output_table’.
+
Note
The allowed names for class values follows object naming rules in PostgreSQL [1]. Quoted identifiers are allowed and should be enclosed in double quotes in the usual way. If for some reason the class values in the examples above were “ReD” and “BluE” then the comma delimited list for ‘class_size’ would be: ‘“ReD”=3000, “BluE”=4000’.
+
+
output_table_size (optional)
+
INTEGER, default NULL. Desired size of the output data set. This parameter is ignored if ‘class_size’ parameter is set to either ‘oversample’ or ‘undersample’ since output table size is already determined. If NULL, the resulting output table size will depend on the settings for the ‘class_size’ parameter (see table below for more details).
+
+
+
grouping_cols (optional)
+
TEXT, default: NULL. A single column or a list of comma-separated columns that defines the strata. When this parameter is NULL, no grouping is used so the sampling is non-stratified, that is, the whole table is treated as a single group.
+
Note
The 'output_table_size' and the 'class_sizes' are defined for the whole table. When grouping is used, these parameters are split evenly for each group. Further, if a specific class value is specified in the 'class_sizes' parameter, that particular class value should be present in each group. If not, an error will be thrown.
+
+
with_replacement (optional)
+
BOOLEAN, default FALSE. Determines whether to sample with replacement or without replacement (default). With replacement means that it is possible that the same row may appear in the sample set more than once. Without replacement means a given row can be selected only once. This parameter affects undersampling only since oversampling is always done with replacement.
+
+
+
keep_null (optional)
+
BOOLEAN, default FALSE. Determines whether to sample rows whose class values are NULL. By default, all rows with NULL class values are ignored. If this is set to TRUE, then NULL is treated as another class value.
+
+
How Output Table Size is Determined
+
The rule of thumb is that if you specify a value for 'output_table_size', then you will generally get an output table of that size, with some minor rounding variations. If you set 'output_table_size' to NULL, then the size of the output table will be calculated depending on what you put for the 'class_size' parameter. The following table shows how the parameters 'class_size' and 'output_table_size' work together:
+
+
+
Case
'class_size'
'output_table_size'
Result
+
+
1
'uniform'
NULL
Resample for uniform class size with output size = input size (i.e., balanced).
+
+
2
'uniform'
10000
Resample for uniform class size with output size = 10K (i.e., balanced).
+
+
3
NULL
NULL
Resample for uniform class size with output size = input size (i.e., balanced). Class_size=NULL has same behavior as ‘uniform’.
+
+
4
NULL
10000
Resample for uniform class size with output size = 10K (i.e., balanced). Class_size=NULL has same behavior as ‘uniform’.
+
+
5
'undersample'
n/a
Undersample such that all class values end up with the same number of observations as the minority.
+
+
6
'oversample'
n/a
Oversample with replacement (always) such that all class values end up with the same number of observations as the majority.
+
+
7
'red=3000'
NULL
Resample red to 3K, leave rest of the class values (blue, green, etc.) as is.
+
+
8
'red=3000, blue=4000'
10000
Resample red to 3K and blue to 4K, divide remaining class values evenly 3K/(n-2) each, where n=number of class values. Note that if red and blue are the only class values, then output table size will be 7K not 10K. (This is the only case where specifying a value for 'output_table_size' may not actually result in an output table of that size.)
+
+
Examples
+
Note that due to the random nature of sampling, your results may look different from those below.
Uniform sampling. All class values will be resampled so that they have the same number of rows. The output data size will be the same as the input data size, ignoring NULL values. Uniform sampling is the default for the 'class_size' parameter so we do not need to explicitly set it:
+DROP TABLE IF EXISTS output_table;
+SELECT madlib.balance_sample(
+ 'flags', -- Source table
+ 'output_table', -- Output table
+ 'mainhue'); -- Class column
+SELECT * FROM output_table ORDER BY mainhue, name;
+
Oversampling. Oversample with replacement such that all class values except NULLs end up with the same number of observations as the majority class. Countries with red flags is the majority class with 10 observations, so other class values will be oversampled to 10 observations:
+DROP TABLE IF EXISTS output_table;
+SELECT madlib.balance_sample(
+ 'flags', -- Source table
+ 'output_table', -- Output table
+ 'mainhue', -- Class column
+ 'oversample'); -- Oversample
+SELECT * FROM output_table ORDER BY mainhue, name;
+
Undersampling. Undersample such that all class values except NULLs end up with the same number of observations as the minority class. Countries with white flags is the minority class with 2 observations, so other class values will be undersampled to 2 observations:
+DROP TABLE IF EXISTS output_table;
+SELECT madlib.balance_sample(
+ 'flags', -- Source table
+ 'output_table', -- Output table
+ 'mainhue', -- Class column
+ 'undersample'); -- Undersample
+SELECT * FROM output_table ORDER BY mainhue, name;
+
Next we set the number of rows for red and blue flags, and also set an output table size. This means that green and white flags will be uniformly sampled to get to the desired output table size:
+DROP TABLE IF EXISTS output_table;
+SELECT madlib.balance_sample(
+ 'flags', -- Source table
+ 'output_table', -- Output table
+ 'mainhue', -- Class column
+ 'red=7, blue=7', -- Want 7 reds and 7 blues
+ 22); -- Desired output table size
+SELECT * FROM output_table ORDER BY mainhue, name;
+
To perform the balance sampling for independent groups, use the 'grouping_cols' parameter. Note below that each group (zone) has a different count of the classes (mainhue), with some groups not containing some class values.
+DROP TABLE IF EXISTS output_table;
+SELECT madlib.balance_sample(
+ 'flags', -- Source table
+ 'output_table', -- Output table
+ 'mainhue', -- Class column
+ NULL, -- Uniform
+ NULL, -- Output table size
+ 'zone' -- Grouping by zone
+);
+SELECT * FROM output_table ORDER BY zone, mainhue;
+
Grouping can be used with class size specification as well. Note below that 'blue=<Integer>' is the only valid class value since 'blue' is the only class value that is present in each group. Further, 'blue=8' will be split between the four groups, resulting in two blue rows for each group.
+DROP TABLE IF EXISTS output_table;
+SELECT madlib.balance_sample(
+ 'flags', -- Source table
+ 'output_table', -- Output table
+ 'mainhue', -- Class column
+ 'blue=8', -- Specified class value size. Rest of the values are outputed as is.
+ NULL, -- Output table size
+ 'zone' -- Group by zone
+);
+SELECT * FROM output_table ORDER BY zone, mainhue;
+
This MADlib method is still in early stage development. Interface and implementation are subject to change.
+
Naive Bayes refers to a stochastic model where all independent variables \( a_1, \dots, a_n \) (often referred to as attributes in this context) independently contribute to the probability that a data point belongs to a certain class \( c \).
+
Naives Bayes classification estimates feature probabilities and class priors using maximum likelihood or Laplacian smoothing. For numeric attributes, Gaussian smoothing can be used to estimate the feature probabilities.These parameters are then used to classify new data.
+
Training Function(s)
+
For data with only categorical attributes, precompute feature probabilities and class priors using the following function:
For data containing both categorical and numeric attributes, use the following form to precompute the Gaussian parameters (mean and variance) for numeric attributes alongside the feature probabilities for categorical attributes and class priors.
The trainingSource is expected to be of the following form:
{TABLE|VIEW} trainingSource (
+ ...
+ trainingClassColumn INTEGER,
+ trainingAttrColumn INTEGER[] OR NUMERIC[] OR FLOAT8[],
+ ...
+)
numericAttrsColumnIndices should be of type TEXT, specified as an array of indices (starting from 1) in the trainingAttrColumn attributes-array that correspond to numeric attributes.
+
The two output tables are:
+
featureProbsName – stores feature probabilities
+
classPriorsName – stores the class priors
+
+
In addition to the above, if the function specifying numeric attributes is used, an additional table numericAttrParamsName is created which stores the Gaussian parameters for the numeric attributes.
This creates the view destName mapping classifyKeyColumn and every single class to the Naive Bayes probability:
+key | class | nb_prob
+ ---+-------+--------
+...
+
Ad Hoc Computation Function
+
With ad hoc execution (no precomputation), the functions create_nb_classify_view() and create_nb_probs_view() can be used in an ad-hoc fashion without the precomputation step. In this case, replace the function arguments
The probabilities computed on the platforms of PostgreSQL and Greenplum database have a small difference due to the nature of floating point computation. Usually this is not important. However, if a data point has
+\[ P(C=c_i \mid A) \approx P(C=c_j \mid A) \]
+
+ for two classes, this data point might be classified into diferent classes on PostgreSQL and Greenplum. This leads to the differences in classifications on PostgreSQL and Greenplum for some data sets, but this should not affect the quality of the results.
+
When two classes have equal and highest probability among all classes, the classification result is an array of these two classes, but the order of the two classes is random.
+
The current implementation of Naive Bayes classification is suitable for discontinuous (categorial) attributes as well as continuous (numeric) attributes.
+For continuous data, a typical assumption, usually used for small datasets, is that the continuous values associated with each class are distributed according to a Gaussian distribution, and the probabilities \( P(A_i = a \mid C=c) \) are estimated using the Gaussian Distribution formula:
+ where \(\mu_c\) and \(\sigma^{2}_c\) are the population mean and variance of the attribute for the class \(c\).
+Another common technique for handling continuous values, which is better for large data sets, is to use binning to discretize the values, and convert the continuous data into categorical bins. This approach is currently not implemented.
+
One can provide floating point data to the Naive Bayes classification function. If the corresponding attribute index is not specified in numericAttrsColumnIndices, floating point numbers will be used as symbolic substitutions for categorial data. In this case, the classification would work best if there are sufficient data points for each floating point attribute. However, if floating point numbers are used as continuous data without the attribute being marked as of type numeric in numericAttrsColumnIndices, no warning is raised and the result may not be as expected.
+
+
+
Examples
+
The following is an extremely simplified example of the above option #1 which can by verified by hand.
+\[ \Pr(A_1 = a_1, \dots, A_n = a_n \mid C = c) = \prod_{i=1}^n \Pr(A_i = a_i \mid C = c) \,. \]
+
+
Naives Bayes classification estimates feature probabilities and class priors using maximum likelihood or Laplacian smoothing. These parameters are then used to classifying new data.
+
A Naive Bayes classifier computes the following formula:
+\[ \text{classify}(a_1, ..., a_n) = \arg\max_c \left\{ \Pr(C = c) \cdot \prod_{i=1}^n \Pr(A_i = a_i \mid C = c) \right\} \]
+
+
where \( c \) ranges over all classes in the training data and probabilites are estimated with relative frequencies from the training set. There are different ways to estimate the feature probabilities \( P(A_i = a \mid C = c) \). The maximum likelihood estimate takes the relative frequencies. That is:
+\[ P(A_i = a \mid C = c) = \frac{\#(c,i,a)}{\#c} \]
+
+
where
+
\( \#(c,i,a) \) denotes the # of training samples where attribute \( i \) is \( a \) and class is \( c \)
+
\( \#c \) denotes the # of training samples where class is \( c \).
+
+
Since the maximum likelihood sometimes results in estimates of "0", you might want to use a "smoothed" estimate. To do this, you add a number of "virtual" samples and make the assumption that these samples are evenly distributed among the values assumed by attribute \( i \) (that is, the set of all values observed for attribute \( a \) for any class):
+
+\[ P(A_i = a \mid C = c) = \frac{\#(c,i,a) + s}{\#c + s \cdot \#i} \]
+
+
where
+
\( \#i \) denotes the # of distinct values for attribute \( i \) (for all classes)
+
\( s \geq 0 \) denotes the smoothing factor.
+
+
The case \( s = 1 \) is known as "Laplace smoothing". The case \( s = 0 \) trivially reduces to maximum-likelihood estimates.
+
Literature
+
[1] Tom Mitchell: Machine Learning, McGraw Hill, 1997. Book chapter Generativ and Discriminative Classifiers: Naive Bayes and Logistic Regression available at: http://www.cs.cmu.edu/~tom/NewChapters.html
Given a graph and a source vertex, the breadth-first search (BFS) algorithm finds all nodes reachable from the source vertex by searching / traversing the graph in a breadth-first manner.
TEXT. Name of the table containing the vertex data for the graph. Must contain the column specified in the 'vertex_id' parameter below.
+
+
+
vertex_id
+
TEXT, default = 'id'. Name of the column in 'vertex_table' containing vertex ids. The vertex ids can be of type INTEGER or BIGINT with no duplicates. They do not need to be contiguous.
+
+
+
edge_table
+
TEXT. Name of the table containing the edge data. The edge table must contain columns for source vertex and destination vertex. Column naming convention is described below in the 'edge_args' parameter. In addition to vertex columns, if grouping is used then the columns specified in the 'grouping_cols' parameter must be present.
+
+
+
edge_args
+
TEXT. A comma-delimited string containing multiple named arguments of the form "name=value". The following parameters are supported for this string argument:
+
src (INTEGER or BIGINT): Name of the column containing the source vertex ids in the edge table. Default column name is 'src'. (This is not to be confused with the 'source_vertex' argument passed to the BFS function.)
+
dest (INTEGER or BIGINT): Name of the column containing the destination vertex ids in the edge table. Default column name is 'dest'.
+
+
+
+
source_vertex
+
INTEGER or BIGINT. The source vertex id for the algorithm to start. This vertex id must exist in the 'vertex_id' column of 'vertex_table'.
+
+
+
out_table
+
TEXT. Name of the table to store the result of BFS. It contains a row for every vertex that is reachable from the source_vertex. In the presence of grouping columns, only those edges are used for which there are no NULL values in any grouping column. The output table will have the following columns (in addition to the grouping columns):
+
vertex_id : The id for any node reachable from source_vertex in addition to the source_vertex. Will use the input parameter 'vertex_id' for column naming.
+
dist : The distance in number of edges (or hops) from the source_vertex to where this vertex is located.
+
parent : The parent of this vertex in BFS traversal of the graph from source_vertex. Will use 'parent' for column naming. For the case where vertex_id = source_vertex, the value for parent is NULL.
+
+
A summary table named <out_table>_summary is also created. This is an internal table that keeps a record of the input parameters.
+
+
+
max_distance (optional)
+
INT, default = NULL. Maximum distance to traverse from the source vertex. When this value is null, traverses until reaches leaf node. E.g., if set to 1 will return only adjacent vertices, if set to 7 will return vertices up to a maximum distance of 7 vertices away.
+
+
+
directed (optional)
+
BOOLEAN, default = FALSE. If TRUE the graph will be treated as directed, else it will be treated as an undirected graph.
+
+
+
grouping_cols (optional)
+
TEXT, default = NULL. A comma-separated list of columns used to group the input into discrete subgraphs. These columns must exist in the edge table. When this value is NULL, no grouping is used and a single BFS result is generated.
Note
Expressions are not currently supported for 'grouping_cols'.
+
+
+
Examples
+
+
Create vertex and edge tables to represent the graph:
+DROP TABLE IF EXISTS out, out_summary;
+SELECT madlib.graph_bfs(
+ 'vertex', -- Vertex table
+ NULL, -- Vertix id column (NULL means use default naming)
+ 'edge', -- Edge table
+ NULL, -- Edge arguments (NULL means use default naming)
+ 3, -- Source vertex for BFS
+ 'out'); -- Output table of nodes reachable from source_vertex
+ -- Default values used for the other arguments
+SELECT * FROM out ORDER BY dist,id;
+
In this example, we use max_distance to limit the search distance.
+DROP TABLE IF EXISTS out_max, out_max_summary;
+SELECT madlib.graph_bfs(
+ 'vertex', -- Vertex table
+ NULL, -- Vertix id column (NULL means use default naming)
+ 'edge', -- Edge table
+ NULL, -- Edge arguments (NULL means use default naming)
+ 3, -- Source vertex for BFS
+ 'out_max', -- Output table of nodes reachable from source_vertex
+ 2); -- Maximum distance to traverse from source_vertex
+ -- Default values used for the other arguments
+SELECT * FROM out_max ORDER BY dist,id;
+
Now let's do an example using different column names in the tables (i.e., not the defaults). Create the vertex and edge tables:
+DROP TABLE IF EXISTS vertex_alt, edge_alt;
+CREATE TABLE vertex_alt AS SELECT id AS v_id FROM vertex;
+CREATE TABLE edge_alt AS SELECT src AS n1, dest AS n2 FROM edge;
+
+
Run BFS from vertex 8:
+DROP TABLE IF EXISTS out_alt, out_alt_summary;
+SELECT madlib.graph_bfs(
+ 'vertex_alt', -- Vertex table
+ 'v_id', -- Vertex id column (NULL means use default naming)
+ 'edge_alt', -- Edge table
+ 'src=n1, dest=n2', -- Edge arguments (NULL means use default naming)
+ 8, -- Source vertex for BFS
+ 'out_alt'); -- Output table of nodes reachable from source_vertex
+SELECT * FROM out_alt ORDER BY v_id;
+
Now we show an example where the graph is treated as a directed graph.
+DROP TABLE IF EXISTS out_alt_dir, out_alt_dir_summary;
+SELECT madlib.graph_bfs(
+ 'vertex_alt', -- Vertex table
+ 'v_id', -- Vertex id column (NULL means use default naming)
+ 'edge_alt', -- Edge table
+ 'src=n1, dest=n2', -- Edge arguments (NULL means use default naming)
+ 8, -- Source vertex for BFS
+ 'out_alt_dir', -- Output table of nodes reachable from source_vertex
+ NULL, -- Maximum distance to traverse from source_vertex
+ TRUE); -- Flag for specifying directed graph
+SELECT * FROM out_alt_dir ORDER BY v_id;
+
On a Greenplum cluster, the edge table should be distributed by the source vertex id column for better performance.
+
The graph_bfs function is a SQL implementation of the well-known breadth-first search algorithm [1] modified appropriately for a relational database. It will find any node in the graph reachable from the 'source_vertex' only once. If a node is reachable by many different paths from the 'source_vertex' (i.e. has more than one parent), then only one of those parents is present in the output table. The BFS result will, in general, be different for different choices of 'source_vertex'.
Matrix \( \boldsymbol A \) is assumed to be stored in a table where each row consists of at least two columns: array containing values of a given row, row number:
The Clustered Variance module adjusts standard errors for clustering. For example, replicating a dataset 100 times should not increase the precision of parameter estimates, but performing this procedure with the IID assumption will actually do this. Another example is in economics of education research, it is reasonable to expect that the error terms for children in the same class are not independent. Clustering standard errors can correct for this.
+
The MADlib Clustered Variance module includes functions to calculate linear, logistic, and multinomial logistic regression problems.
+
Clustered Variance Linear Regression Training Function
+
The clustered variance linear regression training function has the following syntax.
TEXT. The name of the table containing the input data.
+
+
+
out_table
+
VARCHAR. Name of the generated table containing the output model. The output table contains the following columns.
+
+
coef
DOUBLE PRECISION[]. Vector of the coefficients of the regression.
+
+
std_err
DOUBLE PRECISION[]. Vector of the standard error of the coefficients.
+
+
t_stats
DOUBLE PRECISION[]. Vector of the t-stats of the coefficients.
+
+
p_values
DOUBLE PRECISION[]. Vector of the p-values of the coefficients.
+
+
A summary table named <out_table>_summary is also created, which is the same as the summary table created by linregr_train function. Please refer to the documentation for linear regression for details.
+
+
+
+
dependent_varname
+
TEXT. An expression to evaluate for the dependent variable.
+
independent_varname
+
TEXT. An Expression to evalue for the independent variables.
+
clustervar
+
TEXT. A comma-separated list of the columns to use as cluster variables.
+
grouping_cols (optional)
+
TEXT, default: NULL. Not currently implemented. Any non-NULL value is ignored. An expression list used to group the input dataset into discrete groups, running one regression per group. Similar to the SQL GROUP BY clause. When this value is null, no grouping is used and a single result model is generated.
+
+
Clustered Variance Logistic Regression Training Function
+
The clustered variance logistic regression training function has the following syntax.
TEXT. The name of the table containing the input data.
+
out_table
+
VARCHAR. Name of the generated table containing the output model. The output table has the following columns:
+
+
coef
Vector of the coefficients of the regression.
+
+
std_err
Vector of the standard error of the coefficients.
+
+
z_stats
Vector of the z-stats of the coefficients.
+
+
p_values
Vector of the p-values of the coefficients.
+
+
A summary table named <out_table>_summary is also created, which is the same as the summary table created by logregr_train function. Please refer to the documentation for logistic regression for details.
+
+
+
dependent_varname
+
TEXT. An expression to evaluate for the dependent variable.
+
independent_varname
+
TEXT. An expression to evaluate for the independent variable.
+
clustervar
+
TEXT. A comma-separated list of columns to use as cluster variables.
+
grouping_cols (optional)
+
TEXT, default: NULL. Not yet implemented. Any non-NULL values are ignored. An expression list used to group the input dataset into discrete groups, running one regression per group. Similar to the SQL GROUP BY clause. When this value is NULL, no grouping is used and a single result model is generated.
+
max_iter (optional)
+
INTEGER, default: 20. The maximum number of iterations that are allowed.
+
optimizer (optional)
+
TEXT, default: 'irls'. The name of the optimizer to use:
+
+'newton' or 'irls': Iteratively reweighted least squares
+
+'cg': conjugate gradient
+
+'igd': incremental gradient descent.
+
+
+
tolerance (optional)
+
FLOAT8, default: 0.0001 The difference between log-likelihood values in successive iterations that should indicate convergence. A zero disables the convergence criterion, so that execution stops after n Iterations have completed.
+
verbose_mode (optional)
+
BOOLEAN, default FALSE. Provides verbose_mode output of the results of training.
+
+
Clustered Variance Multinomial Logistic Regression Training Function
TEXT. The name of the table containing the input data.
+
out_table
+
TEXT. The name of the table where the regression model will be stored. The output table has the following columns:
+
+
category
The category.
+
+
ref_category
The refererence category used for modeling.
+
+
coef
Vector of the coefficients of the regression.
+
+
std_err
Vector of the standard error of the coefficients.
+
+
z_stats
Vector of the z-stats of the coefficients.
+
+
p_values
Vector of the p-values of the coefficients.
+
+
A summary table named <out_table>_summary is also created, which is the same as the summary table created by mlogregr_train function. Please refer to the documentation for multinomial logistic regression for details.
+
+
dependent_varname
+
TEXT. An expression to evaluate for the dependent variable.
+
independent_varname
+
TEXT. An expression to evaluate for the independent variable.
+
cluster_varname
+
TEXT. A comma-separated list of columns to use as cluster variables.
+
ref_category (optional)
+
INTEGER. Reference category in the range [0, num_category).
+
groupingvarng_cols (optional)
+
TEXT, default: NULL. Not yet implemented. Any non-NULL values are ignored. A comma-separated list of columns to use as grouping variables.
+
optimizer_params (optional)
+
TEXT, default: NULL, which uses the default values of optimizer parameters: max_iter=20, optimizer='newton', tolerance=1e-4. It should be a string that contains pairs of 'key=value' separated by commas.
+
verbose_mode (optional)
+
BOOLEAN, default FALSE. If TRUE, detailed information is printed when computing logistic regression.
+
+
Clustered Variance for Cox Proportional Hazards model
+
The clustered robust variance estimator function for the Cox Proportional Hazards model has the following syntax.
Note that we need to manually include an intercept term in the independent variable expression. The NULL value of groupingvar means that there is no grouping in the calculation.
+
+
Technical Background
+
Assume that the data can be separated into \(m\) clusters. Usually this can be done by grouping the data table according to one or multiple columns.
+
The estimator has a similar form to the usual sandwich estimator
where \(G_m\) is the set of rows that belong to the same cluster.
+
We can compute the quantities of \(B\) and \(A\) for each cluster during one scan through the data table in an aggregate function. Then sum over all clusters to the full \(B\) and \(A\) in the outside of the aggregate function. At last, the matrix mulplitications are done in a separate function on the master node.
+
When multinomial logistic regression is computed before the multinomial clustered variance calculation, it uses a default reference category of zero and the regression coefficients are included in the output table. The regression coefficients in the output are in the same order as multinomial logistic regression function, which is described below. For a problem with \( K \) dependent variables \( (1, ..., K) \) and \( J \) categories \( (0, ..., J-1) \), let \( {m_{k,j}} \) denote the coefficient for dependent variable \( k \) and category \( j \). The output is \( {m_{k_1, j_0}, m_{k_1, j_1} \ldots m_{k_1, j_{J-1}}, m_{k_2, j_0}, m_{k_2, j_1} \ldots m_{k_K, j_{J-1}}} \). The order is NOT CONSISTENT with the multinomial regression marginal effect calculation with function marginal_mlogregr. This is deliberate because the interfaces of all multinomial regressions (robust, clustered, ...) will be moved to match that used in marginal.
Convert feature columns in a table into an array in a single column.
+
Given a table with a number of feature columns, this function will create an output table that contains the feature columns in an array. A summary table will also be created that contains the names of the features combined into array, so that this process can be reversed using the function vec2cols.
TEXT. Name of the table containing the source data.
+
+
+
output_table
+
TEXT. Name of the generated table containing the output.
+
+
+
list_of_features
+
TEXT. Comma-separated string of column names or expressions to put into feature array. Can also be '*' implying all columns are to be put into feature array (except for the ones included in the next argument that lists exclusions). Type casting will be done as per the regular type casting rules of the underlying database. Array columns in the source table are not supported in the 'list_of_features' parameter. Also, all of the features to be included must be of the same type and must not have null values.
+
+
+
list_of_features_to_exclude (optional)
+
TEXT. Default NULL. Comma-separated string of column names to exclude from the feature array. Typically used when 'list_of_features' is set to '*'.
+
+
+
cols_to_output (optional)
+
TEXT. Default NULL. Comma-separated string of column names from the source table to keep in the output table, in addition to the feature array. To keep all columns from the source table, use '*' for this parameter.
+
+
Output table
+ The output table produced by the cols2vec function contains the following columns:
+
+
<...>
Columns from source table, depending on which ones are kept (if any).
+
+
feature_vector
Column that contains the feature array.
+
+
Output summary table
+ A summary table named <output_table>_summary is also created that contains:
+
+
source_table
Name of the table containing the source data.
+
+
list_of_features
List of features to put in vector.
+
+
list_of_features_to_exclude
Features specified by the user to exclude from 'list_of_features'.
+
+
feature_names
Names of the features that were nested (converted to a vector) in the output table.
This also shows that you can exclude features in 'list_of_features_to_exclude' that are in the list of 'list_of_features'. This can be useful if the 'list_of_features' is generated from an expression or subquery.
+
Type casting works as per regular rules of the underlying database. E.g, combining integer and double precisions columns will create a double precision feature vector. For Boolean, do an explicit cast to the target type:
+DROP TABLE IF EXISTS cols2vec_result, cols2vec_result_summary;
+SELECT madlib.cols2vec(
+ 'golf',
+ 'cols2vec_result',
+ 'windy::TEXT, class',
+ NULL,
+ 'id'
+);
+SELECT * FROM cols2vec_result ORDER BY id;
+
A correlation function is the degree and direction of association of two variables—how well one random variable can be predicted from the other. It is a normalized version of covariance. The Pearson correlation coefficient is used here, which has a value between -1 and 1, where 1 implies total positive linear correlation, 0 means no linear correlation, and -1 means total negative linear correlation.
+
This function generates an \(N\)x \(N\) cross correlation matrix for pairs of numeric columns in a source_table. It is square symmetrical with the \( (i,j) \)th element equal to the correlation coefficient between the \(i\)th and the \(j\)th variable. The diagonal elements (correlations of variables with themselves) are always equal to 1.0.
+
We also provide a covariance function which is similar in nature to correlation, and is a measure of the joint variability of two random variables.
+
Covariance and Correlation Functions
+
The correlation function has the following syntax:
TEXT. Name of the table containing the input data.
+
+
+
output_table
+
TEXT. Name of the table containing the cross correlation matrix. The output table has N rows, where N is the number of 'target_cols' in the 'source_table' for which correlation or covariance is being computed. It has the following columns:
+
+
column_position
An automatically generated sequential counter indicating the order of the variable in the 'output_table'.
+
+
variable
Contains the row header for the variables of interest.
+
+
grouping_cols
Contains the grouping columns, if any.
+
+
<...>
The remainder of the table is the NxN correlation matrix for the pairs of variables of interest.
+
+
The output table is arranged as a lower-triangular matrix with the upper triangle set to NULL and the diagonal elements set to 1.0. To obtain the result from the 'output_table' order by 'column_position':
+SELECT * FROM output_table ORDER BY column_position;
+
In addition to output table, a summary table named <output_table>_summary is also created, which has the following columns:
+
+
method
'Correlation' or 'Covariance'
+
+
source_table
VARCHAR. Data source table name.
+
+
output_table
VARCHAR. Output table name.
+
+
column_names
VARCHAR. Column names used for correlation computation, as a comma-separated string.
+
+
grouping_cols
Contains the grouping columns, if any.
+
+
mean_vector
FLOAT8[]. Mean value of column for variables of interest.
+
+
total_rows_processed
BIGINT. Total numbers of rows processed.
+
+
+
+
target_cols (optional)
+
TEXT, default: '*'. A comma-separated list of the columns to correlate. If NULL or '*', results are produced for all numeric columns.
+
+
+
verbose (optional)
+
BOOLEAN, default: FALSE. Print verbose information if TRUE.
+
+
+
grouping_cols (optional)
+
TEXT, default: NULL. A comma-separated list of the columns to group by.
+
+
+
n_groups_per_run (optional)
+
INTEGER, default: 10. Number of groups to process at a time. This parameter is ignored if 'grouping_cols' is not specified. Generally the default value will work fine, but there may be cases (see below) where you will want to experiment with it to reduce execution time and memory usage.
Note
This is a lower level parameter that can potentially be used to improve performance, but should be used with caution. It is designed to handle the case where you have a large number of groups. In general, increasing 'n_groups_per_run' means we construct a larger 'UNION ALL' query which uses more memory and may slow down execution if it gets too big. If you have a large number of groups and a smaller data size, there may be benefits to increasing this value. Conversely, decreasing 'n_groups_per_run' means we issue more 'plpy.execute' commands. This increases overhead and can modestly affect the execution time.
Null values will be replaced by the mean of their respective columns (mean imputation/substitution). Mean imputation is a method in which the missing value on a certain variable is replaced by the mean of the available cases. This method maintains the sample size and is easy to use, but the variability in the data is reduced, so the standard deviations and the variance estimates tend to be underestimated. Please refer to [1] and [2] for details.
+
If the mean imputation method is not suitable for the target use case, it is advised to employ a view that handles the NULL values prior to calling the correlation/covariance functions.
This module implements Cormode-Muthukrishnan CountMin sketches on integer values, implemented as a user-defined aggregate. It also provides scalar functions over the sketches to produce approximate counts, order statistics, and histograms.
+
Syntax
+
Get a sketch of a selected column specified by col_name.
+cmsketch( col_name )
+
+
Get the number of rows where col_name = p, computed from the sketch obtained from cmsketch.
+cmsketch_count( cmsketch,
+ p )
+
+
Get the number of rows where col_name is between m and n inclusive.
+cmsketch_rangecount( cmsketch,
+ m,
+ n )
+
+
Get the kth percentile of col_name where count specifies number of rows. k should be an integer between 1 to 99.
+cmsketch_centile( cmsketch,
+ k,
+ count )
+
+
Get the median of col_name where count specifies number of rows. This is equivalent to cmsketch_centile(cmsketch,50,count).
+cmsketch_median( cmsketch,
+ count )
+
+
Get an n-bucket histogram for values between min and max for the column where each bucket has approximately the same width. The output is a text string containing triples {lo, hi, count} representing the buckets; counts are approximate.
+cmsketch_width_histogram( cmsketch,
+ min,
+ max,
+ n )
+
+
Get an n-bucket histogram for the column where each bucket has approximately the same count. The output is a text string containing triples {lo, hi, count} representing the buckets; counts are approximate. Note that an equi-depth histogram is equivalent to a spanning set of equi-spaced centiles.
+cmsketch_depth_histogram( cmsketch,
+ n )
+
+
+
+
Note
This is a User Defined Aggregate which returns the results when used in a query. Use "CREATE TABLE AS ", with the UDA as subquery if the results are to be stored. This is unlike the usual MADlib stored procedure interface which places the results in a table instead of returning it.
+
Examples
+
+
Generate some data.
+CREATE TABLE data(class INT, a1 INT);
+INSERT INTO data SELECT 1,1 FROM generate_series(1,10000);
+INSERT INTO data SELECT 1,2 FROM generate_series(1,15000);
+INSERT INTO data SELECT 1,3 FROM generate_series(1,10000);
+INSERT INTO data SELECT 2,5 FROM generate_series(1,1000);
+INSERT INTO data SELECT 2,6 FROM generate_series(1,1000);
+
+
Count number of rows where a1 = 2 in each class. Store results in a table.
+CREATE TABLE sketch_count AS
+SELECT class,
+ cmsketch_count( cmsketch( a1 ), 2 )
+FROM data GROUP BY data.class;
+SELECT * FROM sketch_count;
+
Proportional-Hazard models enable the comparison of various survival models. These survival models are functions describing the probability of a one-item event (prototypically, this event is death) with respect to time. The interval of time before the occurrence of death can be called the survival time. Let T be a random variable representing the survival time, with a cumulative probability function P(t). Informally, P(t) is the probability that death has happened before time t.
+
Training Function
+
Following is the syntax for the coxph_train() training function:
TEXT. The name of the table containing input data.
+
output_table
+
TEXT. The name of the table where the output model is saved. The output is saved in the table named by the output_table argument. It has the following columns:
+
+
coef
FLOAT8[]. Vector of the coefficients.
+
+
loglikelihood
FLOAT8. Log-likelihood value of the MLE estimate.
+
+
std_err
FLOAT8[]. Vector of the standard error of the coefficients.
+
+
stats
FLOAT8[]. Vector of the statistics of the coefficients.
+
+
p_values
FLOAT8[]. Vector of the p-values of the coefficients.
+
+
hessian
FLOAT8[]. The Hessian matrix computed using the final solution.
+
+
num_iterations
INTEGER. The number of iterations performed by the optimizer.
+
+
Additionally, a summary output table is generated that contains a summary of the parameters used for building the Cox model. It is stored in a table named <output_table>_summary. It has the following columns:
+
+
source_table
The source table name.
+
+
dependent_variable
The dependent variable name.
+
+
independent_variable
The independent variable name.
+
+
right_censoring_status
The right censoring status
+
+
strata
The stratification columns
+
+
num_processed
The number of rows that were actually used in the computation.
+
+
num_missing_rows_skipped
The number of rows that were skipped in the computation due to NULL values in them.
+
+
+
+
dependent_variable
+
TEXT. A string containing the name of a column that contains an array of numeric values, or a string expression in the format 'ARRAY[1, x1, x2, x3]', where x1, x2 and x3 are column names. Dependent variables refer to the time of death. There is no need to pre-sort the data.
+
independent_variable
+
TEXT. The name of the independent variable.
+
right_censoring_status (optional)
+
TEXT, default: TRUE for all observations. A string containing an expression that evaluates to the right-censoring status for the observation—TRUE if the observation is not censored and FALSE if the observation is censored. The string could contain the name of the column containing the right-censoring status, a fixed Boolean expression (i.e., 'true', 'false', '0', '1') that applies to all observations, or a Boolean expression such as 'column_name < 10' that can be evaluated for each observation.
+
strata (optional)
+
VARCHAR, default: NULL, which does not do any stratifications. A string of comma-separated column names that are the strata ID variables used to do stratification.
+
optimizer_params (optional)
+
VARCHAR, default: NULL, which uses the default values of optimizer parameters: max_iter=100, optimizer=newton, tolerance=1e-8, array_agg_size=10000000, sample_size=1000000. It should be a string that contains 'key=value' pairs separated by commas. The meanings of these parameters are:
+
+
max_iter — The maximum number of iterations. The computation stops if the number of iterations exceeds this, which usually means that there is no convergence.
+
optimizer — The optimization method. Right now, "newton" is the only one supported.
+
tolerance — The stopping criteria. When the difference between the log-likelihoods of two consecutive iterations is smaller than this number, the computation has already converged and stops.
+
array_agg_size — To speed up the computation, the original data table is cut into multiple pieces, and each pieces of the data is aggregated into one big row. In the process of computation, the whole big row is loaded into memory and thus speed up the computation. This parameter controls approximately how many numbers we want to put into one big row. Larger value of array_agg_size may speed up more, but the size of the big row cannot exceed 1GB due to the restriction of PostgreSQL databases.
+
sample_size — To cut the data into approximate equal pieces, we first sample the data, and then find out the break points using this sampled data. A larger sample_size produces more accurate break points.
+
+
+
+
Proportional Hazards Assumption Test Function
+
The cox_zph() function tests the proportional hazards assumption (PHA) of a Cox regression.
+
Proportional-hazard models enable the comparison of various survival models. These PH models, however, assume that the hazard for a given individual is a fixed proportion of the hazard for any other individual, and the ratio of the hazards is constant across time. MADlib does not currently have support for performing any transformation of the time to compute the correlation.
+
The cox_zph() function is used to test this assumption by computing the correlation of the residual of the coxph_train() model with time.
+
Following is the syntax for the cox_zph() function:
+cox_zph(cox_model_table, output_table)
+
Arguments
+
cox_model_table
+
TEXT. The name of the table containing the Cox Proportional-Hazards model.
+
+
+
output_table
+
TEXT. The name of the table where the test statistics are saved. The output table is named by the output_table argument and has the following columns:
+
+
covariate
TEXT. The independent variables.
+
+
rho
FLOAT8[]. Vector of the correlation coefficients between survival time and the scaled Schoenfeld residuals.
+
+
chi_square
FLOAT8[]. Chi-square test statistic for the correlation analysis.
+
+
p_value
FLOAT8[]. Two-side p-value for the chi-square statistic.
+
+
+
+
Additionally, the residual values are outputted to the table named output_table_residual. The table contains the following columns:
+
+
<dep_column_name>
FLOAT8. Time values (dependent variable) present in the original source table.
+
+
residual
FLOAT8[]. Difference between the original covariate values and the expectation of the covariates obtained from the coxph_train model.
+
+
scaled_residual
Residual values scaled by the variance of the coefficients.
+
+
Notes
+
+
Table names can be optionally schema qualified (current_schemas() is used if a schema name is not provided) and table and column names should follow case-sensitivity and quoting rules per the database. For instance, 'mytable' and 'MyTable' both resolve to the same entity—'mytable'. If mixed-case or multi-byte characters are desired for entity names then the string should be double-quoted; in this case the input would be '"MyTable"'.
The prediction function is provided to calculate the linear predictionors, risk or the linear terms for the given prediction data. It has the following syntax:
TEXT. Name of the table containing the prediction data.
+
+
+
id_col_name
+
TEXT. Name of the id column in the source table.
+
+
+
output_table
+
TEXT. Name of the table to store the prediction results in. The output table is named by the output_table argument and has the following columns:
+
+
id
TEXT. The id column name from the source table.
+
+
predicted_result
DOUBLE PRECISION. Result of prediction based of the value of the prediction type parameter.
+
+
+
+
pred_type
+
TEXT, OPTIONAL. Type of prediction. This can be one of 'linear_predictors', 'risk', or 'terms'. DEFAULT='linear_predictors'.
+
'linear_predictors' calculates the dot product of the independent variables and the coefficients.
+
'risk' is the exponentiated value of the linear prediction.
+
'terms' correspond to the linear terms obtained by multiplying the independent variables with their corresponding coefficients values (without further calculating the sum of these terms)
+
+
+
+
reference
+
TEXT, OPTIONAL. Reference level to use for centering predictions. Can be one of 'strata', 'overall'. DEFAULT='strata'. Note that R uses 'sample' instead of 'overall' when referring to the overall mean value of the covariates as being the reference level.
+
+
Examples
+
+
Create an input data set.
+DROP TABLE IF EXISTS sample_data;
+CREATE TABLE sample_data (
+ id INTEGER NOT NULL,
+ grp DOUBLE PRECISION,
+ wbc DOUBLE PRECISION,
+ timedeath INTEGER,
+ status BOOLEAN
+);
+COPY sample_data FROM STDIN WITH DELIMITER '|';
+ 0 | 0 | 1.45 | 35 | t
+ 1 | 0 | 1.47 | 34 | t
+ 3 | 0 | 2.2 | 32 | t
+ 4 | 0 | 1.78 | 25 | t
+ 5 | 0 | 2.57 | 23 | t
+ 6 | 0 | 2.32 | 22 | t
+ 7 | 0 | 2.01 | 20 | t
+ 8 | 0 | 2.05 | 19 | t
+ 9 | 0 | 2.16 | 17 | t
+ 10 | 0 | 3.6 | 16 | t
+ 11 | 1 | 2.3 | 15 | t
+ 12 | 0 | 2.88 | 13 | t
+ 13 | 1 | 1.5 | 12 | t
+ 14 | 0 | 2.6 | 11 | t
+ 15 | 0 | 2.7 | 10 | t
+ 16 | 0 | 2.8 | 9 | t
+ 17 | 1 | 2.32 | 8 | t
+ 18 | 0 | 4.43 | 7 | t
+ 19 | 0 | 2.31 | 6 | t
+ 20 | 1 | 3.49 | 5 | t
+ 21 | 1 | 2.42 | 4 | t
+ 22 | 1 | 4.01 | 3 | t
+ 23 | 1 | 4.91 | 2 | t
+ 24 | 1 | 5 | 1 | t
+\.
+
Computing predictions using cox model. (This example uses the original data table to perform the prediction. Typically a different test dataset with the same features as the original training dataset would be used.)
+\x off
+-- Display the linear predictors for the original dataset
+DROP TABLE IF EXISTS sample_pred;
+SELECT madlib.coxph_predict('sample_cox',
+ 'sample_data',
+ 'id',
+ 'sample_pred');
+SELECT * FROM sample_pred;
+
+-- Display the relative risk for the original dataset
+DROP TABLE IF EXISTS sample_pred;
+SELECT madlib.coxph_predict('sample_cox',
+ 'sample_data',
+ 'id',
+ 'sample_pred',
+ 'risk');
+SELECT * FROM sample_pred;
+
Generally, proportional-hazard models start with a list of \( \boldsymbol n \) observations, each with \( \boldsymbol m \) covariates and a time of death. From this \( \boldsymbol n \times m \) matrix, we would like to derive the correlation between the covariates and the hazard function. This amounts to finding the parameters \( \boldsymbol \beta \) that best fit the model described below.
+
Let us define:
+
\( \boldsymbol t \in \mathbf R^{m} \) denote the vector of observed dependent variables, with \( n \) rows.
+
\( X \in \mathbf R^{m} \) denote the design matrix with \( m \) columns and \( n \) rows, containing all observed vectors of independent variables \( \boldsymbol x_i \) as rows.
+
\( R(t_i) \) denote the set of observations still alive at time \( t_i \)
+
+
Note that this model does not include a constant term, and the data cannot contain a column of 1s.
Using this score function and Hessian matrix, the partial likelihood can be maximized using the Newton-Raphson algorithm. Breslow's method is used to resolved tied times of deaths. The time of death for two records are considered "equal" if they differ by less than 1.0e-6
+
The inverse of the Hessian matrix, evaluated at the estimate of \( \boldsymbol \beta \), can be used as an approximate variance-covariance matrix for the estimate, and used to produce approximate standard errors for the regression coefficients.
The Wald \( p \)-value for coefficient \( i \) gives the probability (under the assumptions inherent in the Wald test) of seeing a value at least as extreme as the one observed, provided that the null hypothesis ( \( c_i = 0 \)) is true. Letting \( F \) denote the cumulative density function of a standard normal distribution, the Wald \( p \)-value for coefficient \( i \) is therefore
where \( Z \) is a standard normally distributed random variable.
+
The condition number is computed as \( \kappa(H) \) during the iteration immediately preceding convergence (i.e., \( A \) is computed using the coefficients of the previous iteration). A large condition number (say, more than 1000) indicates the presence of significant multicollinearity.
+
Literature
+
A somewhat random selection of nice write-ups, with valuable pointers into further literature:
If number of ties in the source table is very large, a memory allocation error may be raised. The limitation is about \((10^8 / m)\), where \(m\) is number of featrues. For instance, if there are 100 featrues, the number of ties should be fewer than 1 million.
A conditional random field (CRF) is a type of discriminative, undirected probabilistic graphical model. A linear-chain CRF is a special type of CRF that assumes the current state depends only on the previous state.
+
Feature extraction modules are provided for text-analysis tasks such as part-of-speech (POS) tagging and named-entity resolution (NER). Currently, six feature types are implemented:
+
+
Edge Feature: transition feature that encodes the transition feature weight from current label to next label.
+
Start Feature: fired when the current token is the first token in a sequence.
+
End Feature: fired when the current token is the last token in a sequence.
+
Word Feature: fired when the current token is observed in the trained dictionary.
+
Unknown Feature: fired when the current token is not observed in the trained dictionary for at least a certain number of times (default 1).
+
Regex Feature: fired when the current token can be matched by a regular expression.
+
+
A Viterbi implementation is also provided to get the best label sequence and the conditional probability \( \Pr( \text{best label sequence} \mid \text{sequence}) \).
+
Following steps are required for CRF Learning and Inference:
The function takes train_segment_tbl and regex_tbl as input and does feature generation generating three tables dictionary_tbl, train_feature_tbl and train_featureset_tbl, that are required as an input for CRF training.
TEXT. Name of the training segment table. The table is expected to have the following columns:
+
+
doc_id
INTEGER. Document id column
+
+
start_pos
INTEGER. Index of a particular term in the respective document
+
+
seg_text
TEXT. Term at the respective start_pos in the document
+
+
label
INTEGER. Label id for the term corresponding to the actual label from label_tbl
+
+
+
regex_tbl
+
TEXT. Name of the regular expression table. The table is expected to have the following columns:
+
+
pattern
TEXT. Regular Expression
+
+
name
TEXT. Regular Expression name
+
+
+
label_tbl
+
TEXT. Name of the table containing unique labels and their id's. The table is expected to have the following columns:
+
+
id
INTEGER. Unique label id. NOTE: Must range from 0 to total number of labels in the table - 1.
+
+
label
TEXT. Label name
+
+
+
dictionary_tbl
+
TEXT. Name of the dictionary table to be created containing unique terms along with their counts. The table will have the following columns:
+
+
token
TEXT. Contains all the unique terms found in train_segment_tbl
+
+
total
INTEGER. Respective counts for the terms
+
+
+
train_feature_tbl
+
+
+
TEXT. Name of the training feature table to be created. The table will have the following columns:
+
+
doc_id
INTEGER. Document id
+
+
f_size
INTEGER. Feature set size. This value will be same for all the tuples in the table
+
+
sparse_r
DOUBLE PRECISION[]. Array union of individual single state features (previous label, label, feature index, start position, training existance indicator), ordered by their start position.
+
+
dense_m
DOUBLE PRECISION[]. Array union of (previous label, label, feature index, start position, training existance indicator) of edge features ordered by start position.
+
+
sparse_m
DOUBLE PRECISION[]. Array union of (feature index, previous label, label) of edge features ordered by feature index.
+
+
+
+
train_featureset_tbl
+
TEXT. Name of the table to be created containing distinct featuresets generated from training feature extraction. The table will have the following columns:
ARRAY. Feature value. The value is of the form [L1, L2]
+ - If L1 = -1: represents single state feature with L2 being the current label id.
+ - If L1 != -1: represents transition feature with L1 be the previous label and L2 be the current label.
+
+
+
+
+
Linear Chain CRF Training Function
The function takes train_feature_tbl and train_featureset_tbl tables generated in the training feature generation steps as input along with other required parameters and produces two output tables crf_stats_tbl and crf_weights_tbl.
To find the best labels for a test set using the trained linear CRF model, repeat steps #1-2 and generate the test features, except instead of creating a new dictionary, use the dictionary generated from the training set.
\( F_m(\boldsymbol x, \boldsymbol y) = \sum_{i=1}^n f_m(y_i,y_{i-1},x_i) \) is a global feature function that is a sum along a sequence \( \boldsymbol x \) of length \( n \)
+
\( f_m(y_i,y_{i-1},x_i) \) is a local feature function dependent on the current token label \( y_i \), the previous token label \( y_{i-1} \), and the observation \( x_i \)
+
\( \lambda_m \) is the corresponding feature weight
+
\( Z_\lambda(\boldsymbol x) \) is an instance-specific normalizer
is found since the maximum likelihood is reached when the empirical average of the global feature vector equals its model expectation. The MADlib implementation uses limited-memory BFGS (L-BFGS), a limited-memory variation of the Broyden–Fletcher–Goldfarb–Shanno (BFGS) update, a quasi-Newton method for unconstrained optimization.
+
\(E_{p_\lambda(Y|x)}[F(x,Y)]\) is found by using a variant of the forward-backward algorithm:
For security reasons there are controls on custom functions in MADlib. You must be a superuser to create custom functions because they could theoretically allow execution of any untrusted Python code. Regular users with MADlib USAGE permission can use existing custom functions but cannot create new ones or update existing ones. See references [1] and [2] for information on privileges in Greenplum and PostgreSQL.
+
This function loads custom Python functions into a table for use by deep learning algorithms.
+
Custom functions can be useful if, for example, you need loss functions or metrics that are not built into the standard libraries. The functions to be loaded must be in the form of serialized Python objects created using Dill, which extends Python's pickle module to the majority of the built-in Python types [3].
+
Custom functions can also be used to return top k categorical accuracy in the case that you want a different k value than the default from tensorflow.keras. This module includes a helper function to create the custom function automatically for a specified k.
+
There is also a utility function to delete a function from the table.
+
Note
Do not specify a schema for the argument 'object_table' containing the Python objects, because the 'object_table' is automatically put in the MADlib schema. Also, any subsequent SQL queries on this table by regular users must specify '<madlib_schema>.object_table' in the usual way.
VARCHAR. Table to load serialized Python objects. If this table does not exist, it will be created. If this table already exists, a new row is inserted into the existing table. Do not specify schema as part of the object table name, since it will be put in the MADlib schema automatically.
+
+
+
object
+
BYTEA. PostgreSQL binary data type of the Python object. Object must be created with the Dill package for serializing Python objects.
+
Note
The Dill package must be installed on all segments of the database cluster [3].
+
+
name
+
TEXT. Name of the object. Must be unique identifier in the table, since this name is used when passing the object to Keras.
+
+
+
description (optional)
+
TEXT, default: NULL. Free text string to provide a description, if desired.
+
+
+
+
Output table
+ The output table contains the following columns:
+
+
id
SERIAL. Object ID.
+
+
name
TEXT PRIMARY KEY. Name of the object.
+
+
description
TEXT. Description of the object (free text).
+
+
object
BYTEA. Serialized Python object stored as a PostgreSQL binary data type.
+
+
Delete Function
+
Delete by id:
+delete_custom_function(
+ object_table,
+ id
+)
+
Or alternatively by name:
+delete_custom_function(
+ object_table,
+ name
+)
+
Arguments
+
object_table
+
VARCHAR. Table containing Python object to be deleted. Do not specify schema as part of the object table name.
+
id
+
INTEGER. The id of the object to be deleted.
+
name
+
TEXT. Name of the object to be deleted.
+
+
Top k Accuracy Function
+
Create and load a custom function for a specific k into the custom functions table. The Keras accuracy parameter 'top_k_categorical_accuracy' returns top 5 accuracy by default [4]. If you want a different top k value, use this helper function to create a custom Python function to compute the top k accuracy that you specify.
+
+load_top_k_accuracy_function(
+ object_table,
+ k
+ )
+
Arguments
+
object_table
+
VARCHAR. Table to load serialized Python objects. If this table does not exist, it will be created. If this table already exists, a new row is inserted into the existing table. Do not specify schema as part of the object table name, since it will be put in the MADlib schema automatically.
+
+
+
k
+
INTEGER. k value for the top k accuracy that you want.
+
+
+
+
Output table
+ The output table contains the following columns:
+
+
id
SERIAL. Object ID.
+
+
name
TEXT PRIMARY KEY. Name of the object. Generated with the following pattern: top_(k)_accuracy.
+
+
description
TEXT. Description of the object.
+
+
object
BYTEA. Serialized Python object stored as a PostgreSQL binary data type.
+
+
Examples
+
Load object using psycopg2. Psycopg is a PostgreSQL database adapter for the Python programming language. Note need to use the psycopg2.Binary() method to pass as bytes.
+# import database connector psycopg2 and create connection cursor
+import psycopg2 as p2
+conn = p2.connect('postgresql://gpadmin:8000/madlib')
+cur = conn.cursor()
+# import Dill and define functions
+import dill
+# custom loss
+def squared_error(y_true, y_pred):
+ import keras.backend as K
+ return K.square(y_pred - y_true)
+pb_squared_error=dill.dumps(squared_error)
+# custom metric
+def rmse(y_true, y_pred):
+ import keras.backend as K
+ return K.sqrt(K.mean(K.square(y_pred - y_true), axis=-1))
+pb_rmse=dill.dumps(rmse)
+# call load function
+cur.execute("DROP TABLE IF EXISTS madlib.custom_function_table")
+cur.execute("SELECT madlib.load_custom_function('custom_function_table', %s,'squared_error', 'squared error')", [p2.Binary(pb_squared_error)])
+cur.execute("SELECT madlib.load_custom_function('custom_function_table', %s,'rmse', 'root mean square error')", [p2.Binary(pb_rmse)])
+conn.commit()
+
List table to see objects:
+SELECT id, name, description FROM madlib.custom_function_table ORDER BY id;
+
+ id | name | description
+----+---------------+------------------------
+ 1 | squared_error | squared error
+ 2 | rmse | root mean square error
+
+
Load object using a PL/Python function. First define the objects:
+CREATE OR REPLACE FUNCTION custom_function_squared_error()
+RETURNS BYTEA AS
+$$
+import dill
+def squared_error(y_true, y_pred):
+ import keras.backend as K
+ return K.square(y_pred - y_true)
+pb_squared_error=dill.dumps(squared_error)
+return pb_squared_error
+$$ language plpython3u;
+CREATE OR REPLACE FUNCTION custom_function_rmse()
+RETURNS BYTEA AS
+$$
+import dill
+def rmse(y_true, y_pred):
+ import keras.backend as K
+ return K.sqrt(K.mean(K.square(y_pred - y_true), axis=-1))
+pb_rmse=dill.dumps(rmse)
+return pb_rmse
+$$ language plpython3u;
+
This MADlib method is still in early stage development. Interface and implementation are subject to change.
+
Density-based spatial clustering of applications with noise (DBSCAN) is a data clustering algorithm designed to discover clusters of arbitrary shape [1,2]. It places minimum requirements on domain knowledge to determine input parameters and has good efficiency on large databases.
+
Given a set of points, DBSCAN groups together points that are closely packed with many nearby neighbors (core), and marks points that lie alone in low-density regions with few nearby neighbors (border). Other points in the dataset that are not core or border are considered as noise. This method tends to be good for data which contains clusters of similar density.
+
Currently only a brute force approach is implemented, suitable for small datasets. Other approaches for larger datasets will be implemented in a future release.
TEXT. Name of the table containing the input data points.
+
+
+
output_table
+
TEXT. Name of the table containing the clustering results.
+
+
+
id
+
TEXT. Name of a column or expression containing a unique integer id for each training point.
+
+
+
point
+
TEXT. Name of the column with point coordinates in array form, or an expression that evaluates to an array of point coordinates.
+
+
+
eps
+
FLOAT8. Maximum distance between two samples for one to be considered in the neighborhood of the other. (Note this is not a maximum bound on the distances of points within a cluster.) This is an important parameter to choose appropriately and should consider both the nature of the data set and the distance function.
+
+
+
min_samples (optional)
+
INTEGER, default: 5. Number of samples in a neighborhood for a point to be considered as a core point. This includes the point itself. This parameter controls how tolerant the algorithm is towards noise, so on noisy data sets it may be useful to increase the magnitude of this parameter.
+
Note
The parameters 'eps' and 'min_samples' together define the density of a cluster. A core point is where there exist 'min_samples' other points within a distance of 'eps', which are defined as neighbors of the core point. A higher value of 'min_samples' or a lower value of 'eps' indicate that higher density is needed to form a cluster.
+
+
metric (optional)
+
TEXT, default: 'squared_dist_norm2'. The name of the function used to calculate the distance between data points. The following distance functions can be used:
+
+dist_norm1: 1-norm/Manhattan (element-wise median). MADlib does not provide a median aggregate function for performance reasons.
+dist_angle: angle (element-wise mean of normalized points)
+
+
dist_tanimoto: tanimoto (element-wise mean of normalized points)
+
+
+
+
+
algorithm (optional)
+
TEXT, default: 'optimized'. The name of the algorithm used to compute clusters.
+
+
brute_force: Brute force can be slow and should not be used for large datasets. However, it does have less initial setup overhead than the parllel-optimized algorithm, which may make it faster for some cases involving small datasets. You can also use a short form "b" or "brute" etc. to select brute force.
+
+
+
+
optimized: This uses an rtree index to accelerate range_queries while performing the cluster detection. It is designed with gpdb in mind, in which case it intelligently partitions the space into a number of connected regions, runs DBSCAN in parallel on each region, and then merges the results together. By default, the maximum number of regions it will consider using is equal to the number of segments in the database, but if you suspect it may be spending too much time on the segmentation, this can be limited further by setting the max_segmentation_depth parameter to something lower. This should decrease the segmentation overhead, but will also decrease the amount of parallelism. If the dataset is relatively small, another option to consider is brute force, which has very little overhead but won't scale as well. For large enough datasets, and an appropriate choice of eps, this algorithm should be able to run in O((N/S) log N) time where N is the number of rows (points) in the input and S is the number of regions (often equal to the number of segments, but may be less). (Worst case for brute force is O(N^2).) eps should be chosen based on the density of the dataset, where DBSCAN works best for datasets where the clusters have a roughly uniform density. If eps is chosen too large, then the runtime will explode and nearly every point will be considered connected to every other point in one large cluster. Best practice is to start with a relatively small eps, so it can return the first result faster; if it looks like there are too many small clusters, increase eps and allow it to run for longer.
+
+
+
+
+
+
max_segmentation_depth (optional)
+
INTEGER, default: <number of="" segments>="">. The number of regions to segment the data. See optimized for usage.
+
+
+
+
Output
+ The output table for DBSCAN module has the following columns:
+
+
id
SMALLINT|INTEGER|BIGINT. A column or expression which specifies unique ids for each row in the training dataset. Must evaluate to an integer type.
+
+
cluster_id
INTEGER. The cluster id of each classified data point.
+
+
is_core_point
BOOLEAN. Indicates if the training data point is core If it is not a core point and listed in the output table, it is a border point. Noise points are not shown in this table.
+
+
point
TEXT. The coordinates of each point classified
+
+
Cluster Assignment
+
After clustering, the cluster assignment for each data point can be computed:
Run DBSCAN using the brute force method with a Euclidean distance function:
+DROP TABLE IF EXISTS dbscan_result, dbscan_result_summary;
+SELECT madlib.dbscan(
+ 'dbscan_train_data', -- source table
+ 'dbscan_result', -- output table
+ 'pid', -- point id column
+ 'points', -- data point
+ 1.75, -- epsilon
+ 4, -- min samples
+ 'dist_norm2', -- metric
+ 'brute_force'); -- algorithm
+SELECT * FROM dbscan_result ORDER BY pid;
+
+ pid | cluster_id | is_core_point | __points__
+-----+------------+---------------+------------
+ 1 | 0 | t | {1,1}
+ 2 | 0 | t | {2,1}
+ 3 | 0 | t | {1,2}
+ 4 | 0 | t | {2,2}
+ 6 | 1 | f | {3,9}
+ 7 | 1 | t | {3,10}
+ 8 | 1 | t | {4,10}
+ 9 | 1 | t | {4,11}
+ 10 | 1 | f | {5,10}
+ 13 | 2 | t | {10,6}
+ 14 | 2 | t | {9,5}
+ 15 | 2 | t | {10,5}
+ 16 | 2 | t | {11,5}
+ 17 | 2 | t | {9,4}
+ 18 | 2 | t | {10,4}
+ 19 | 2 | t | {11,4}
+ 20 | 2 | t | {10,3}
+(17 rows)
+
There are three clusters created. All points are core points except for 6 and 10 which are border points. The noise points do not show up in the output table. If you want to see the noise points you can use a query like:
+SELECT l.* FROM dbscan_train_data l WHERE NOT EXISTS
+ (SELECT NULL FROM dbscan_result r WHERE r.pid = l.pid)
+ ORDER BY l.pid;
+
Find the cluster assignment for the test data points:
+SELECT madlib.dbscan_predict(
+ 'dbscan_result', -- from DBSCAN run
+ 'dbscan_test_data', -- test dataset
+ 'pid', -- point id column
+ 'points', -- data point
+ 'dbscan_predict_out' -- output table
+ );
+SELECT * FROM dbscan_predict_out ORDER BY pid;
+
[1] Martin Ester, Hans-Peter Kriegel, Jörg Sander, Xiaowei Xu, "A Density-Based Algorithm for Discovering Clusters
+in Large Spatial Databases with Noise", KDD-96 Proceedings, https://www.aaai.org/Papers/KDD/1996/KDD96-037.pdf
+
[2] Erich Schubert, Jörg Sander, Martin Ester, Hans-Peter Kriegel, Xiaowei Xu, "DBSCAN Revisited, Revisited: Why and How You Should (Still) Use DBSCAN", ACM Transactions on Database Systems, July 2017, Article No. 19, https://dl.acm.org/doi/10.1145/3068335
+
+
+
+
+
+
Generated on Mon Aug 28 2023 08:03:58 for MADlib by
+
+ 1.8.13
A decision tree is a supervised learning method that can be used for classification and regression. It consists of a structure in which internal nodes represent tests on attributes, and the branches from nodes represent the result of those tests. Each leaf node is a class label and the paths from root to leaf nodes define the set of classification or regression rules.
+
Training Function
We implement the decision tree using the CART algorithm introduced by Breiman et al. [1]. The training function has the following syntax:
TEXT. Name of the table containing the training data.
+
+
+
output_table_name
+
TEXT. Name of the generated table containing the model. If a table with the same name already exists, an error will be returned. A summary table named <output_table_name>_summary is also created. A cross-validation table <output_table_name>_cv may also be created. These are described later on this page.
+
+
+
id_col_name
+
TEXT. Name of the column containing id information in the training data. This is a mandatory argument and is used for prediction and cross-validation. The values are expected to be unique for each row.
+
+
+
dependent_variable
+
TEXT. Name of the column that contains the output (response) for training. Boolean, integer and text types are considered to be classification outputs, while double precision values are considered to be regression outputs. The response variable for a classification tree can be multinomial, but the time and space complexity of the training function increases linearly as the number of response classes increases.
+
+
+
list_of_features
+
TEXT. Comma-separated string of column names or expressions to use as predictors. Can also be a '*' implying all columns are to be used as predictors (except for the ones included in the next argument that lists exclusions). The types of the features can be mixed: boolean, integer, and text columns are considered categorical and double precision columns are considered continuous. Categorical variables are not encoded and used as is in the training.
+
Array columns can also be included in the list, where the array is expanded to treat each element of the array as a feature.
+
Note that not every combination of the levels of a categorical variable is checked when evaluating a split. The levels of the non-integer categorical variable are ordered by the entropy of the variable in predicting the response. The split at each node is evaluated between these ordered levels. Integer categorical variables, however, are simply ordered by their value.
+
+
+
list_of_features_to_exclude (optional)
+
TEXT. Comma-separated string of column names to exclude from the predictors list. If the dependent_variable is an expression (including cast of a column name), then this list should include the columns present in the dependent_variable expression, otherwise those columns will be included in the features. The names in this parameter should be identical to the names used in the table and quoted appropriately.
+
+
+
split_criterion (optional)
+
TEXT, default = 'gini' for classification, 'mse' for regression. Impurity function to compute the feature to use to split a node. Supported criteria are 'gini', 'entropy', 'misclassification' for classification trees. For regression trees, split_criterion of 'mse' (mean-squared error) is always used, irrespective of the input for this argument. Refer to reference [1] for more information on impurity measures.
+
+
+
grouping_cols (optional)
+
TEXT, default: NULL. Comma-separated list of column names to group the data by. This will produce multiple decision trees, one for each group.
+
+
+
weights (optional)
+
TEXT. Column name containing numerical weights for each observation. Can be any value greater than 0 (does not need to be an integer). This can be used to handle the case of unbalanced data sets. The weights are used to compute a weighted average in the output leaf node. For classification, the contribution of a row towards the vote of its corresponding level is multiplied by the weight (weighted mode). For regression, the output value of the row is multiplied by the weight (weighted mean).
+
+
+
max_depth (optional)
+
INTEGER, default: 7. Maximum depth of any node of the final tree, with the root node counted as depth 0. A deeper tree can lead to better prediction but will also result in longer processing time and higher memory usage. Current allowed maximum is 100.
+
+
+
min_split (optional)
+
INTEGER, default: 20. Minimum number of observations that must exist in a node for a split to be attempted. The best value for this parameter depends on the number of tuples in the dataset.
+
+
+
min_bucket (optional)
+
INTEGER, default: min_split/3. Minimum number of observations in any terminal node. If only one of min_bucket or min_split is specified, min_split is set to min_bucket*3 or min_bucket to min_split/3, as appropriate.
+
+
+
num_splits (optional)
+
INTEGER, default: 20. Continuous-valued features are binned into discrete quantiles to compute split boundaries. Uniform binning is used. This global parameter is used to compute the resolution of splits for continuous features. Higher number of bins will lead to better prediction, but will also result in longer processing time and higher memory usage.
+
+
+
pruning_params (optional)
+
TEXT. Comma-separated string of key-value pairs giving the parameters for pruning the tree.
+
+
cp
Default: 0. Complexity parameter. A split on a node is attempted only if it decreases the overall lack of fit by a factor of 'cp', otherwise the split is pruned away. This value is used to create an initial tree before running cross-validation (see below).
+
+
+
+
n_folds
Default: 0 (i.e. no cross-validation). Number of cross-validation folds to use to compute the best value of cp. To perform cross-validation, a positive value of n_folds (2 or more) should be specified. An additional output table <model_table>_cv is created containing the values of evaluated cp and the cross-validation error statistics. The tree returned in the output table corresponds to the cp with the lowest cross-validation error (we pick the maximum cp if multiple values have same error).
+
The list of cp values is automatically computed by parsing through the tree initially trained on the complete dataset. The tree output is a subset of this initial tree corresponding to the best computed cp.
+
+
+
+
+
+
null_handling_params (optional)
+
TEXT. Comma-separated string of key-value pairs controlling the behavior of various features handling missing values. One of the following can be used if desired (not both):
+
+
max_surrogates
Default: 0. Number of surrogates to store for each node. One approach to handling NULLs is to use surrogate splits for each node. A surrogate variable enables you to make better use of the data by using another predictor variable that is associated (correlated) with the primary split variable. The surrogate variable comes into use when the primary predictior value is NULL. Surrogate rules implemented here are based on reference [1].
+
+
null_as_category
Default: FALSE. Whether to treat NULL as a valid level for categorical features. FALSE means that NULL is not a valid level, which is probably the most common sitation.
+
If set to TRUE, NULL values are considered a categorical value and placed at the end of the ordering of categorical levels. Placing at the end ensures that NULL is never used as a value to split a node on. One reason to make NULL a category is that it allows you to predict on categorical levels that were not in the training data by lumping them into an "other bucket."
+
This parameter is ignored for continuous-valued features.
+
+
+
+
+
verbosity (optional)
+
BOOLEAN, default: FALSE. Provides verbose output of the training result.
+
+
+
Output
+
+
The model table produced by the training function contains the following columns:
+
+
+
<...>
Grouping columns, if provided as input, in the same types as the training table. This could be multiple columns depending on the grouping_cols input.
+
+
tree
BYTEA8. Trained decision tree model stored in binary format (not human readable).
+
+
cat_levels_in_text
TEXT[]. Ordered levels (values) of categorical variables corresponding to the categorical features in the 'list_of_features' argument above. Used to help interpret the trained decision tree. For example, if the categorical features specified are weather_outlook and windy in that order, then 'cat_levels_in_text' might be [overcast, rain, sunny, False, True].
+
+
cat_n_levels
INTEGER[]. Number of levels for each categorical variable. Used to help interpret the trained decision tree. In the example from above, 'cat_n_levels' would be [3, 2] since there are 3 levels for weather_outlook and 2 levels windy.
+
+
impurity_var_importance
DOUBLE PRECISION[]. Impurity importance of each variable. The order of the variables is the same as that of the 'independent_varnames' column in the summary table (see below).
+
The impurity importance of any feature is the decrease in impurity by a node containing the feature as a primary split, summed over the whole tree. If surrogates are used, then the importance value includes the impurity decrease scaled by the adjusted surrogate agreement. Importance values are displayed as raw values as per the 'split_criterion' parameter. To see importance values normalized to sum to 100 across all variables, use the importance display helper function described later on this page. Please refer to [1] for more information on variable importance.
+
+
+
+
tree_depth
INTEGER. The maximum depth the tree obtained after training (root has depth 0).
+
+
+
+
pruning_cp
DOUBLE PRECISION. The cost complexity parameter used for pruning the trained tree(s). This could be different than the cp value input using the pruning_params if cross-validation is used.
+
+
+
+
A summary table named <output_table_name>_summary is also created at the same time, which has the following columns:
+
+
method
TEXT. 'tree_train'
+
+
+
+
is_classification
BOOLEAN. TRUE if the decision trees are for classification, FALSE if for regression.
+
+
+
+
source_table
TEXT. The data source table name.
+
+
+
+
model_table
TEXT. The model table name.
+
+
+
+
id_col_name
TEXT. The ID column name.
+
+
+
+
list_of_features
TEXT. The list_of_features inputed to the 'tree_train' procedure.
+
+
+
+
list_of_features_to_exclude
TEXT. The list_of_features_to_exclude inputed to the 'tree_train' procedure.
+
+
+
+
dependent_varname
TEXT. The dependent variable.
+
+
+
+
independent_varnames
TEXT. The independent variables. These are the features used in the training of the decision tree.
+
+
+
+
cat_features
TEXT. The list of categorical feature names as a comma-separated string.
+
+
+
+
con_features
TEXT. The list of continuous feature names as a comma-separated string.
+
+
+
+
grouping_cols
TEXT. Names of grouping columns.
+
+
+
+
num_all_groups
INTEGER. Number of groups in decision tree training.
+
+
+
+
num_failed_groups
INTEGER. Number of failed groups in decision tree training.
+
+
+
+
total_rows_processed
BIGINT. Total numbers of rows processed in all groups.
+
+
+
+
total_rows_skipped
BIGINT. Total numbers of rows skipped in all groups due to missing values or failures.
+
+
+
+
dependent_var_levels
TEXT. For classification, the distinct levels of the dependent variable.
+
+
+
+
dependent_var_type
TEXT. The type of dependent variable.
+
+
+
+
input_cp
DOUBLE PRECISION. The complexity parameter (cp) used for pruning the trained tree(s) before cross-validation is run. This is same as the cp value input using the pruning_params.
+
+
+
+
independent_var_types
TEXT. A comma separated string for the types of independent variables.
+
+
+
+
n_folds
BIGINT. Number of cross-validation folds used.
+
+
+
+
null_proxy
TEXT. Describes how NULLs are handled. If NULL is not treated as a separate categorical variable, this will be NULL. If NULL is treated as a separate categorical value, this will be set to "__NULL__"
+
+
A cross-validation table called <output_table_name>_cv is created if 'n_folds' is set in the 'pruning_params'. It has the following columns:
+
+
cp
DOUBLE PRECISION. Complexity parameter.
+
+
+
+
cv_error_avg
DOUBLE PRECISION. Average error resulting from cp value.
+
+
+
+
cv_error_stdev
DOUBLE PRECISION. Standard deviation resulting from cp value.
+
+
Note
+
Many of the parameters are designed to be similar to the popular R package 'rpart'. An important distinction between rpart and the MADlib function is that for both response and feature variables, MADlib considers integer values as categorical values, while rpart considers them as continuous. To use integers as continuous, cast them to double precision.
+
Integer values are ordered by value for computing the split boundaries. Cast to TEXT if the entropy-based ordering method is desired.
+
When cross-validation is not used (n_folds=0), each tree output is pruned by the input cost complexity (cp). With cross-validation, the input cp is the minimum value of all the explored values of 'cp'. During cross-validation, we train an initial tree using the provided cp and explore all possible sub-trees (up to a single-node tree) to compute the optimal sub-tree. The optimal sub-tree and the 'cp' corresponding to this optimal sub-tree is placed in the output_table, with the columns named as tree and pruning_cp respectively.
+
+
+
Run-time and Memory Usage
+
The number of features and the number of class values per categorical feature have a direct impact on run-time and memory. In addition, here is a summary of the main parameters in the training function that affect run-time and memory:
+
+
+
Parameter
Run-time
Memory
Notes
+
+
'max_depth'
High
High
Deeper trees can take longer to run and use more memory.
+
+
'min_split'
No or little effect, unless very small.
No or little effect, unless very small.
If too small, can impact run-time by building trees that are very thick.
+
+
'min_bucket'
No or little effect, unless very small.
No or little effect, unless very small.
If too small, can impact run-time by building trees that are very thick.
+
+
'num_splits'
High
High
Depends on number of continuous variables. Effectively adds more features as the binning becomes more granular.
+
+
If you experience long run-times or are hitting memory limits, consider reducing one or more of these parameters. One approach when building a decision tree model is to start with a low maximum depth value and use suggested defaults for other parameters. This will give you a sense of run-time and test set accuracy. Then you can change maximum depth in a systematic way as required to improve accuracy.
+
Prediction Function
The prediction function estimates the conditional mean given a new predictor. It has the following syntax:
TEXT. Name of the table containing the decision tree model. This should be the output table returned from tree_train.
+
+
+
new_data_table
+
TEXT. Name of the table containing prediction data. This table is expected to contain the same features that were used during training. The table should also contain id_col_name used for identifying each row.
+
+
+
output_table
+
TEXT. Name of the table to output prediction results. If this table already exists, an error is returned. The table contains the id_col_name column giving the 'id' for each prediction and the prediction columns for the dependent variable.
+
If type = 'response', then the table has a single additional column with the prediction value of the response. The type of this column depends on the type of the response variable used during training.
+
If type = 'prob', then the table has multiple additional columns, one for each possible value of the response variable. The columns are labeled as 'estimated_prob_dep_value', where dep_value represents each value of the response variable.
+
+
+
type (optional)
+
TEXT, optional, default: 'response'. For regression trees, the output is always the predicted value of the dependent variable. For classification trees, the type variable can be 'response', giving the classification prediction as output, or 'prob', giving the class probabilities as output. For each value of the dependent variable, a column with the probabilities is added to the output table.
+
+
Tree Display
The display function outputs a graph representation of the decision tree. The output can either be in the popular 'dot' format that can be visualized using various programs including those in the GraphViz package, or in a simple text format. The details of the text format are output with the tree.
An additional display function is provided to output the surrogate splits chosen for each internal node:
+tree_surr_display(tree_model)
+
This output contains the list of surrogate splits for each internal node. The nodes are sorted in ascending order by id. This is equivalent to viewing the tree in a breadth-first manner. For each surrogate, we output the surrogate split (variable and threshold) and also give the number of rows that were common between the primary split and the surrogate split. Finally, the number of rows present in the majority branch of the primary split is also shown. Only surrogates that perform better than this majority branch are included in the surrogate list. When the primary variable has a NULL value the surrogate variables are used in order to compute the split for that node. If all surrogates variables are NULL, then the majority branch is used to compute the split for a tuple.
+
Arguments
+
tree_model
+
TEXT. Name of the table containing the decision tree model.
+
dot_format (optional)
+
BOOLEAN, default = TRUE. Output can either be in a dot format or a text format. If TRUE, the result is in the dot format, else output is in text format.
+
verbosity (optional)
+
BOOLEAN, default = FALSE. If set to TRUE, the dot format output will contain additional information (impurity, sample size, number of weighted rows for each response variable, classification or prediction if the tree was pruned at this level)
+
+
The output is always returned as a 'TEXT'. For the dot format, the output can be redirected to a file on the client side and then rendered using visualization programs.
+
To export the dot format result to an external file, use the method below. Please note that you should use unaligned table output mode for psql with '-A' flag, or else you may get an error when you try to convert the dot file to another format for viewing (e.g., PDF). And inside the psql client, both '\t' and '\o' should be used:
+
+> # under bash
+> psql -A my_database
+# -- in psql now
+# \t
+# \o test.dot -- export to a file
+# select madlib.tree_display('tree_out');
+# \o
+# \t
+
After the dot file has been generated, use third-party plotting software to plot the trees in a nice format:
+> # under bash, convert the dot file into a PDF file
+> dot -Tpdf test.dot > test.pdf
+> xpdf test.pdf&
+
Please see the examples below for more details on the contents of the tree output formats.
+
An additional display function is provided to output the surrogate splits chosen for each internal node:
+tree_surr_display(tree_model)
+
Importance Display
+
This is a helper function that creates a table to more easily view impurity variable importance values for a given model table. This function rescales the importance values to represent them as percentages i.e. importance values are scaled to sum to 100.
Predict output categories. For the purpose of this example, we use the same data that was used for training:
+\x off
+DROP TABLE IF EXISTS prediction_results;
+SELECT madlib.tree_predict('train_output', -- tree model
+ 'dt_golf', -- new data table
+ 'prediction_results', -- output table
+ 'response'); -- show response
+SELECT g.id, class, estimated_class FROM prediction_results p,
+dt_golf g WHERE p.id = g.id ORDER BY g.id;
+
+ id | class | estimated_class
+----+------------+-----------------
+ 1 | Don't Play | Don't Play
+ 2 | Don't Play | Don't Play
+ 3 | Play | Play
+ 4 | Play | Play
+ 5 | Play | Play
+ 6 | Don't Play | Don't Play
+ 7 | Play | Play
+ 8 | Don't Play | Don't Play
+ 9 | Play | Play
+ 10 | Play | Play
+ 11 | Play | Play
+ 12 | Play | Play
+ 13 | Play | Play
+ 14 | Don't Play | Don't Play
+(14 rows)
+
To display the probabilities associated with each value of the dependent variable, set the 'type' parameter to 'prob':
+DROP TABLE IF EXISTS prediction_results;
+SELECT madlib.tree_predict('train_output', -- tree model
+ 'dt_golf', -- new data table
+ 'prediction_results', -- output table
+ 'prob'); -- show probability
+SELECT g.id, class, "estimated_prob_Don't Play", "estimated_prob_Play"
+FROM prediction_results p, dt_golf g WHERE p.id = g.id ORDER BY g.id;
+
+ id | class | estimated_prob_Don't Play | estimated_prob_Play
+----+------------+---------------------------+---------------------
+ 1 | Don't Play | 1 | 0
+ 2 | Don't Play | 1 | 0
+ 3 | Play | 0 | 1
+ 4 | Play | 0 | 1
+ 5 | Play | 0 | 1
+ 6 | Don't Play | 1 | 0
+ 7 | Play | 0 | 1
+ 8 | Don't Play | 1 | 0
+ 9 | Play | 0 | 1
+ 10 | Play | 0 | 1
+ 11 | Play | 0 | 1
+ 12 | Play | 0 | 1
+ 13 | Play | 0 | 1
+ 14 | Don't Play | 1 | 0
+(14 rows)
+
+ -------------------------------------
+ - Each node represented by 'id' inside ().
+ - Each internal nodes has the split condition at the end, while each
+ leaf node has a * at the end.
+ - For each internal node (i), its child nodes are indented by 1 level
+ with ids (2i+1) for True node and (2i+2) for False node.
+ - Number of (weighted) rows for each response variable inside [].'
+ The response label order is given as ['"\'Don\'t Play\'"', '"\'Play\'"'].
+ For each leaf, the prediction is given after the '-->'
+ -------------------------------------
+ (0)[5 9] "OUTLOOK" in {overcast}
+ (1)[0 4] * --> "Play"
+ (2)[5 5] temperature <= 75
+ (5)[3 5] temperature <= 65
+ (11)[1 0] * --> "Don't Play"
+ (12)[2 5] temperature <= 70
+ (25)[0 3] * --> "Play"
+ (26)[2 2] temperature <= 72
+ (53)[2 0] * --> "Don't Play"
+ (54)[0 2] * --> "Play"
+ (6)[2 0] * --> "Don't Play"
+ -------------------------------------
+
Here are some details on how to interpret the tree display above:
+
Node numbering starts at 0 for the root node and would be contiguous 1,2...n if the tree was completely full (no pruning). Since the tree has been pruned, the node numbering is not contiguous.
+
The order of values [x y] indicates the number of weighted rows that correspond to ["Don't play" "Play"] before the node test. For example, at (root) node 0, there are 5 rows for "Don't play" and 9 rows for "Play" in the raw data.
+
If we apply the test of "OUTLOOK" being overcast, then the True (yes) result is leaf node 1 which is "Play". There are 0 "Don't play" rows and 4 "Play" rows that correspond to this case (overcast). In other words, if it is overcast, you always play golf. If it is not overcast, you may or may not play golf, depending on the rest of the tree.
+
The remaining 5 "Don't play" rows and 5 "Play rows" are then tested at node 2 on temperature<=75. The False (no) result is leaf node 6 which is "Don't Play". The True (yes) result proceeds to leaf node 5 to test on temperature<=65. And so on down the tree.
+
Creating a dot format visualization of the tree, as described below, can help with following the decision flows.
One important difference to note about the dot format above is how categorical variable tests are displayed:
+
In the text format of the tree, the node 0 test is "OUTLOOK" in {overcast}, but in the dot format of the tree, the same node 0 test reads "\"OUTLOOK" <= overcast". This is because in dot format for categorical variables, the '<=' symbol represents the location in the array 'cat_levels_in_text' from the output table for the "OUTLOOK" levels. The array is ['overcast', 'rain', 'sunny', 'False', 'True'] with the first 3 entries corresponding to "OUTLOOK" and the last 2 entries corresponding to 'windy'. So the test "\"OUTLOOK" <= overcast" means all "OUTLOOK" levels to the left of, and including, 'overcast'. In this case there are no levels to the left of 'overcast' in the array so it is simply a test on whether it is overcast or not.
+
If there was a test "\"OUTLOOK" <= rain", this would include both 'overcast' and 'rain', since 'overcast' is to the left of 'rain' in the array.
+
If there was a test "windy <= True", this would include both 'False' and 'True', since 'False' is to the left of 'True' in the array.
+
+
+
Now create a dot format display of the tree with additional information:
The additional information in each node is: impurity, sample size, number of weighted rows for each response variable, and classification if the tree was pruned at this level. If your tree is not too big, you may wish to convert the dot format to PDF or another format for better visualization of the tree structure.
+
Arrays of features. Categorical and continuous features can be array columns, in which case the array is expanded to treat each element of the array as a feature:
+DROP TABLE IF EXISTS train_output, train_output_summary;
+SELECT madlib.tree_train('dt_golf', -- source table
+ 'train_output', -- output model table
+ 'id', -- id column
+ 'class', -- response
+ '"Temp_Humidity", clouds_airquality', -- features
+ NULL::text, -- exclude columns
+ 'gini', -- split criterion
+ NULL::text, -- no grouping
+ NULL::text, -- no weights, all observations treated equally
+ 5, -- max depth
+ 3, -- min split
+ 1, -- min bucket
+ 10 -- number of bins per continuous variable
+ );
+
View the output table (excluding the tree which is in binary format):
+\x on
+SELECT pruning_cp, cat_levels_in_text, cat_n_levels, impurity_var_importance, tree_depth FROM train_output;
+
The first 4 levels correspond to cloud ceiling and the next 3 levels correspond to air quality.
+
Weighting observations. Use the 'weights' parameter to adjust a row's vote to balance the dataset. In our example, the weights are somewhat random but show that a different decision tree is create compared to the case where no weights are used:
+DROP TABLE IF EXISTS train_output, train_output_summary;
+SELECT madlib.tree_train('dt_golf', -- source table
+ 'train_output', -- output model table
+ 'id', -- id column
+ 'class', -- response
+ '"OUTLOOK", temperature, windy', -- features
+ NULL::text, -- exclude columns
+ 'gini', -- split criterion
+ NULL::text, -- no grouping
+ 'observation_weight', -- weight observations
+ 5, -- max depth
+ 3, -- min split
+ 1, -- min bucket
+ 10 -- number of bins per continuous variable
+ );
+SELECT madlib.tree_display('train_output', FALSE);
+
+ -------------------------------------
+ - Each node represented by 'id' inside ().
+ - Each internal nodes has the split condition at the end, while each
+ leaf node has a * at the end.
+ - For each internal node (i), its child nodes are indented by 1 level
+ with ids (2i+1) for True node and (2i+2) for False node.
+ - Number of (weighted) rows for each response variable inside [].'
+ The response label order is given as ['"Don\'t Play"', '"Play"'].
+ For each leaf, the prediction is given after the '-->'
+ -------------------------------------
+ (0)[17 19] temperature <= 75
+ (1)[ 7 16] temperature <= 72
+ (3)[ 7 10] temperature <= 70
+ (7)[ 1 8.5] * --> "Play"
+ (8)[ 6 1.5] "OUTLOOK" in {overcast}
+ (17)[ 0 1.5] * --> "Play"
+ (18)[6 0] * --> "Don't Play"
+ (4)[0 6] * --> "Play"
+ (2)[10 3] "OUTLOOK" in {overcast}
+ (5)[0 3] * --> "Play"
+ (6)[10 0] * --> "Don't Play"
+
+
+
Decision Tree Regression Examples
+
+
Load input data related to fuel consumption and 10 aspects of automobile design and performance for 32 automobiles (1973–74 models). Data was extracted from the 1974 Motor Trend US magazine.
View the normalized impurity importance table using the helper function:
+\x off
+DROP TABLE IF EXISTS imp_output;
+SELECT madlib.get_var_importance('train_output','imp_output');
+SELECT * FROM imp_output ORDER BY impurity_var_importance DESC;
+
Predict regression output for the same data and compare with original:
+\x off
+DROP TABLE IF EXISTS prediction_results;
+SELECT madlib.tree_predict('train_output',
+ 'mt_cars',
+ 'prediction_results',
+ 'response');
+SELECT s.id, mpg, estimated_mpg, mpg-estimated_mpg as delta
+FROM prediction_results p,
+mt_cars s WHERE s.id = p.id ORDER BY id;
+
+ -------------------------------------
+ - Each node represented by 'id' inside ().
+ - Each internal nodes has the split condition at the end, while each
+ leaf node has a * at the end.
+ - For each internal node (i), its child nodes are indented by 1 level
+ with ids (2i+1) for True node and (2i+2) for False node.
+ - Number of rows and average response value inside []. For a leaf node, this is the prediction.
+ -------------------------------------
+ (0)[32, 20.0906] cyl in {4}
+ (1)[11, 26.6636] wt <= 2.2
+ (3)[6, 30.0667] *
+ (4)[5, 22.58] *
+ (2)[21, 16.6476] disp <= 258
+ (5)[7, 19.7429] *
+ (6)[14, 15.1] qsec <= 17.42
+ (13)[10, 15.81] qsec <= 16.9
+ (27)[5, 14.78] *
+ (28)[5, 16.84] *
+ (14)[4, 13.325] *
+ -------------------------------------
+(1 row)
+
+
Display the surrogate variables that are used to compute the split for each node when the primary variable is NULL:
The 'cyl' parameter in the data set has two tuples with NULL values (id = 9 and id = 18). In the prediction based on this tree, the surrogate splits for the cyl in {4} split in node 0 are used to predict those two tuples. The splits are used in descending order until a surrogate variable is found that is not NULL. In this case, the two tuples have non-NULL values for disp, hence the disp <= 146.7 split is used to make the prediction. If all the surrogate variables are NULL then the majority branch would be followed.
+
+
Now let's use cross validation to select the best value of the complexity parameter cp:
+DROP TABLE IF EXISTS train_output, train_output_summary, train_output_cv;
+SELECT madlib.tree_train('mt_cars', -- source table
+ 'train_output', -- output model table
+ 'id', -- id column
+ 'mpg', -- dependent variable
+ '*', -- features
+ 'id, hp, drat, am, gear, carb', -- exclude columns
+ 'mse', -- split criterion
+ NULL::text, -- no grouping
+ NULL::text, -- no weights, all observations treated equally
+ 10, -- max depth
+ 8, -- min split
+ 3, -- number of bins per continuous variable
+ 10, -- number of splits
+ 'n_folds=3' -- pruning parameters for cross validation
+ );
+
View the output table (excluding the tree which is in binary format). The input cp value was 0 (default) and the best 'pruning_cp' value turns out to be 0 as well in this small example:
+\x on
+SELECT pruning_cp, cat_levels_in_text, cat_n_levels, impurity_var_importance, tree_depth FROM train_output;
+
Predict for data not previously seen by assuming NULL value as the default:
+\x off
+DROP TABLE IF EXISTS table_test;
+CREATE TABLE table_test (
+ id integer,
+ country text,
+ city text,
+ weather text,
+ expected_response text
+);
+INSERT INTO table_test VALUES
+(1,'IN','MUM','cloudy','a'),
+(2,'US','HOU','humid','b'),
+(3,'US','NY','sunny','c'),
+(4,'US','NY','rainy','d');
+
+DROP TABLE IF EXISTS prediction_results;
+SELECT madlib.tree_predict('train_output',
+ 'table_test',
+ 'prediction_results',
+ 'response');
+SELECT s.id, expected_response, estimated_response
+FROM prediction_results p, table_test s
+WHERE s.id = p.id ORDER BY id;
+
+ id | expected_response | estimated_response
+----+-------------------+--------------------
+ 1 | a | a
+ 2 | b | b
+ 3 | c | c
+ 4 | d | d
+(4 rows)
+
There is only training data for country 'US' so the response for country 'IN' is 'a', corresponding to a NULL (not 'US') country level. Likewise, any city in the 'US' that is not 'NY' will predict response 'b', corresponding to a NULL (not 'NY') city level.
+ -------------------------------------
+ - Each node represented by 'id' inside ().
+ - Each internal nodes has the split condition at the end, while each
+ leaf node has a * at the end.
+ - For each internal node (i), its child nodes are indented by 1 level
+ with ids (2i+1) for True node and (2i+2) for False node.
+ - Number of rows and average response value inside []. For a leaf node, this is the prediction.
+ -------------------------------------
+ (0)[1 1 1 1] city in {NY}
+ (1)[0 0 1 1] weather in {rainy}
+ (3)[0 0 0 1] * --> "d"
+ (4)[0 0 1 0] * --> "c"
+ (2)[1 1 0 0] country in {US}
+ (5)[0 1 0 0] * --> "b"
+ (6)[1 0 0 0] * --> "a"
+ -------------------------------------
+(1 row)
+
+
+
Literature
[1] Breiman, Leo; Friedman, J. H.; Olshen, R. A.; Stone, C. J. (1984). Classification and Regression Trees. Monterey, CA: Wadsworth & Brooks/Cole Advanced Books & Software.
The linear systems module implements solution methods for systems of consistent linear equations. Systems of linear equations take the form:
+\[ Ax = b \]
+
+
where \(x \in \mathbb{R}^{n}\), \(A \in \mathbb{R}^{m \times n} \) and \(b \in \mathbb{R}^{m}\). We assume that there are no rows of \(A\) where all elements are zero. The algorithms implemented in this module can handle large dense linear systems. Currently, the algorithms implemented in this module solve the linear system by a direct decomposition. Hence, these methods are known as direct method.
Each row represents a single equation. The right_hand_side column refers to the right hand side of the equations while the left_hand_side column refers to the multipliers on the variables on the left hand side of the same equations.
+
+
+
tbl_result
+
TEXT. The name of the table where the output is saved. The output is stored in the table named by the tbl_result argument. It contains the following columns:
+
+
solution
FLOAT8[]. The solution variables in the same order as that provided as input in the 'left_hand_side' column name of the source_table
+
+
residual_norm
FLOAT8. The scaled residual norm, defined as \( \frac{|Ax - b|}{|b|} \). This value is an indication of the accuracy of the solution.
+
+
iters
INTEGER. Number of iterations required by the algorithm (only applicable for iterative algorithms). The output is NULL for 'direct' methods.
+
+
+
+
row_id
+
TEXT. The name of the column storing the 'row id' of the equations.
+
For a system with N equations, the row_id's must be a continuous range of integers from \( 0 \ldots n-1 \).
+
+
+
LHS
+
TEXT. The name of the column storing the 'left hand side' of the equations, stored as an array.
+
+
+
RHS
+
TEXT. The name of the column storing the 'right hand side' of the equations.
+
+
+
grouping_cols (optional)
+
TEXT, default: NULL. Group by column names. Not currently implemented. Any non-NULL value is ignored.
+
optimizer (optional)
+
TEXT, default: 'direct'. The type of optimizer.
+
+
+
optimizer_params (optional)
+
TEXT, default: NULL. Optimizer specific parameters.
+
+
+
Optimizer Parameters
+
For each optimizer, there are specific parameters that can be tuned for better performance.
+
+
algorithm (default: householderqr)
+
There are several algorithms that can be classified as 'direct' methods of solving linear systems. MADlib dense linear system solvers provide various algorithmic options for users.
+
The following table provides a guideline on the choice of algorithm based on conditions on the A matrix, speed of the algorithms and numerical stability.
There are three main steps in order to run deep learning workloads with MADlib:
+
Preparation, which includes data preprocessing and model definition. Data preprocessing is required to format training data for use by frameworks like Keras and TensorFlow that support mini-batching as an optimization option. Model definition involves describing model architectures (and optionally custom functions) and loading them into tables.
+
Model training, either one model at a time or multiple models in parallel. In the latter case, you will need to define the configurations for the multiple models that you want to train - this can be done manually or in an automated way using autoML methods. The trained models can then be used for evaluation and inference.
Finds the solution to the function \( \boldsymbol Ax = \boldsymbol b \), where \(A\) is a symmetric, positive-definite matrix and \(x\) and \( \boldsymbol b \) are vectors.
Constructs a classification model from a dataset where each attribute independently contributes to the probability that a data point belongs to a category.
This module implements elastic net regularization [1] for linear and logistic regression. Regularization is a technique often used to prevent overfitting.
TEXT. The name of the table containing the training data.
+
+
+
tbl_result
+
TEXT. Name of the output table containing output model. The output table produced by the elastic_net_train() function has the following columns:
+
+
regress_family
The regression type: 'gaussian' or 'binomial'.
+
+
features
Array of features (independent variables) passed to the algorithm.
+
+
features_selected
Array of features selected by the algorithm.
+
+
coef_nonzero
Coefficients of the selected features.
+
+
coef_all
Coefficients of all features, both selected and unselected.
+
+
intercept
Intercept for the model.
+
+
log_likelihood
Log of the likelihood value produced by the algorithm.
+
+
standardize
BOOLEAN. If data has been normalized, will be set to TRUE.
+
+
iteration_run
The number of iterations executed.
+
+
+
+
col_dep_var
+
TEXT. An expression for the dependent variable.
+
Note
Both col_dep_var and col_ind_var can be valid PostgreSQL expressions. For example, col_dep_var = 'log(y+1)', and col_ind_var = 'array[exp(x[1]), x[2], 1/(1+x[3])]'. In the binomial case, you can use a Boolean expression, for example, col_dep_var = 'y < 0'.
+
+
col_ind_var
+
TEXT. An expression for the independent variables. Use '*' to specify all columns of tbl_source except those listed in the excluded string described below. If col_dep_var is a column name, it is automatically excluded from the independent variables. However, if col_dep_var is a valid PostgreSQL expression, any column names used within the expression are only excluded if they are explicitly listed in the excluded argument. Therefore, it is a good idea to add all column names involved in the dependent variable expression to the excluded string.
+
+
+
regress_family
+
TEXT. For regression type, specify either 'gaussian' ('linear') or 'binomial' ('logistic').
+
+
+
alpha
+
FLOAT8. Elastic net control parameter with a value in the range [0, 1]. A value of 1 means L1 regularization, and a value of 0 means L2 regularization.
+
+
+
lambda_value
+
FLOAT8. Regularization parameter (must be positive).
+
+
+
standardize (optional)
+
BOOLEAN, default: TRUE. Whether to normalize the data or not. Setting to TRUE usually yields better results and faster convergence.
+
+
+
grouping_col (optional)
+
TEXT, default: NULL. A single column or a list of comma-separated columns that divides the input data into discrete groups, resulting in one regression per group. When this value is NULL, no grouping is used and a single model is generated for all data.
+
Note
Expressions are not currently supported for 'grouping_col'.
+
+
optimizer (optional)
+
TEXT, default: 'fista'. Name of optimizer, either 'fista' or 'igd'. FISTA [2] is an algorithm with a fast global rate of convergence for solving linear inverse problems. Incremental gradient descent (IGD) is a stochastic approach to minimizing an objective function [4].
+
+
+
optimizer_params (optional)
+
TEXT, default: NULL. Optimizer parameters, delimited with commas. These parameters differ depending on the value of optimizer parameter. See the descriptions below for details.
+
+
+
excluded (optional)
+
TEXT, default: NULL. If the col_ind_var input is '*' then excluded can be provided as a comma-delimited list of column names that are to be excluded from the features. For example, 'col1, col2'. If the col_ind_var is an array, excluded must be a list of the integer array positions to exclude, for example '1,2'. If this argument is NULL or an empty string, no columns are excluded.
+
+
+
max_iter (optional)
+
INTEGER, default: 1000. The maximum number of iterations allowed.
+
+
+
tolerance (optional)
+
FLOAT8, default: 1e-6. This is the criterion to stop iterating. Both the 'fista' and 'igd' optimizers compute the difference between the log likelihood of two consecutive iterations, and when the difference is smaller than tolerance or the iteration number is larger than max_iter, the computation stops.
+
+
Other Parameters
+
For optimizer_params, there are several parameters that can be supplied in a string containing a comma-delimited list of name-value pairs . All of these named parameters are optional and use the format "<param_name> = <value>".
+
The parameters described below are organized by category: warmup, cross validation and optimization.
Default: FALSE. If warmup is TRUE, a series of strictly descending lambda values are used, which end with the lambda value that the user wants to calculate. A larger lambda gives a sparser solution, and the sparse solution is then used as the initial guess for the next lambda's solution, which can speed up the computation for the next lambda. For larger data sets, this can sometimes accelerate the whole computation and may in fact be faster than computation with only a single lambda value.
+
+
+
warmup_lambdas
+
Default: NULL. Set of lambda values to use when warmup is TRUE. The default is NULL, which means that lambda values will be automatically generated.
+
+
+
warmup_lambda_no
+
Default: 15. Number of lambda values used in warm-up. If warmup_lambdas is not NULL, this value is overridden by the number of provided lambda values.
+
+
+
warmup_tolerance
+
The value of tolerance used during warmup. The default value is the same as the tolerance argument described above.
+
+
Cross validation parameters
Note
Please note that for performance reasons, warmup is disabled whenever cross validation is used. Also, cross validation is not supported if grouping is used.
Hyperparameter optimization can be carried out using the built-in cross validation mechanism, which is activated by assigning a value greater than 1 to the parameter n_folds.
+
The cross validation scores are the mean and standard deviation of the accuracy when predicted on the validation fold, averaged over all folds and all rows. For classification, the accuracy metric used is the ratio of correct classifications. For regression, the accuracy metric used is the negative of mean squared error (negative to make it a concave problem, thus selecting max means the highest accuracy). Cross validation scores are written out to a separate table with the user specified name given in the 'validation_result' parameter.
+
The values of a parameter to cross validate should be provided in a list. For example, to regularize with the L1 norm and use a lambda value from the set {0.3, 0.4, 0.5}, include 'lambda_value={0.3, 0.4, 0.5}'. Note that the use of '{}' and '[]' are both valid here.
+
+
n_folds
+
Default: 0. Number of folds (k). Must be at least 2 to activate cross validation. If a value of k > 2 is specified, each fold is then used as a validation set once, while the other k - 1 folds form the training set.
+
+
+
validation_result
+
Default: NULL. Name of the table to store the cross validation results, including the values of parameters and their averaged error values. The table is only created if the name is not NULL.
+
+
+
lambda_value
+
Default: NULL. Set of regularization values to be used for cross validation. The default is NULL, which means that lambda values will be automatically generated.
+
+
+
n_lambdas
+
Default: 15. Number of lambdas to cross validate over. If a list of lambda values is not provided in the lambda_value set above, this parameter can be used to autogenerate the set of lambdas. If the lambda_value set is not NULL, this value is overridden by the number of provided lambda values.
+
Note
If you want to cross validate over alpha only and not lambda, then set lambda_value to NULL and n_lambdas to 0. In this case, cross validation will be done on the set of alpha values specified in the next parameter. The lambda value used will be the one specified in the main function call at the top of this page.
+
+
alpha
+
Elastic net control parameter. This is a list of values to apply cross validation on. (Note that alpha values are not autogenerated.) If not specified, the alpha value used will be the one specified in the main function call at the top of this page.
Default: 4.0. Initial backtracking step size. At each iteration, the algorithm first tries stepsize = max_stepsize, and if it does not work out, it then tries a smaller step size, stepsize = stepsize/eta, where eta must be larger than 1. At first glance, this seems to perform repeated iterations for even one step, but using a larger step size actually greatly increases the computation speed and minimizes the total number of iterations. A careful choice of max_stepsize can decrease the computation time by more than 10 times.
+
+
+
eta
+
Default: 2.0 If stepsize does not work, stepsize/eta is tried. Must be greater than 1.
+
+
+
use_active_set
+
Default: FALSE. If use_active_set is TRUE, an active-set method is used to speed up the computation. Considerable speedup is obtained by organizing the iterations around the active set of features—those with nonzero coefficients. After a complete cycle through all the variables, we iterate only on the active set until convergence. If another complete cycle does not change the active set, we are done. Otherwise, the process is repeated.
+
+
+
activeset_tolerance
+
The value of tolerance used during active set calculation. The default value is the same as the tolerance argument described above.
+
+
+
random_stepsize
+
Default: FALSE. Whether to add some randomness to the step size. Sometimes, this can speed up the calculation.
The actual stepsize used for current step is (previous stepsize) / exp(step_decay). The default value is 0, which means that a constant stepsize is used in IGD.
+
+
+
threshold
+
Default: 1e-10. When a coefficient is really small, set this coefficient to be 0.
+
Due to the stochastic nature of SGD, we can only obtain very small values for the fitting coefficients. Therefore, threshold is needed at the end of the computation to screen out tiny values and hard-set them to zeros. This is accomplished as follows: (1) multiply each coefficient with the standard deviation of the corresponding feature; (2) compute the average of absolute values of re-scaled coefficients; (3) divide each rescaled coefficient with the average, and if the resulting absolute value is smaller than threshold, set the original coefficient to zero.
+
+
+
parallel
+
Whether to run the computation on multiple segments. The default is TRUE.
+
SGD is a sequential algorithm in nature. When running in a distributed manner, each segment of the data runs its own SGD model and then the models are averaged to get a model for each iteration. This averaging might slow down the convergence speed, but it affords the ability to process large datasets on a cluster of machines. This algorithm, therefore, provides the parallel option to allow you to choose whether to do parallel computation.
+
+
+
Prediction Function
+
Per-Tuple Prediction
+
The prediction function returns a double value for the Gaussian family and a Boolean value for the Binomial family.
DOUBLE PRECISION[]. Fitting coefficients, usually coef_all or coef_nonzero.
+
intercept
+
DOUBLE PRECISION. Intercept for the model.
+
ind_var
+
DOUBLE PRECISION[]. Independent variables that correspond to coefficients. Use features column in tbl_result for coef_all, and features_selected for coef_nonzero. See the examples for this case below.
Note
Unexpected results or errors may be returned in the case that this argument ind_var is not specified properly.
+
+
+
For the binomial family, there is a function (elastic_net_binomial_prob()) that outputs the probability of the instance being TRUE:
Alternatively, you can use another prediction function that stores the prediction result in a table (elastic_net_predict()). This is useful if you want to use elastic net together with the general cross validation function.
Use the prediction function to evaluate residuals:
+\x off
+SELECT madlib.elastic_net_predict(
+ 'houses_en1', -- Model table
+ 'houses', -- New source data table
+ 'id', -- Unique ID associated with each row
+ 'houses_en1_prediction' -- Table to store prediction result
+ );
+SELECT houses.id,
+ houses.price,
+ houses_en1_prediction.prediction,
+ houses.price - houses_en1_prediction.prediction AS residual
+FROM houses_en1_prediction, houses
+WHERE houses.id = houses_en1_prediction.id ORDER BY id;
+
We can still use the prediction function with coef_all to evaluate residuals:
+\x off
+SELECT id, price, predict, price - predict AS residual
+FROM (
+ SELECT
+ houses.*,
+ madlib.elastic_net_gaussian_predict(
+ m.coef_all, -- All coefficients
+ m.intercept, -- Intercept
+ ARRAY[tax,bath,size] -- All features
+ ) AS predict
+ FROM houses, houses_en2 m) s
+ORDER BY id;
+
+
We can speed up the prediction function with coef_nonzero to evaluate residuals. This requires the user to examine the feature_selected column in the result table to construct the correct set of independent variables to provide to the prediction function:
+\x off
+SELECT id, price, predict, price - predict AS residual
+FROM (
+ SELECT
+ houses.*,
+ madlib.elastic_net_gaussian_predict(
+ m.coef_nonzero, -- Non-zero coefficients
+ m.intercept, -- Intercept
+ ARRAY[tax,size] -- Features corresponding to non-zero coefficients
+ ) AS predict
+ FROM houses, houses_en2 m) s
+ORDER BY id;
+
The two queries above will result in same residuals:
Reuse the houses table above. Here we use 3-fold cross validation with 3 automatically generated lambda values and 3 specified alpha values. (This can take some time to run since elastic net is effectively being called 27 times for these combinations, then a 28th time for the whole dataset.)
It is stronglyrecommended that you run elastic_net_train() on a subset of the data with a limited max_iter before applying it to the full data set with a large max_iter. In the pre-run, you can adjust the parameters to get the best performance and then apply the best set of parameters to the whole data set.
+
Technical Background
+
Elastic net regularization seeks to find a weight vector that, for any given training example set, minimizes:
where \(L\) is the metric function that the user wants to minimize. Here \( \alpha \in [0,1] \) and \( lambda \geq 0 \). If \(alpha = 0\), we have the ridge regularization (known also as Tikhonov regularization), and if \(\alpha = 1\), we have the LASSO regularization.
+
For the Gaussian response family (or linear model), we have
and then minimize with the regularization terms. At the end of the calculation, the orginal scales will be restored and an intercept term will be obtained at the same time as a by-product.
+
Note that fitting after scaling is not equivalent to directly fitting.
[2] Beck, A. and M. Teboulle (2009), A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. on Imaging Sciences 2(1), 183-202.
+
[3] Shai Shalev-Shwartz and Ambuj Tewari, Stochastic Methods for L1 Regularized Loss Minimization. Proceedings of the 26th International Conference on Machine Learning, Montreal, Canada, 2009.
Categorical variables [1] require special attention in regression analysis because, unlike dichotomous or continuous variables, they cannot be entered into the regression equation just as they are. For example, if you have a variable called race that is coded with 1=Hispanic, 2=Asian, 3=Black, 4=White, then entering race in your regression will look at the linear effect of the race variable, which is probably not what you intended. Instead, categorical variables like this need to be coded into a series of indicator variables which can then be entered into the regression model. There are a variety of coding systems that can be used for coding categorical variables, including one-hot, dummy, effects, orthogonal, and Helmert.
+
We currently support one-hot and dummy coding techniques.
+
Dummy coding is used when a researcher wants to compare other groups of the predictor variable with one specific group of the predictor variable. Often, the specific group to compare with is called the reference group.
+
One-hot encoding is similar to dummy coding except it builds indicator (0/1) columns (cast as numeric) for each value of each category. Only one of these columns could take on the value 1 for each row (data point). There is no reference category for this function.
VARCHAR. Name of the table containing the source categorical data to encode.
+
+
+
output_table
+
VARCHAR. Name of the result table.
+
Note
If there are index columns in the 'source_table' specified by the parameter 'row_id' (see below), then the output table will contain only the index columns 'row_id' and the encoded columns. If the parameter 'row_id' is not specified, then all columns from the 'source_table', with the exception of the original columns that have been encoded, will be included in the 'output_table'.
+
+
categorical_cols
+
VARCHAR. Comma-separated string of column names of categorical variables to encode. Can also be '*' meaning all columns are to be encoded, except the ones specified in 'categorical_cols_to_exclude' and 'row_id'. Please note that all Boolean, integer and text columns are considered categorical columns and will be encoded when ‘*’ is specified for this argument.
+
+
+
categorical_cols_to_exclude (optional)
+
VARCHAR. Comma-separated string of column names to exclude from the categorical variables to encode. Applicable only if 'categorical_cols' = '*'.
+
+
+
row_id (optional)
+
VARCHAR. Comma-separated column name(s) corresponding to the primary key(s) of the source table. This parameter determines the format of the 'output_table' as described above. If 'categorical_cols' = '*', these columns will be excluded from encoding (but will be included in the output table).
+
Note
If you want to see both the raw categorical variable and its encoded form in the output_table, then include the categorical variable in the 'row_id' parameter. However, this will not work if you specify '*' for the parameter 'categorical_cols', because in this case 'row_id' columns will not be encoded at all.
+
+
top (optional)
+
VARCHAR. default: NULL. If integer, encodes the top n values by frequency. If float in the range (0.0, 1.0), encodes the specified fraction of values by frequency (e.g., 0.1 means top 10%). Can be specified as a global for all categorical columns, or as a dictionary with separate 'top' values for each categorical variable. Set to NULL to encode all levels (values) for all categorical columns.
+
+
+
value_to_drop (optional)
+
VARCHAR. Default: NULL.
+
+
For dummy coding, indicate the desired value (reference) to drop for each categorical variable. Can be specified as a global for all categorical columns, or a comma-separated string containing items of the form 'name=value', where 'name' is the column name and 'value' is the reference value to be dropped.
+
Set to NULL for one-hot encoding (default)
+
+
Note
If you specify both 'value_to_drop' and 'top' parameters, the 'value_to_drop' will be applied first (takes priority), then 'top' will be applied to the remaining values.
+
+
encode_null (optional)
+
BOOLEAN. default: FALSE. Whether NULL should be treated as one of the values of the categorical variable. If TRUE, then an indicator variable is created corresponding to the NULL value. If FALSE, then all encoded values for that variable will be set to 0.
+
+
+
output_type (optional)
+
VARCHAR. default: 'column'. This parameter controls the output format of the indicator variables. If 'column', a column is created for each indicator variable. PostgreSQL limits the number of columns in a table. If the total number of indicator columns exceeds the limit, then make this parameter either 'array' to combine the indicator columns into an array or 'svec' to cast the array output to 'madlib.svec' type.
+
Since the array output for any single tuple would be sparse (only one non-zero entry for each categorical column), the 'svec' output would be most efficient for storage. The 'array' output is useful if the array is used for post-processing, including concatenating with other non-categorical features.
+
The order of the array is the same as specified in 'categorical_cols'. A dictionary will be created when 'output_type' is 'array' or 'svec' to define an index into the array. The dictionary table will be given the name of the 'output_table' appended by '_dictionary'.
+
+
+
output_dictionary (optional)
+
BOOLEAN. default: FALSE. This parameter is used to handle auto-generated column names that exceed the PostgreSQL limit of 63 bytes.
+
+
If TRUE, column names will include numerical IDs and will create a dictionary table called 'output_table_dictionary' ('output_table' appended with '_dictionary').
+
If FALSE, will auto-generate column names in the usual way unless the limit of 63 bytes will be exceeded. In this case, a dictionary output file will be created and a message given to the user.
+
+
+
+
distributed_by (optional)
+
VARCHAR. default: NULL. Columns to use for the distribution policy of the output table. When NULL, either 'row_id' is used as distribution policy (when provided), or else the distribution policy of 'source_table' will be used. This argument does not apply to PostgreSQL platforms.
+
+
NULL: By default, the distribution policy of the source_table will be used.
+
Comma-separated column names: Column(s) to be used for the distribution key.
+
RANDOMLY: Use random distribution policy (only if there does not exist a column named 'randomly').
Create new table with one-hot encoding. The column 'sex' is replaced by three columns encoding the values 'F', 'M' and 'I'. Null values are not encoded by default:
+DROP TABLE IF EXISTS abalone_out, abalone_out_dictionary;
+SELECT madlib.encode_categorical_variables (
+ 'abalone', -- Source table
+ 'abalone_out', -- Output table
+ 'sex' -- Categorical columns
+ );
+SELECT * FROM abalone_out ORDER BY id;
+
Encode all categorical variables in the source table. Also, specify the column 'id' as the index (primary key) which changes the output table to include only the index and the encoded variables:
+DROP TABLE IF EXISTS abalone_out, abalone_out_dictionary;
+SELECT madlib.encode_categorical_variables (
+ 'abalone', -- Source table
+ 'abalone_out', -- Output table
+ '*', -- Categorical columns
+ NULL, -- Categorical columns to exclude
+ 'id' -- Index columns
+ );
+SELECT * FROM abalone_out ORDER BY id;
+
Now let's encode only the top values and group others into a miscellaneous bucket column. Top values can be global across all columns or specified by column. As an example of the latter, here are the top 2 'sex' values and the top 50% of 'rings' values:
+DROP TABLE IF EXISTS abalone_out, abalone_out_dictionary;
+SELECT madlib.encode_categorical_variables (
+ 'abalone', -- Source table
+ 'abalone_out', -- Output table
+ '*', -- Categorical columns
+ NULL, -- Categorical columns to exclude
+ 'id', -- Index columns
+ 'sex=2, rings=0.5' -- Top values
+ );
+SELECT * FROM abalone_out ORDER BY id;
+
If you want to see both the raw categorical variable and its encoded form in the output_table, then include the categorical variable(s) in the index parameter. (Remember that this will not work if you specify '*' for the parameter 'categorical_cols', because in this case 'row_id' columns will not be encoded at all.)
+DROP TABLE IF EXISTS abalone_out, abalone_out_dictionary;
+SELECT madlib.encode_categorical_variables (
+ 'abalone', -- Source table
+ 'abalone_out', -- Output table
+ 'sex, rings', -- Categorical columns
+ NULL, -- Categorical columns to exclude
+ 'id, sex, rings' -- Index columns
+ );
+SELECT * FROM abalone_out ORDER BY id;
+
For dummy encoding, let's make the 'I' value from the 'sex' variable as the reference. Here we use the 'value_to_drop' parameter:
+DROP TABLE IF EXISTS abalone_out, abalone_out_dictionary;
+SELECT madlib.encode_categorical_variables (
+ 'abalone', -- Source table
+ 'abalone_out', -- Output table
+ '*', -- Categorical columns
+ 'rings', -- Categorical columns to exclude
+ 'id', -- Index columns
+ NULL, -- Top value
+ 'sex=I' -- Value to drop for dummy encoding
+ );
+SELECT * FROM abalone_out ORDER BY id;
+
You can also use a logical expression in the categorical columns, which will be passed as boolean, and in the output table there will be two columns with name logical_expression_true and logical_expression_false:
+DROP TABLE IF EXISTS abalone_out, abalone_out_dictionary;
+SELECT madlib.encode_categorical_variables (
+ 'abalone', -- Source table
+ 'abalone_out', -- Output table
+ 'height>.10' -- Categorical columns
+ );
+SELECT * FROM abalone_out ORDER BY id;
+
fmsketch_dcount can be run on a column of any type. It returns an approximation to the number of distinct values (a la COUNT(DISTINCT x)), but faster and approximate. Like any aggregate, it can be combined with a GROUP BY clause to do distinct counts per group.
+
Syntax
+
Get the number of distinct values in a designated column.
+fmsketch_dcount( col_name )
+
Note
This is a User Defined Aggregate which returns the results when used in a query. Use "CREATE TABLE AS ", with the UDA as subquery if the results are to be stored. This is unlike the usual MADlib stored procedure interface which places the results in a table instead of returning it.
+
Examples
+
Generate some data.
+CREATE TABLE data(class INT, a1 INT);
+INSERT INTO data SELECT 1,1 FROM generate_series(1,10000);
+INSERT INTO data SELECT 1,2 FROM generate_series(1,15000);
+INSERT INTO data SELECT 1,3 FROM generate_series(1,10000);
+INSERT INTO data SELECT 2,5 FROM generate_series(1,1000);
+INSERT INTO data SELECT 2,6 FROM generate_series(1,1000);
+
+
Find the distinct number of values for each class.
+SELECT class, fmsketch_dcount(a1)
+FROM data
+GROUP BY data.class;
+
[1] P. Flajolet and N.G. Martin. Probabilistic counting algorithms for data base applications, Journal of Computer and System Sciences 31(2), pp 182-209, 1985. http://algo.inria.fr/flajolet/Publications/FlMa85.pdf
Generalized linear models extends ordinary linear regression by allowing the response variable to follow a more general set of distributions (rather than simply Gaussian distributions), and for a general family of functions of the response variable (the link function) to vary linearly with the predicted values (rather than assuming that the response itself must vary linearly).
+
For example, data of counts would typically be modeled with a Poisson distribution and a log link, while binary outcomes would typically be modeled with a Bernoulli distribution (or binomial distribution, depending on exactly how the problem is phrased) and a log-odds (or logit) link function.
+
Currently, the implemented distribution families are
TEXT. The name of the table containing the training data.
+
+
+
model_table
+
TEXT. Name of the generated table containing the model.
+
The model table produced by glm contains the following columns:
+
+
+
<...>
Text. Grouping columns, if provided in input. This could be multiple columns depending on the grouping_col input.
+
+
+
+
coef
FLOAT8. Vector of the coefficients in linear predictor.
+
+
+
+
log_likelihood
FLOAT8. The log-likelihood \( l(\boldsymbol \beta) \). We use the maximum likelihood estimate of dispersion parameter to calculate the log-likelihood while R and Python use deviance estimate and Pearson estimate respectively.
+
+
+
+
std_err
FLOAT8[]. Vector of the standard error of the coefficients.
+
+
+
+
z_stats or t_stats
FLOAT8[]. Vector of the z-statistics (in Poisson distribtuion and Binomial distribution) or the t-statistics (in all other distributions) of the coefficients.
+
+
+
+
p_values
FLOAT8[]. Vector of the p-values of the coefficients.
+
+
+
+
dispersion
FLOAT8. The dispersion value (Pearson estimate). When family=poisson or family=binomial, the dispersion is always 1.
+
+
+
+
num_rows_processed
BIGINT. Numbers of rows processed.
+
+
+
+
num_rows_skipped
BIGINT. Numbers of rows skipped due to missing values or failures.
+
+
+
+
num_iterations
INTEGER. The number of iterations actually completed. This would be different from the nIterations argument if a tolerance parameter is provided and the algorithm converges before all iterations are completed.
+
+
A summary table named <model_table>_summary is also created at the same time, which has the following columns:
+
+
method
'glm'
+
+
+
+
source_table
The data source table name.
+
+
+
+
model_table
The model table name.
+
+
+
+
dependent_varname
The dependent variable.
+
+
+
+
independent_varname
The independent variables
+
+
+
+
family_params
A string that contains family parameters, and has the form of 'family=..., link=...'
+
+
+
+
grouping_col
Name of grouping columns.
+
+
+
+
optimizer_params
A string that contains optimizer parameters, and has the form of 'optimizer=..., max_iter=..., tolerance=...'
+
+
+
+
num_all_groups
Number of groups in glm training.
+
+
+
+
num_failed_groups
Number of failed groups in glm training.
+
+
+
+
total_rows_processed
BIGINT. Total numbers of rows processed in all groups.
+
+
+
+
total_rows_skipped
BIGINT. Total numbers of rows skipped in all groups due to missing values or failures.
+
+
+
+
+
+
dependent_varname
+
TEXT. Name of the dependent variable column.
+
+
+
independent_varname
+
TEXT. Expression list to evaluate for the independent variables. An intercept variable is not assumed. It is common to provide an explicit intercept term by including a single constant 1 term in the independent variable list.
+
+
+
family_params (optional)
+
TEXT, Parameters for distribution family. Currently, we support
+
(1) family=poisson and link=[log or identity or sqrt].
+
(2) family=gaussian and link=[identity or log or inverse]. And when family=gaussian and link=identity, the GLM model is exactly the same as the linear regression.
+
(3) family=gamma and link=[inverse or identity or log].
+
(4) family=inverse_gaussian and link=[sqr_inverse or log or identity or inverse].
+
(5) family=binomial and link=[probit or logit].
+
+
+
grouping_col (optional)
+
TEXT, default: NULL. An expression list used to group the input dataset into discrete groups, running one regression per group. Similar to the SQL "GROUP BY" clause. When this value is NULL, no grouping is used and a single model is generated.
+
+
+
optim_params (optional)
+
TEXT, default: 'max_iter=100,optimizer=irls,tolerance=1e-6'. Parameters for optimizer. Currently, we support tolerance=[tolerance for relative error between log-likelihoods], max_iter=[maximum iterations to run], optimizer=irls.
+
+
+
verbose (optional)
+
BOOLEAN, default: FALSE. Provides verbose output of the results of training.
+
+
+
Note
For p-values, we just return the computation result directly. Other statistical packages, like 'R', produce the same result, but on printing the result to screen, another format function is used and any p-value that is smaller than the machine epsilon (the smallest positive floating-point number 'x' such that '1 + x != 1') will be printed on screen as "< xxx" (xxx is the value of the machine epsilon). Although the results may look different, they are in fact the same.
+
Prediction Function
The prediction function is provided to estimate the conditional mean given a new predictor. It has the following syntax:
+glm_predict(coef,
+ col_ind_var
+ link)
+
+
Arguments
+
coef
+
DOUBLE PRECISION[]. Model coefficients obtained from glm().
+
+
+
col_ind_var
+
New predictor, as a DOUBLE array. This should be the same length as the array obtained by evaluation of the 'independent_varname' argument in glm().
+
+
+
link
+
link function, as a string. This should match the link function the user inputted in glm().
Alternatively, unnest the arrays in the results for easier reading of output:
+\x off
+SELECT unnest(coef) as coefficient,
+ unnest(std_err) as standard_error,
+ unnest(z_stats) as z_stat,
+ unnest(p_values) as pvalue
+FROM glm_model;
+
+
Predicting dependent variable using GLM model. (This example uses the original data table to perform the prediction. Typically a different test dataset with the same features as the original training dataset would be used for prediction.)
+\x off
+-- Display predicted mean value on the original dataset
+SELECT
+ w.id,
+ madlib.glm_predict(
+ coef,
+ ARRAY[1, "wool_B", "tension_M", "tension_H"]::float8[],
+ 'log') AS mu
+FROM warpbreaks_dummy w, glm_model m
+ORDER BY w.id;
+
+-- Display predicted counts (which are predicted mean values rounded to the nearest integral value) on the original dataset
+SELECT
+ w.id,
+ madlib.glm_predict_poisson(
+ coef,
+ ARRAY[1, "wool_B", "tension_M", "tension_H"]::float8[],
+ 'log') AS poisson_count
+FROM warpbreaks_dummy w, glm_model m
+ORDER BY w.id;
+
All table names can be optionally schema qualified (current_schemas() would be searched if a schema name is not provided) and all table and column names should follow case-sensitivity and quoting rules per the database. (For instance, 'mytable' and 'MyTable' both resolve to the same entity, i.e. 'mytable'. If mixed-case or multi-byte characters are desired for entity names then the string should be double-quoted; in this case the input would be '"MyTable"').
+
Currently implementation uses Newton's method and, according to performance tests, when number of features are over 1000, this GLM function could be running slowly.
+
Functions in Linear Regression is prefered to GLM with family=gaussian,link=identity, as the former require only a single pass over the training data. In addition, if user expects to use robust variance, clustered variance, or marginal effects on top of the trained model, functions in Linear Regression and Logistic Regression should be used.
This utility function reports the number and type of GPUs attached to hosts on the database cluster.
+
This can be useful when determining which segments to use for training deep neural nets. For example, for economic reasons you may wish to set up a heterogeneous clusters with GPUs only on some of the hosts, not all of them. This utility can help you identify where the GPUS are and direct the compute to those locations only for model training.
+
GPU Confuguration
+
+gpu_configuration( output_table,
+ source
+ )
+
Arguments
+
output_table
+
TEXT. Name of the output table to write out the GPU information.
+
+
+
source (optional)
+
TEXT, default: 'tensorflow'. Source for determining GPU configuration. Using 'tensorflow' returns a description based on what TensorFlow reports. Using 'nvidia' returns a description based on what the Nvidia Systems Management Interface (nvidia-smi) reports [1]. Note that MADlib and Keras will use the TensorFlow information; the lower level nvidia-smi info is provided for convenience.
+
Note
If the command 'nvidia-smi -L' returns an error, we do not pass through the error message, but instead will show no GPUs for that host in the output table. You may want to run nvidia-smi from the command line to see error and informational messages.
+
+
+
Output
+ The output table contains the following:
+
+
hostname
TEXT. Name of the host machine in the cluster. Does not include master or mirrors. For PostgreSQL this will always return 'localhost'.
+
+
gpu_descr
TEXT. String reported by TensorFlow or nvidia-smi. The formats are different and shown in the examples below.
+
+
Examples
+
+
Get GPU configuration as per TensorFlow:
+DROP TABLE IF EXISTS host_gpu_mapping_tf;
+SELECT * FROM madlib.gpu_configuration('host_gpu_mapping_tf');
+SELECT * FROM host_gpu_mapping_tf ORDER BY hostname, gpu_descr;
+
+ hostname | gpu_descr
+----------+------------------------------------------------------------------------------------------
+ phoenix0 | device: 0, name: Tesla P100-PCIE-16GB, pci bus id: 0000:00:04.0, compute capability: 6.0
+ phoenix0 | device: 1, name: Tesla P100-PCIE-16GB, pci bus id: 0000:00:05.0, compute capability: 6.0
+ phoenix0 | device: 2, name: Tesla P100-PCIE-16GB, pci bus id: 0000:00:06.0, compute capability: 6.0
+ phoenix0 | device: 3, name: Tesla P100-PCIE-16GB, pci bus id: 0000:00:07.0, compute capability: 6.0
+ phoenix1 | device: 0, name: Tesla P100-PCIE-16GB, pci bus id: 0000:00:04.0, compute capability: 6.0
+ phoenix1 | device: 1, name: Tesla P100-PCIE-16GB, pci bus id: 0000:00:05.0, compute capability: 6.0
+ phoenix3 | device: 0, name: Tesla P100-PCIE-16GB, pci bus id: 0000:00:04.0, compute capability: 6.0
+ phoenix3 | device: 1, name: Tesla P100-PCIE-16GB, pci bus id: 0000:00:05.0, compute capability: 6.0
+ phoenix3 | device: 2, name: Tesla P100-PCIE-16GB, pci bus id: 0000:00:06.0, compute capability: 6.0
+ phoenix3 | device: 3, name: Tesla P100-PCIE-16GB, pci bus id: 0000:00:07.0, compute capability: 6.0
+ phoenix4 | device: 0, name: Tesla P100-PCIE-16GB, pci bus id: 0000:00:04.0, compute capability: 6.0
+ phoenix4 | device: 1, name: Tesla P100-PCIE-16GB, pci bus id: 0000:00:05.0, compute capability: 6.0
+ phoenix4 | device: 2, name: Tesla P100-PCIE-16GB, pci bus id: 0000:00:06.0, compute capability: 6.0
+ phoenix4 | device: 3, name: Tesla P100-PCIE-16GB, pci bus id: 0000:00:07.0, compute capability: 6.0
+(14 rows)
+
In this heterogeneous cluster there are 4 GPUs attached to hosts 0, 3 and 4. There are 2 GPUs attached to host 1 and no GPUs attached to host 2.
+
Get GPU configuration as per nvidia-smi:
+DROP TABLE IF EXISTS host_gpu_mapping_nvidia;
+SELECT * FROM madlib.gpu_configuration('host_gpu_mapping_nvidia', -- output table
+ 'nvidia' -- source for GPU info
+ );
+SELECT * FROM host_gpu_mapping_nvidia ORDER BY hostname, gpu_descr;
+
+ hostname | gpu_descr
+----------+------------------------------------------------------------------------------
+ phoenix0 | GPU 0: Tesla P100-PCIE-16GB (UUID: GPU-f2ccc77e-2501-f6ee-4754-069dda256fb2)
+ phoenix0 | GPU 1: Tesla P100-PCIE-16GB (UUID: GPU-b1fc40ca-c7c6-bc86-f20f-6e9a62cda3f8)
+ phoenix0 | GPU 2: Tesla P100-PCIE-16GB (UUID: GPU-d93bb21b-96f9-7c1d-3bab-cdd92b7bbc9d)
+ phoenix0 | GPU 3: Tesla P100-PCIE-16GB (UUID: GPU-2d79c4a8-479e-2f33-39f8-3ba80b63f830)
+ phoenix1 | GPU 0: Tesla P100-PCIE-16GB (UUID: GPU-0af6bb1e-5b5b-4988-ad3a-a917e9584702)
+ phoenix1 | GPU 1: Tesla P100-PCIE-16GB (UUID: GPU-d824c976-a8aa-ef26-a13c-9a9a7fe86bfd)
+ phoenix3 | GPU 0: Tesla P100-PCIE-16GB (UUID: GPU-3681d0b6-1ec6-0453-fd81-29d88e549cd9)
+ phoenix3 | GPU 1: Tesla P100-PCIE-16GB (UUID: GPU-d4b1f2e7-b238-ac9a-bbfe-918adeb69472)
+ phoenix3 | GPU 2: Tesla P100-PCIE-16GB (UUID: GPU-42a32ef1-a60c-e599-c8cf-0e669111ab6f)
+ phoenix3 | GPU 3: Tesla P100-PCIE-16GB (UUID: GPU-1cce09c4-6856-8031-be0b-8e8bbf9a10f3)
+ phoenix4 | GPU 0: Tesla P100-PCIE-16GB (UUID: GPU-a71bdc18-fdd5-ba25-617e-19b23cc8e827)
+ phoenix4 | GPU 1: Tesla P100-PCIE-16GB (UUID: GPU-f9d13688-7fe6-a029-24d1-985a5659f18f)
+ phoenix4 | GPU 2: Tesla P100-PCIE-16GB (UUID: GPU-06a7f54b-c07a-e87a-20d6-09bd99b19531)
+ phoenix4 | GPU 3: Tesla P100-PCIE-16GB (UUID: GPU-af3b32f3-8bd9-cb75-a8fb-25253b9da926)
+(14 rows)
+
+
To get a fuller picture at the segment level, combine with the Greenplum catalog table 'gp_segment_configuration' which contains information about segment instance configuration [2]. Here is an example of this table filtering out master and mirrors:
+SELECT * FROM gp_segment_configuration WHERE role='p' AND content>=0 ORDER BY hostname, dbid;
+
+ dbid | content | role | preferred_role | mode | status | port | hostname | address | replication_port
+------+---------+------+----------------+------+--------+-------+----------+----------+------------------
+ 2 | 0 | p | p | c | u | 40000 | phoenix0 | phoenix0 | 70000
+ 3 | 1 | p | p | c | u | 40001 | phoenix0 | phoenix0 | 70001
+ 4 | 2 | p | p | c | u | 40002 | phoenix0 | phoenix0 | 70002
+ 5 | 3 | p | p | c | u | 40003 | phoenix0 | phoenix0 | 70003
+ 6 | 4 | p | p | c | u | 40000 | phoenix1 | phoenix1 | 70000
+ 7 | 5 | p | p | c | u | 40001 | phoenix1 | phoenix1 | 70001
+ 8 | 6 | p | p | c | u | 40002 | phoenix1 | phoenix1 | 70002
+ 9 | 7 | p | p | c | u | 40003 | phoenix1 | phoenix1 | 70003
+ 10 | 8 | p | p | c | u | 40000 | phoenix2 | phoenix2 | 70000
+ 11 | 9 | p | p | c | u | 40001 | phoenix2 | phoenix2 | 70001
+ 12 | 10 | p | p | c | u | 40002 | phoenix2 | phoenix2 | 70002
+ 13 | 11 | p | p | c | u | 40003 | phoenix2 | phoenix2 | 70003
+ 14 | 12 | p | p | c | u | 40000 | phoenix3 | phoenix3 | 70000
+ 15 | 13 | p | p | c | u | 40001 | phoenix3 | phoenix3 | 70001
+ 16 | 14 | p | p | c | u | 40002 | phoenix3 | phoenix3 | 70002
+ 17 | 15 | p | p | c | u | 40003 | phoenix3 | phoenix3 | 70003
+ 18 | 16 | p | p | c | u | 40000 | phoenix4 | phoenix4 | 70000
+ 19 | 17 | p | p | c | u | 40001 | phoenix4 | phoenix4 | 70001
+ 20 | 18 | p | p | c | u | 40002 | phoenix4 | phoenix4 | 70002
+ 21 | 19 | p | p | c | u | 40003 | phoenix4 | phoenix4 | 70003
+(20 rows)
+
Now join this table with the GPU resources table to create a table containing a list of all segments on hosts with GPUs attached:
+DROP TABLE IF EXISTS segments_to_use;
+CREATE TABLE segments_to_use AS
+ SELECT DISTINCT dbid, hostname FROM gp_segment_configuration JOIN host_gpu_mapping_tf USING (hostname)
+ WHERE role='p' AND content>=0;
+SELECT * FROM segments_to_use ORDER BY hostname, dbid;
+
This function computes the average of the shortest paths between each pair of vertices. Average path length is based on "reachable target vertices", so it ignores infinite-length paths between vertices that are not connected. This means the function will average the path lengths in each connected component. If the user requires the average path length of a particular component, the weakly_connected_components function may be used to isolate the relevant vertices.
+
Note
This function assumes a valid output from a prior APSP run - both the APSP table and the associated output summary table. APSP is a computationally expensive algorithm because it finds the shortest path between all nodes in the graph. The worst case run-time for this implementation is O(V^2 * E) where V is the number of vertices and E is the number of edges. In practice, run-time will be generally be much less than this, depending on the graph.
Create a graph with 2 groups and find APSP for each group:
+DROP TABLE IF EXISTS edge_gr;
+CREATE TABLE edge_gr AS
+(
+ SELECT *, 0 AS grp FROM edge
+ UNION
+ SELECT *, 1 AS grp FROM edge WHERE src_id < 6 AND dest_id < 6
+);
+INSERT INTO edge_gr VALUES
+(4,5,-20,1);
+
+
Find APSP for all groups:
+DROP TABLE IF EXISTS out_gr, out_gr_summary;
+SELECT madlib.graph_apsp(
+ 'vertex', -- Vertex table
+ NULL, -- Vertex id column (NULL means use default naming)
+ 'edge_gr', -- Edge table
+ 'src=src_id, dest=dest_id, weight=edge_weight',
+ 'out_gr', -- Output table of shortest paths
+ 'grp' -- Grouping columns
+);
+
+
Find the average path length in every group
+DROP TABLE IF EXISTS out_gr_path;
+SELECT madlib.graph_avg_path_length('out_gr', 'out_gr_path');
+SELECT * FROM out_gr_path ORDER BY grp;
+
The Closeness function returns various closeness centrality measures and the k-degree for given subset of vertices. The closeness measures are the inverse of the sum, the inverse of the average, and the sum of inverses of the shortest distances to all reachable target vertices (excluding the source vertex).
+
Note
The closeness measures require a valid output from a prior APSP run - both the APSP table and the associated output summary table. APSP is a computationally expensive algorithm because it finds the shortest path between all nodes in the graph. The worst case run-time for this implementation is O(V^2 E) where V is the number of vertices and E is the number of edges. In practice, run-time will be generally be much less than this, depending on the graph.
TEXT. Name of the output table generated by a prior run of all pairs shortest path (APSP).
+
+
+
out_table
+
TEXT. Name of the table to store the closeness measures. It contains a row for every vertex of every group and have the following columns (in addition to the grouping columns):
+
inverse_sum_dist: Inverse of the sum of shortest distances to all reachable vertices.
+
inverse_average_dist: Inverse of the average of shortest distances to all reachable vertices.
+
sum_inverse_dist: Sum of the inverse of shortest distances to all reachable vertices.
+
k_degree: Total number of reachable vertices.
+
+
+
+
vertex_filter_expr (optional)
+
TEXT, default = NULL. Valid PostgreSQL expression that describes the vertices to generate closeness measures for. If this parameter is not specified, closeness measures are generated for all vertices in the apsp table. You can think of this input parameter as being like a WHERE clause.
+
Some example inputs:
+
If you want a short list of vertices, say 1, 2 and 3:
vertex_id IN (1, 2, 3)
+
If you want a range of vertices between 1000 and 2000:
vertix_id BETWEEN 1000 AND 2000
+
If you want a set of vertices from a separate table satisfying to a condition
EXISTS (SELECT vertex_id FROM vertices_of_interest
+ WHERE vertex_id > 5000 AND condition = 'xyz')
+
+
+
+
+
+
Examples
+
+
Create vertex and edge tables to represent the graph:
Create a graph with 2 groups and find APSP for each group:
+DROP TABLE IF EXISTS edge_gr;
+CREATE TABLE edge_gr AS
+(
+ SELECT *, 0 AS grp FROM edge
+ UNION
+ SELECT *, 1 AS grp FROM edge WHERE src_id < 6 AND dest_id < 6
+);
+INSERT INTO edge_gr VALUES
+(4,5,-20,1);
+
+
Find APSP for all groups:
+DROP TABLE IF EXISTS out_gr, out_gr_summary;
+SELECT madlib.graph_apsp(
+ 'vertex', -- Vertex table
+ NULL, -- Vertex id column (NULL means use default naming)
+ 'edge_gr', -- Edge table
+ 'src=src_id, dest=dest_id, weight=edge_weight',
+ 'out_gr', -- Output table of shortest paths
+ 'grp' -- Grouping columns
+);
+
+
Compute closeness measure for vertex 0 to vertex 5 in every group
+DROP TABLE IF EXISTS out_gr_path;
+SELECT madlib.graph_closeness('out_gr', 'out_gr_closeness', 'src_id >= 0 and src_id <=5');
+SELECT * FROM out_gr_closeness ORDER BY grp;
+
Diameter is defined as the longest of all shortest paths in a graph.
+
Note
This function assumes a valid output from a prior APSP run - both the APSP table and the associated output summary table. APSP is a computationally expensive algorithm because it finds the shortest path between all nodes in the graph. The worst case run-time for this implementation is O(V^2 * E) where V is the number of vertices and E is the number of edges. In practice, run-time will be generally be much less than this, depending on the graph.
+
Diameter
+graph_diameter( apsp_table,
+ output_table
+ )
+
+
Arguments
+
apsp_table
+
TEXT. Name of the output table generated by a prior run of all pairs shortest path (APSP).
+
+
+
out_table
+
TEXT. Name of the table to store the diameter. It contains a row for every group, the diameter value and the two vertices that are the farthest apart.
+
+
+
+
Examples
+
+
Create vertex and edge tables to represent the graph:
Create a graph with 2 groups and find APSP for each group:
+DROP TABLE IF EXISTS edge_gr;
+CREATE TABLE edge_gr AS
+(
+ SELECT *, 0 AS grp FROM edge
+ UNION
+ SELECT *, 1 AS grp FROM edge WHERE src_id < 6 AND dest_id < 6
+);
+INSERT INTO edge_gr VALUES
+(4,5,-20,1);
+
+
Find APSP for all groups:
+DROP TABLE IF EXISTS out_gr, out_gr_summary;
+SELECT madlib.graph_apsp(
+ 'vertex', -- Vertex table
+ NULL, -- Vertex id column (NULL means use default naming)
+ 'edge_gr', -- Edge table
+ 'src=src_id, dest=dest_id, weight=edge_weight',
+ 'out_gr', -- Output table of shortest paths
+ 'grp' -- Grouping columns
+);
+
+
Find the diameter of graph in every group
+DROP TABLE IF EXISTS out_gr_path;
+SELECT madlib.graph_diameter('out_gr', 'out_gr_diameter');
+SELECT * FROM out_gr_diameter ORDER BY grp;
+
This function computes the degree of each node. The node degree is the number of edges adjacent to that node. The node in-degree is the number of edges pointing in to the node and node out-degree is the number of edges pointing out of the node.
TEXT. Name of the table containing the vertex data for the graph. Must contain the column specified in the 'vertex_id' parameter below.
+
+
+
vertex_id
+
TEXT, default = 'id'. Name of the column in 'vertex_table' containing vertex ids. The vertex ids can be of type INTEGER or BIGINT with no duplicates. They do not need to be contiguous.
+
+
+
edge_table
+
TEXT. Name of the table containing the edge data. The edge table must contain columns for source vertex, destination vertex and edge weight. Column naming convention is described below in the 'edge_args' parameter.
+
+
+
edge_args
+
TEXT. A comma-delimited string containing multiple named arguments of the form "name=value". The following parameters are supported for this string argument:
+
src (INTEGER or BIGINT): Name of the column containing the source vertex ids in the edge table. Default column name is 'src'.
+
dest (INTEGER or BIGINT): Name of the column containing the destination vertex ids in the edge table. Default column name is 'dest'.
+
weight (FLOAT8): Name of the column containing the edge weights in the edge table. Default column name is 'weight'.
+
+
+
+
out_table
+
TEXT. Name of the table to store the result. It contains a row for every vertex of every group and has the following columns (in addition to the grouping columns):
+
vertex: The id for the source vertex. Will use the input vertex column 'id' for column naming.
+
indegree: Number of incoming edges to the vertex.
+
outdegree: Number of outgoing edges from the vertex.
+
+
+
+
grouping_cols
+
TEXT, default = NULL. List of columns used to group the input into discrete subgraphs. These columns must exist in the edge table. When this value is null, no grouping is used and a single result is generated.
+
+
Examples
+
+
Create vertex and edge tables to represent the graph:
Create a graph with 2 groups and find degrees for each group:
+DROP TABLE IF EXISTS edge_gr;
+CREATE TABLE edge_gr AS
+(
+ SELECT *, 0 AS grp FROM edge
+ UNION
+ SELECT *, 1 AS grp FROM edge WHERE src_id < 6 AND dest_id < 6
+);
+INSERT INTO edge_gr VALUES
+(4,5,-20,1);
+
+
Find in-out degrees for all groups:
+DROP TABLE IF EXISTS out_gr;
+SELECT madlib.graph_vertex_degrees(
+ 'vertex', -- Vertex table
+ NULL, -- Vertex id column (NULL means use default naming)
+ 'edge_gr', -- Edge table
+ 'src=src_id, dest=dest_id, weight=edge_weight',
+ 'out_gr', -- Output table of shortest paths
+ 'grp' -- Grouping columns
+);
+SELECT * FROM out_gr WHERE id < 2 ORDER BY grp, id;
+
Given a graph, the HITS (Hyperlink-Induced Topic Search) algorithm outputs the authority score and hub score of every vertex, where authority estimates the value of the content of the page and hub estimates the value of its links to other pages. This algorithm was originally developed to rate web pages [1].
TEXT. Name of the table containing the vertex data for the graph. Must contain the column specified in the 'vertex_id' parameter below.
+
+
+
vertex_id
+
TEXT, default = 'id'. Name of the column in 'vertex_table' containing vertex ids. The vertex ids can be of type INTEGER or BIGINT with no duplicates. They do not need to be contiguous.
+
+
+
edge_table
+
TEXT. Name of the table containing the edge data. The edge table must contain columns for source vertex and destination vertex.
+
+
+
edge_args
+
TEXT. A comma-delimited string containing multiple named arguments of the form "name=value". The following parameters are supported for this string argument:
+
src (INTEGER or BIGINT): Name of the column containing the source vertex ids in the edge table. Default column name is 'src'.
+
dest (INTEGER or BIGINT): Name of the column containing the destination vertex ids in the edge table. Default column name is 'dest'.
+
+
+
+
out_table
+
TEXT. Name of the table to store the result of HITS. It will contain a row for every vertex from 'vertex_table' with the following columns:
+
vertex_id : The id of a vertex. Will use the input parameter 'vertex_id' for column naming.
+
authority : The vertex authority score.
+
hub : The vertex hub score.
+
grouping_cols : Grouping column values (if any) associated with the vertex_id.
+
+
A summary table is also created that contains information regarding the number of iterations required for convergence. It is named by adding the suffix '_summary' to the 'out_table' parameter.
+
+
+
max_iter (optional)
+
INTEGER, default: 100. The maximum number of iterations allowed. Each iteration consists of both authority and hub phases.
+
+
+
threshold (optional)
+
FLOAT8, default: (1/number of vertices * 1000). Threshold must be set to a value between 0 and 1, inclusive of end points. If the difference between two consecutive iterations of authority AND two consecutive iterations of hub is smaller than 'threshold', then the computation stops. That is, both authority and hub value differences must be below the specified threshold for the algorithm to stop. If you set the threshold to 0, then you will force the algorithm to run for the full number of iterations specified in 'max_iter'.
+
+
+
grouping_cols (optional)
+
TEXT, default: NULL. A single column or a list of comma-separated columns that divides the input data into discrete groups, resulting in one distribution per group. When this value is NULL, no grouping is used and a single model is generated for all data.
Note
Expressions are not currently supported for 'grouping_cols'.
+
+
+
Examples
+
+
Create vertex and edge tables to represent the graph:
The algorithm stopped at 20 iterations even though the convergence for threshold of 0.00001 is at 25 iterations. This is because max_iter was set to 20.
+
Running HITS with grouping column and default values for max_iter and threshold. Add more rows to the edge table to create different graphs based on the user_id column.
On a Greenplum cluster, the edge table should be distributed by the source vertex id column for better performance.
+
This implementation of the HITS algorithm supports multigraph and each duplicated edge is considered for counting when calculating authority and hub scores.
This version of encoding categorical variables has been deprecated. The new module with more capability can be found here Encoding Categorical Variables
Categorical variables require special attention in regression analysis because, unlike dichotomous or continuous variables, they cannot be entered into the regression equation just as they are. For example, if you have a variable called race that is coded 1 = Hispanic, 2 = Asian, 3 = Black, 4 = White, then entering race in your regression will look at the linear effect of race, which is probably not what you intended. Instead, categorical variables like this need to be recoded into a series of indicator variables which can then be entered into the regression model. There are a variety of coding systems (also called as contrasts) that can be used when coding categorical variables. including dummy, effects, orthogonal, and helmert coding.
+
We currently only support the dummy coding technique. Dummy coding is used when a researcher wants to compare other groups of the predictor variable with one specific group of the predictor variable. Often, the specific group to compare with is called the reference group.
VARCHAR. Name of the source table, containing data for categorical variables.
+
output_table
+
VARCHAR. Name of result table. The output table has the same columns as the original table, adding new indicator variable columns for each categorical column. The column name for the indicator variable is 'categorical column name'_'categorical value'.
+
categorical_cols
+
VARCHAR. Comma-separated string of column names of categorical variables that need to be dummy-coded.
+
keep_null (optional)
+
BOOLEAN. default: FALSE. Whether 'NULL' should be treated as one of the categories of the categorical variable. If True, then an indicator variable is created corresponding to the NULL value. If False, then all indicator variables for that record will be set to NULL.
+
distributed_by (optional)
+
VARCHAR. default: NULL. Columns to use for the distribution policy of the output table. When NULL, the distribution policy of 'source_table' will be used. This argument is not available for POSTGRESQL platforms.
This preprocessor prepares training data for deep learning.
+
It packs multiple training examples into the same row for frameworks like Keras and TensorFlow that support mini-batching as an optimization option. The advantage of using mini-batching is that it can perform better than stochastic gradient descent because it uses more than one training example at a time, typically resulting in faster and smoother convergence [1].
+
In the case of image processing, images can be represented as an array of numbers where each element represents grayscale, RGB or other channel values for each pixel in the image. It is standard practice to normalize the image data before training. The normalizing constant in this module is parameterized, so it can be set depending on the format of image data used.
+
This preprocessor also sets the distribution rules for the training data. For example, you may only want to train models on segments that reside on hosts that are GPU enabled.
+
There are two versions of the preprocessor: training_preprocessor_dl() preprocesses input data to be used for training a deep learning model, while validation_preprocessor_dl() preprocesses validation data used for model evaluation.
TEXT. Name of the table containing training dataset. Can also be a view.
+
+
+
output_table
+
TEXT. Name of the output table from the training preprocessor which will be used as input to algorithms that support mini-batching. Note that the arrays packed into the output table are shuffled and normalized, by dividing each element in the independent variable array by the optional 'normalizing_const' parameter. For performance reasons, packed arrays are converted to PostgreSQL bytea format, which is a variable-length binary string.
+
In the case a validation data set is used (see later on this page), this output table is also used as an input to the validation preprocessor so that the validation and training data are both preprocessed in an identical manner.
+
+
+
dependent_varname
+
TEXT. Name of the dependent variable column. In the case that there are multiple dependent variable columns, representing a multi-output neural network, put the columns as a comma separated list, e.g., 'dep_var1, dep_var2, dep_var3'.
Note
The mini-batch preprocessor automatically 1-hot encodes dependent variables of all types. The exception is numeric array types (integer and float), where we assume these are already 1-hot encoded, so these will just be passed through as is.
+
+
independent_varname
+
TEXT. Name of the independent variable column. The column must be a numeric array type. In the case that there are multiple independent variable columns, representing a multi-input neural network, put the columns as a comma separated list, e.g., 'indep_var1, indep_var2, indep_var3'.
+
+
+
buffer_size (optional)
+
INTEGER, default: computed. Buffer size is the number of rows from the source table that are packed into one row of the preprocessor output table. In the case of images, the source table will have one image per row, and the output table will have multiple images per row. The default value is computed considering the sizes of the source table and images and the number of segments in the database cluster.
Note
Using the default for 'buffer_size' will produce buffers that are relatively large, which generally results in the fastest fit() runtime with Keras. Setting a smaller buffer size may cause the preprocessor to run faster (although this is not guaranteed, since it depends on database cluster size, data set, and other factors). But since preprocessing is usually a one-time operation and fit() is called many times, by default buffer sizes are optimized to make fit() as fast as possible. Note that specifying a 'buffer_size' does not guarantee that exact value will be used. Actual buffer size is adjusted to avoid data skew, which adversely impacts fit() runtime.
+
+
normalizing_const (optional)
+
REAL, default: 1.0. The normalizing constant to divide each value in the 'independent_varname' array by. For example, you would use 255 for this value if the image data is in the form 0-255.
+
+
+
num_classes (optional)
+
INTEGER[], default: NULL. Number of class labels of each dependent variable for 1-hot encoding. If NULL, the 1-hot encoded array length will be equal to the number of distinct class values found in the input table.
+
+
+
distribution_rules (optional)
+
TEXT, default: 'all_segments'. Specifies how to distribute the 'output_table'. This is important for how the fit function will use resources on the cluster. The default 'all_segments' means the 'output_table' will be distributed to all segments in the database cluster.
+
If you specify 'gpu_segments' then the 'output_table' will be distributed to all segments that are on hosts that have GPUs attached. This will make maximum use of GPU resources when training a deep learning model.
+
You can also specify the name of a resources table containing the segments to be used for training. This table must contain a column called 'dbid' that specifies the segment id from the 'gp_segment_configuration' table [2]. Refer to the utility function GPU Configuration for more information on how to identify segments attached to hosts that are GPU enabled.
TEXT. Name of the table containing validation dataset. Can also be a view.
+
+
+
output_table
+
TEXT. Name of the output table from the validation preprocessor which will be used as input to algorithms that support mini-batching. The arrays packed into the output table are normalized using the same normalizing constant from the training preprocessor as specified in the 'training_preprocessor_table' parameter described below. Validation data is not shuffled. For performance reasons, packed arrays are converted to PostgreSQL bytea format, which is a variable-length binary string.
+
+
+
dependent_varname
+
TEXT. Name of the dependent variable column.
Note
The mini-batch preprocessor automatically 1-hot encodes dependent variables of all types. The exception is numeric array types (integer and float), where we assume these are already 1-hot encoded, so these will just be passed through as is.
+
+
independent_varname
+
TEXT. Name of the independent variable column. The column must be a numeric array type.
+
+
+
training_preprocessor_table
+
TEXT. The output table obtained by running training_preprocessor_dl(). Validation data is preprocessed in the same way as training data, i.e., same normalizing constant and dependent variable class values. Note that even if the validation dataset is missing some of the class values completely, this parameter will ensure that the ordering and labels still match with training dataset.
+
+
+
buffer_size (optional)
+
INTEGER, default: computed. Buffer size is the number of rows from the source table that are packed into one row of the preprocessor output table. In the case of images, the source table will have one image per row, and the output table will have multiple images per row. The default value is computed considering the sizes of the source table and images and the number of segments in the database cluster.
Note
Using the default for 'buffer_size' will produce buffers that are relatively large, which generally results in the fastest fit() runtime with Keras. Setting a smaller buffer size may cause the preprocessor to run faster (although this is not guaranteed, since it depends on database cluster size, data set, and other factors). But since preprocessing is usually a one-time operation and fit() is called many times, by default buffer sizes are optimized to make fit() as fast as possible. Note that specifying a 'buffer_size' does not guarantee that exact value will be used. Actual buffer size is adjusted to avoid data skew, which adversely impacts fit() runtime.
+
+
distribution_rules (optional)
+
TEXT, default: 'all_segments'. Specifies how to distribute the 'output_table'. This is important for how the fit function will use resources on the cluster. The default 'all_segments' means the 'output_table' will be distributed to all segments in the database cluster.
+
If you specify 'gpu_segments' then the 'output_table' will be distributed to all segments that are on hosts that have GPUs attached. This will make maximum use of GPU resources when training a deep learning model.
+
You can also specify the name of a resources table containing the segments to be used for training. This table must contain a column called 'dbid' that specifies the segment id from the 'gp_segment_configuration' table [2]. Refer to the utility function GPU Configuration for more information on how to identify segments attached to hosts that are GPU enabled.
BYTEA. Packed array of independent variables in PostgreSQL bytea format. Arrays of independent variables packed into the output table are normalized by dividing each element in the independent variable array by the optional 'normalizing_const' parameter. Training data is shuffled, but validation data is not.
+
+
<dependent_varname>
BYTEA. Packed array of dependent variables in PostgreSQL bytea format. The dependent variable is always one-hot encoded as an integer array. For now, we are assuming that input_preprocessor_dl() will be used only for classification problems using deep learning. So the dependent variable is one-hot encoded, unless it's already a numeric array in which case we assume it's already one-hot encoded and just cast it to an integer array.
+
+
<independent_varname>_shape
INTEGER[]. Shape of the independent variable array after preprocessing. The first element is the number of images packed per row, and subsequent elements will depend on how the image is described (e.g., channels first or last).
+
+
<dependent_varname>_shape
INTEGER[]. Shape of the dependent variable array after preprocessing. The first element is the number of images packed per row, and the second element is the number of class values.
+
+
buffer_id
INTEGER. Unique id for each row in the packed table.
Type of the dependent variable from the source table.
+
+
<dependent_varname>_class_values
The dependent level values that one-hot encoding maps to for the dependent variable.
+
+
buffer_size
Buffer size used in preprocessing step.
+
+
normalizing_const
The value used to normalize the input image data.
+
+
num_classes
Number of dependent levels the one-hot encoding is created for. NULLs are padded at the end if the number of distinct class levels found in the input data is less than the 'num_classes' parameter specified in training_preprocessor_dl().
+
+
distribution_rules
This is the list of segment id's in the form of 'dbid' describing how the 'output_table' is distributed, as per the 'distribution_rules' input parameter. If the 'distribution_rules' parameter is set to 'all_segments', then this will also be set to 'all_segments'.
+
+
__internal_gpu_config__
For internal use. (Note: this is the list of segment id's where data is distributed in the form of 'content' id, which is different from 'dbid' [2].)
+
+
Examples
+
Create an artificial 2x2 resolution color image data set with 3 possible classifications. The RGB values are per-pixel arrays:
For small datasets like in this example, buffer size is mainly determined by the number of segments in the database. For a Greenplum database with 2 segments, there will be 2 rows with a buffer size of 26. For PostgresSQL, there would be only one row with a buffer size of 52 since it is a single node database. For larger data sets, other factors go into computing buffers size besides number of segments. Here is the packed output table of training data for our simple example:
+SELECT rgb_shape, species_shape, buffer_id FROM image_data_packed ORDER BY buffer_id;
+
Run the preprocessor for the validation dataset. In this example, we use the same images for validation to demonstrate, but normally validation data is different than training data:
We could choose to use a different buffer size compared to the training_preprocessor_dl run (but generally don't need to). Other parameters such as num_classes and normalizing_const that were passed to training_preprocessor_dl are automatically inferred using the image_data_packed param that is passed. Here is the packed output table of validation data for our simple example:
+SELECT rgb_shape, species_shape, buffer_id FROM val_image_data_packed ORDER BY buffer_id;
+
Load data in another format. Create an artificial 2x2 resolution color image data set with 3 possible classifications. The RGB values are unrolled into a flat array:
+DROP TABLE IF EXISTS image_data;
+CREATE TABLE image_data AS (
+SELECT ARRAY[
+ (random() * 256)::integer, -- R values
+ (random() * 256)::integer,
+ (random() * 256)::integer,
+ (random() * 256)::integer,
+ (random() * 256)::integer, -- G values
+ (random() * 256)::integer,
+ (random() * 256)::integer,
+ (random() * 256)::integer,
+ (random() * 256)::integer, -- B values
+ (random() * 256)::integer,
+ (random() * 256)::integer,
+ (random() * 256)::integer
+ ] as rgb, ('{cat,dog,bird}'::text[])[ceil(random()*3)] as species
+FROM generate_series(1, 52)
+);
+SELECT * FROM image_data;
+
Run the preprocessor for the validation dataset. In this example, we use the same images for validation to demonstrate, but normally validation data is different than training data:
Using distribution rules to specify how to distribute the 'output_table'. This is important for how the fit function will use resources on the cluster. To distribute to all segments on hosts with GPUs attached:
+DROP TABLE IF EXISTS image_data_packed, image_data_packed_summary;
+SELECT madlib.training_preprocessor_dl('image_data', -- Source table
+ 'image_data_packed', -- Output table
+ 'species', -- Dependent variable
+ 'rgb', -- Independent variable
+ NULL, -- Buffer size
+ 255, -- Normalizing constant
+ NULL, -- Number of classes
+ 'gpu_segments' -- Distribution rules
+ );
+\x on
+SELECT * FROM image_data_packed_summary;
+
This module allows you to use SQL to call deep learning models designed in Keras [1], which is a high-level neural network API written in Python.
+
Keras can run on top of different backends and the one that is currently supported by MADlib is TensorFlow [2]. The implementation in MADlib is designed to train a single model across multiple segments (workers) in Greenplum database. (PostgreSQL is also supported.) Alternatively, to train multiple models at the same time for model architecture search or hyperparameter tuning, you can use the methods in Train Multiple Models.
+
The main use case supported is classification using sequential models, which are made up of a linear stack of layers. This includes multilayer perceptrons (MLPs) and convolutional neural networks (CNNs). Regression is not currently supported.
+
Before using Keras in MADlib you will need to preprocess your training and evaluation datasets by using the method called Preprocess Data which is a utility that prepares data for use by models that support mini-batch as an optimization option. This is a one-time operation and you would only need to re-run the preprocessor if your input data has changed. The advantage of using mini-batching is that it can perform better than stochastic gradient descent because it uses more than one training example at a time, typically resulting faster and smoother convergence [3].
+
You can also do inference on models that have not been trained in MADlib, but rather imported from an external source. This is described in the section called "Predict BYOM" below, where "BYOM" stands for "Bring Your Own Model."
+
Note that the following MADlib functions are targeting a specific TensorFlow kernel version (1.14). Using a newer or older version may or may not work as intended.
+
MADlib's deep learning methods are designed to use the TensorFlow package and its built in Keras functions. To ensure consistency, please use tensorflow.keras objects (models, layers, etc.) instead of importing Keras and using its objects.
+
Note
CUDA GPU memory cannot be released until the process holding it is terminated. When a MADlib deep learning function is called with GPUs, Greenplum internally creates a process (called a slice) which calls TensorFlow to do the computation. This process holds the GPU memory until one of the following two things happen: query finishes and user logs out of the Postgres client/session; or, query finishes and user waits for the timeout set by gp_vmem_idle_resource_timeout. The default value for this timeout is 18 sec [8]. So the recommendation is: log out/reconnect to the session after every GPU query; or wait for gp_vmem_idle_resource_timeout before you run another GPU query (you can also set it to a lower value).
+
Fit
The fit (training) function has the following format:
TEXT. Name of the table containing the training data. This is the name of the output table from the data preprocessor. Independent and dependent variables are specified in the preprocessor step which is why you do not need to explictly state them here as part of the fit function.
+
+
+
model
+
TEXT. Name of the output table containing the model. Details of the output table are shown below.
+
+
+
model_arch_table
+
TEXT. Name of the table containing the model architecture and (optionally) initial weights to use for training.
+
+
+
model_id
+
INTEGER. This is the id in 'model_arch_table' containing the model architecture and (optionally) initial weights to use for training.
+
+
+
compile_params
+
TEXT. Parameters passed to the compile method of the Keras model class [4]. These parameters will be passed through as is so they must conform to the Keras API definition. As an example, you might use something like: loss='categorical_crossentropy', optimizer='adam', metrics=['acc']. The mandatory parameters that must be specified are 'optimizer' and 'loss'. Others are optional and will use the default values as per Keras if not specified here. Also, when specifying 'loss' and 'metrics' do not include the module and submodule prefixes like loss='losses.categorical_crossentropy' or optimizer='keras.optmizers.adam'.
+
Note
+
Custom loss functions and custom metrics can be used as defined in Define Custom Functions. List the custom function name and provide the name of the table where the serialized Python objects reside using the parameter 'object_table' below.
+
The following loss function is not supported: sparse_categorical_crossentropy. The following metrics are not supported: sparse_categorical_accuracy, sparse_top_k_categorical_accuracy.
+
The Keras accuracy parameter top_k_categorical_accuracy returns top 5 accuracy by default. If you want a different top k value, use the helper function Top k Accuracy Function to create a custom Python function to compute the top k accuracy that you want.
+
+
+
+
+
fit_params
+
TEXT. Parameters passed to the fit method of the Keras model class [4]. These will be passed through as is so they must conform to the Keras API definition. As an example, you might use something like: batch_size=128, epochs=4. There are no mandatory parameters so if you specify NULL, it will use all default values as per Keras.
+
Note
Callbacks are not currently supported except for TensorBoard which you can specify in the usual way, e.g., callbacks=[TensorBoard(log_dir="/tmp/logs/fit")]
+
+
num_iterations
+
INTEGER. Number of iterations to train.
+
+
+
use_gpus (optional)
+
BOOLEAN, default: FALSE (i.e., CPU). Determines whether GPUs are to be used for training the neural network. Set to TRUE to use GPUs.
+
Note
This parameter must not conflict with how the distribution rules are set in the preprocessor function. For example, if you set a distribution rule to use certain segments on hosts that do not have GPUs attached, you will get an error if you set ‘use_gpus’ to TRUE. Also, we have seen some memory related issues when segments share GPU resources. For example, if you have 1 GPU per segment host and your cluster has 4 segments per segment host, it means that all 4 segments will share the same GPU on each host. The current recommended configuration is 1 GPU per segment.
+
+
validation_table (optional)
+
TEXT, default: none. Name of the table containing the validation dataset. Note that the validation dataset must be preprocessed in the same way as the training dataset, so this is the name of the output table from running the data preprocessor on the validation dataset. Using a validation dataset can mean a longer training time, depending on its size. This can be controlled using the 'metrics_compute_frequency' paremeter described below.
+
+
+
metrics_compute_frequency (optional)
+
INTEGER, default: once at the end of training after 'num_iterations'. Frequency to compute per-iteration metrics for the training dataset and validation dataset (if specified). There can be considerable cost to computing metrics every iteration, especially if the training dataset is large. This parameter is a way of controlling the frequency of those computations. For example, if you specify 5, then metrics will be computed every 5 iterations as well as at the end of training after 'num_iterations'. If you use the default, metrics will be computed only once after 'num_iterations' have completed.
+
+
+
warm_start (optional)
+
BOOLEAN, default: FALSE. Initalize weights with the coefficients from the last call of the fit function. If set to TRUE, weights will be initialized from the model table generated by the previous training run.
+
Note
The warm start feature works based on the name of the model output table from a previous training run. When using warm start, do not drop the model output table or the model output summary table before calling the fit function, since these are needed to obtain the weights from the previous run. If you are not using warm start, the model output table and the model output table summary must be dropped in the usual way before calling the training function.
+
+
name (optional)
+
TEXT, default: NULL. Free text string to identify a name, if desired.
+
+
+
description (optional)
+
TEXT, default: NULL. Free text string to provide a description, if desired.
+
+
+
object_table (optional)
+
TEXT, default: NULL. Name of the table that contains the custom functions. Note that this table must be created using the Define Custom Functions method. Do not qualify with a schema name, since schema will be automatically pulled from the function definition.
+
+
Output tables
+ The model table produced by fit contains the following columns:
+
+
model_weights
BYTEA8. Byte array containing the weights of the neural net.
+
+
model_arch
TEXT. A JSON representation of the model architecture used in training.
+
+
A summary table named <model>_summary is also created, which has the following columns:
+
+
source_table
Source table used for training.
+
+
model
Model output table produced by training.
+
+
dependent_varname
Dependent variable column from the original source table in the data preprocessing step.
+
+
independent_varname
Independent variables column from the original source table in the data preprocessing step.
+
+
model_arch_table
Name of the table containing the model architecture and (optionally) the initial model weights.
+
+
model_id
The id of the model in the model architecture table used for training.
+
+
compile_params
Compile parameters passed to Keras.
+
+
fit_params
Fit parameters passed to Keras.
+
+
num_iterations
Number of iterations of training completed.
+
+
validation_table
Name of the table containing the validation dataset (if specified).
+
+
metrics_compute_frequency
Frequency that per-iteration metrics are computed for the training dataset and validation dataset.
+
+
name
Name of the training run (free text).
+
+
description
Description of the training run (free text).
+
+
model_type
General identifier for type of model trained. Currently says 'madlib_keras'.
+
+
model_size
Size of the model in KB. Models are stored in 'bytea' data format which is used for binary strings in PostgreSQL type databases.
+
+
start_training_time
Timestamp for start of training.
+
+
end_training_time
Timestamp for end of training.
+
+
metrics_elapsed_time
Array of elapsed time for metric computations as per the 'metrics_compute_frequency' parameter. Useful for drawing a curve showing loss, accuracy or other metrics as a function of time. For example, if 'metrics_compute_frequency=5' this would be an array of elapsed time for every 5th iteration, plus the last iteration. Note that this field reports the time for training + validation, if there is a validation table provided.
+
+
madlib_version
Version of MADlib used.
+
+
num_classes
Count of distinct classes values used for each dependent variable.
+
+
dependent_vartype
Data type for each dependent variable.
+
+
normalizing_constant
Normalizing constant used from the data preprocessing step.
+
+
metrics_type
Metric specified in the 'compile_params'.
+
+
loss_type
Loss specified in the 'compile_params'.
+
+
training_metrics_final
Final value of the training metric after all iterations have completed. The metric reported is the one specified in the 'metrics_type' parameter.
+
+
training_loss_final
Final value of the training loss after all iterations have completed.
+
+
training_metrics
Array of training metrics as per the 'metrics_compute_frequency' parameter. For example, if 'metrics_compute_frequency=5' this would be an array of metrics for every 5th iteration, plus the last iteration.
+
+
training_loss
Array of training losses as per the 'metrics_compute_frequency' parameter. For example, if 'metrics_compute_frequency=5' this would be an array of losses for every 5th iteration, plus the last iteration.
+
+
validation_metrics_final
Final value of the validation metric after all iterations have completed. The metric reported is the one specified in the 'metrics_type' parameter.
+
+
validation_loss_final
Final value of the validation loss after all iterations have completed.
+
+
validation_metrics
Array of validation metrics as per the 'metrics_compute_frequency' parameter. For example, if 'metrics_compute_frequency=5' this would be an array of metrics for every 5th iteration, plus the last iteration.
+
+
validation_loss
Array of validation losses as per the 'metrics_compute_frequency' parameter. For example, if 'metrics_compute_frequency=5' this would be an array of losses for every 5th iteration, plus the last iteration.
+
+
metrics_iters
Array indicating the iterations for which metrics are calculated, as derived from the parameters 'num_iterations' and 'metrics_compute_frequency'. For example, if 'num_iterations=5' and 'metrics_compute_frequency=2', then 'metrics_iters' value would be {2,4,5} indicating that metrics were computed at iterations 2, 4 and 5 (at the end). If 'num_iterations=5' and 'metrics_compute_frequency=1', then 'metrics_iters' value would be {1,2,3,4,5} indicating that metrics were computed at every iteration.
+
+
<dependent_varname>_class_values
Array of class values used for a particular dependent variable. A column will be generated for each dependent variable.
TEXT. Name of the table containing the model to use for validation.
+
+
+
test_table
+
TEXT. Name of the table containing the evaluation dataset. Note that test/validation data must be preprocessed in the same way as the training dataset, so this is the name of the output table from the data preprocessor. Independent and dependent variables are specified in the preprocessor step which is why you do not need to explictly state them here as part of the fit function.
+
+
+
output_table
+
TEXT. Name of table that validation output will be written to. Table contains:
+
+
loss
Loss value on evaluation dataset, where 'loss_type' below identifies the type of loss.
+
+
metric
Metric value on evaluation dataset, where 'metrics_type' below identifies the type of metric.
+
+
metrics_type
Type of metric function that was used in the training step. (It means you cannot have a different metric in evaluate compared to training.)
+
+
loss_type
Type of loss function that was used in the training step. (It means you cannot have a different loss in evaluate compared to training.)
+
+
+
+
+
use_gpus (optional)
+
BOOLEAN, default: FALSE (i.e., CPU). Determines whether GPUs are to be used for training the neural network. Set to TRUE to use GPUs.
+
Note
This parameter must not conflict with how the distribution rules are set in the preprocessor function. For example, if you set a distribution rule to use certain segments on hosts that do not have GPUs attached, you will get an error if you set ‘use_gpus’ to TRUE. Also, we have seen some memory related issues when segments share GPU resources. For example, if you have 1 GPU per segment host and your cluster has 4 segments per segment host, it means that all 4 segments will share the same GPU on each host. The current recommended configuration is 1 GPU per segment.
TEXT. Name of the table containing the model to use for prediction.
+
+
+
test_table
+
TEXT. Name of the table containing the dataset to predict on. Note that test data is not preprocessed (unlike fit and evaluate) so put one test image per row for prediction. Also see the comment below for the 'independent_varname' parameter regarding normalization.
+
+
+
+
id_col
+
TEXT. Name of the id column in the test data table.
+
+
+
independent_varname
+
TEXT. Column with independent variables in the test table. If a 'normalizing_const' is specified when preprocessing the training dataset, this same normalization will be applied to the independent variables used in predict. In the case that there are multiple independent variables, representing a multi-input neural network, put the columns as a comma separated list, e.g., 'indep_var1, indep_var2, indep_var3' in the same way as was done in the preprocessor step for the training data.
+
+
+
output_table
+
TEXT. Name of the table that prediction output will be written to. Table contains:
+
+
id
Gives the 'id' for each prediction, corresponding to each row from the test_table.
+
+
class_name
Name of variable being predicted.
+
+
class_value
Estimated class value.
+
+
prob
Probability of a given class value.
+
+
rank
The rank of a given class based on the ordering of probabilities.
+
+
+
+
+
pred_type (optional)
+
TEXT or INTEGER or DOUBLE PRECISION default: 'prob'. The type and range of output desired. This parameter allows the following options.
+
'response': the actual prediction
+
'prob': the probability value for each class
+
0<value<1: the lower limit for the probability (double precision)
+
1<=value: the lower limit for the rank of the prediction (integer)
+
+
+
+
use_gpus (optional)
+
BOOLEAN, default: FALSE (i.e., CPU). Flag to enable GPU support for training neural network. The number of GPUs to use is determined by the parameters passed to the preprocessor.
+
Note
We have seen some memory related issues when segments share GPU resources. For example, if you provide 1 GPU and your database cluster is set up to have 4 segments per segment host, it means that all 4 segments on a segment host will share the same GPU. The current recommended configuration is 1 GPU per segment.
+
+
+
Predict BYOM (bring your own model)
The predict BYOM function allows you to do inference on models that have not been trained on MADlib, but rather imported from elsewhere. It has the following format:
TEXT. Name of the architecture table containing the model to use for prediction. The model weights and architecture can be loaded to this table by using the Define Model Architectures function.
+
+
+
model_id
+
INTEGER. This is the id in 'model_arch_table' containing the model architecture and model weights to use for prediction.
+
+
+
test_table
+
TEXT. Name of the table containing the dataset to predict on. Note that test data is not preprocessed (unlike fit and evaluate) so put one test image per row for prediction. Set the 'normalizing_const' below for the independent variable if necessary.
+
+
+
id_col
+
TEXT. Name of the id column in the test data table.
+
+
+
independent_varname
+
TEXT. Column with independent variables in the test table. Set the 'normalizing_const' below if necessary.
+
+
+
output_table
+
TEXT. Name of the table that prediction output will be written to. Table contains:
+
+
id
Gives the 'id' for each prediction, corresponding to each row from the test_table.
+
+
class_name
The estimated variable.
+
+
class_value
The estimated class for classification.
+
+
prob
Probability of a given class.
+
+
rank
The rank of a given class based on the ordering of probabilities.
+
+
+
+
+
pred_type (optional)
+
TEXT, default: 'response'. The type of output desired, where 'response' gives the actual prediction and 'prob' gives the probability value for each class.
+
+
+
use_gpus (optional)
+
BOOLEAN, default: FALSE (i.e., CPU). Flag to enable GPU support for training neural network. The number of GPUs to use is determined by the parameters passed to the preprocessor.
+
Note
We have seen some memory related issues when segments share GPU resources. For example, if you provide 1 GPU and your database cluster is set up to have 4 segments per segment host, it means that all 4 segments on a segment host will share the same GPU. The current recommended configuration is 1 GPU per segment.
+
+
class_values (optional)
+
TEXT[], default: NULL. Two dimensional list of class labels that were used while training the model for each dependent variable.
+
Note
If you specify the class values parameter, it must reflect how the dependent variable was 1-hot encoded for training. If you accidently pick another order that does not match the 1-hot encoding, the predictions would be wrong.
+
+
normalizing_const (optional)
+
DOUBLE PRECISION, default: 1.0. The normalizing constant to divide each value in the 'independent_varname' array by. For example, you would use 255 for this value if the image data is in the form 0-255.
+
+
+
dependent_count (optional)
+
INTEGER, default: 1. The number of dependent variables in the model.
Predict. We will use the validation dataset for prediction as well, which is not usual but serves to show the syntax. The prediction is in the 'estimated_class_text' column:
+DROP TABLE IF EXISTS iris_predict;
+SELECT madlib.madlib_keras_predict('iris_model', -- model
+ 'iris_test', -- test_table
+ 'id', -- id column
+ 'attributes', -- independent var
+ 'iris_predict' -- output table
+ );
+SELECT * FROM iris_predict ORDER BY id, rank;
+
+SELECT COUNT(*) FROM iris_predict JOIN iris_test USING (id)
+WHERE iris_predict.class_value != iris_test.class_text AND iris_predict.rank = 1;
+
+ count
+-------+
+ 9
+(1 row)
+
Accuracy:
+SELECT round(count(*)*100/(150*0.2),2) as test_accuracy_percent from
+ (select iris_test.class_text as actual, iris_predict.class_value as estimated
+ from iris_predict inner join iris_test
+ on iris_test.id=iris_predict.id where iris_predict.rank = 1) q
+WHERE q.actual=q.estimated;
+
Predict BYOM. We will use the validation dataset for prediction as well, which is not usual but serves to show the syntax. See Define Model Architectures for details on how to load the model architecture and weights. In this example we will use weights we already have:
Now train using a model from the model architecture table directly without referencing the model table from the MADlib training. Note that if you specify the class values parameter as we do below, it must reflect how the dependent variable was 1-hot encoded for training. In this example the 'training_preprocessor_dl()' in Step 2 above encoded in the order {'Iris-setosa', 'Iris-versicolor', 'Iris-virginica'} so this is the order we pass in the parameter. If we accidently pick another order that does not match the 1-hot encoding, the predictions would be wrong.
+DROP TABLE IF EXISTS iris_predict_byom;
+SELECT madlib.madlib_keras_predict_byom('model_arch_library', -- model arch table
+ 1, -- model arch id
+ 'iris_test', -- test_table
+ 'id', -- id column
+ 'attributes', -- independent var
+ 'iris_predict_byom', -- output table
+ 'response', -- prediction type
+ FALSE, -- use GPUs
+ ARRAY[ARRAY['Iris-setosa', 'Iris-versicolor', 'Iris-virginica']], -- class values
+ 1.0 -- normalizing const
+ );
+SELECT * FROM iris_predict_byom ORDER BY id;
+
The prediction is in the 'estimated_dependent_var' column:
+SELECT COUNT(*) FROM iris_predict_byom JOIN iris_test USING (id)
+WHERE iris_predict_byom.class_value != iris_test.class_text;
+
+ count
+-------+
+ 9
+(1 row)
+
Accuracy:
+SELECT round(count(*)*100/(150*0.2),2) as test_accuracy_percent from
+ (select iris_test.class_text as actual, iris_predict_byom.class_value as estimated
+ from iris_predict_byom inner join iris_test
+ on iris_test.id=iris_predict_byom.id) q
+WHERE q.actual=q.estimated;
+
Validation dataset. Now use a validation dataset and compute metrics every 3rd iteration using the 'metrics_compute_frequency' parameter. This can help reduce run time if you do not need metrics computed at every iteration.
+DROP TABLE IF EXISTS iris_model, iris_model_summary;
+SELECT madlib.madlib_keras_fit('iris_train_packed', -- source table
+ 'iris_model', -- model output table
+ 'model_arch_library', -- model arch table
+ 1, -- model arch id
+ $$ loss='categorical_crossentropy', optimizer='adam', metrics=['categorical_accuracy'] $$, -- compile_params
+ $$ batch_size=5, epochs=3 $$, -- fit_params
+ 10, -- num_iterations
+ FALSE, -- use GPUs
+ 'iris_test_packed', -- validation dataset
+ 3, -- metrics compute frequency
+ FALSE, -- warm start
+ 'Sophie L.', -- name
+ 'Simple MLP for iris dataset' -- description
+ );
+\x on
+SELECT * FROM iris_model_summary;
+
+DROP TABLE IF EXISTS iris_predict;
+SELECT madlib.madlib_keras_predict('iris_model', -- model
+ 'iris_test', -- test_table
+ 'id', -- id column
+ 'attributes', -- independent var
+ 'iris_predict', -- output table
+ 'prob' -- response type
+ );
+\x off
+SELECT * FROM iris_predict ORDER BY id;
+
Warm start. Next, use the warm_start parameter to continue learning, using the coefficients from the run above. Note that we don't drop the model table or model summary table:
+SELECT madlib.madlib_keras_fit('iris_train_packed', -- source table
+ 'iris_model', -- model output table
+ 'model_arch_library', -- model arch table
+ 1, -- model arch id
+ $$ loss='categorical_crossentropy', optimizer='adam', metrics=['categorical_accuracy'] $$, -- compile_params
+ $$ batch_size=5, epochs=3 $$, -- fit_params
+ 5, -- num_iterations
+ FALSE, -- use GPUs
+ 'iris_test_packed', -- validation dataset
+ 1, -- metrics compute frequency
+ TRUE, -- warm start
+ 'Sophie L.', -- name
+ 'Simple MLP for iris dataset' -- description
+ );
+\x on
+SELECT * FROM iris_model_summary;
+
Note that the loss and accuracy values pick up from where the previous run left off.
+
+
Transfer Learning
+
Here we want to start with initial weights from a pre-trained model rather than training from scratch. We also want to use a model architecture with the earlier feature layer(s) frozen to save on training time. The example below is somewhat contrived but gives you the idea of the steps.
+
+
Define and load a model architecture with the 1st hidden layer frozen:
Fetch the weights from a previous MADlib run. (Normally these would be downloaded from a source that trained the same model architecture on a related dataset.)
Refer to the deep learning section of the Apache MADlib wiki [5] for important information including supported libraries and versions.
+
Classification is currently supported, not regression.
+
Reminder about the distinction between warm start and transfer learning. Warm start uses model state (weights) from the model output table from a previous training run - set the 'warm_start' parameter to TRUE in the fit function. Transfer learning uses initial model state (weights) stored in the 'model_arch_table' - in this case set the 'warm_start' parameter to FALSE in the fit function.
+
+
Technical Background
+
For an introduction to deep learning foundations, including MLP and CNN, refer to [6].
+
This module trains a single large model across the database cluster using the bulk synchronous parallel (BSP) approach, with model averaging [7].
+
On the effect of database cluster size: as the database cluster size increases, the per iteration loss will be higher since the model only sees 1/n of the data, where n is the number of segments. However, each iteration runs faster than single node because it is only traversing 1/n of the data. For highly non-convex solution spaces, convergence behavior may diminish as cluster size increases. Ensure that each segment has sufficient volume of data and examples of each class value.
+
Alternatively, to train multiple models at the same time for model architecture search or hyperparameter tuning, you can use the methods in Train Multiple Models, which does not do model averaging and hence may have better covergence efficiency.
[6] Deep Learning, Ian Goodfellow, Yoshua Bengio and Aaron Courville, MIT Press, 2016.
+
[7] "Resource-Efficient and Reproducible Model Selection on Deep Learning Systems," Supun Nakandala, Yuhao Zhang, and Arun Kumar, Technical Report, Computer Science and Engineering, University of California, San Diego https://adalabucsd.github.io/papers/TR_2019_Cerebro.pdf.
This function loads model architectures and weights into a table for use by deep learning algorithms.
+
Model architecture is in JSON form and model weights are in the form of PostgreSQL binary data types (bytea). If the output table already exists, a new row is inserted into the table so it can act as a repository for multiple model architectures and weights.
+
There is also a function to delete a model from the table.
+
MADlib's deep learning methods are designed to use the TensorFlow package and its built in Keras functions. To ensure consistency, please use tensorflow.keras objects (models, layers, etc.) instead of importing Keras and using its objects.
VARCHAR. Output table to load keras model architecture and weights.
+
+
+
model_arch
+
JSON. JSON of the model architecture to load.
Note
Please note that every input layer must have the 'input_shape' stated explicitly in the model architecture. MADlib has this requirement because, in some cases, the JSON representation may not have the input shape by default and it has to be read from the JSON for fit() type functions.
+
+
+
model_weights (optional)
+
bytea. Model weights to load as a PostgreSQL binary data type.
+
+
+
name (optional)
+
TEXT, default: NULL. Free text string to provide a name, if desired.
+
+
+
description (optional)
+
TEXT, default: NULL. Free text string to provide a description, if desired.
+
+
+
+
Output table
+ The output table contains the following columns:
+
+
model_id
SERIAL PRIMARY KEY. Model ID.
+
+
model_arch
JSON. JSON blob of the model architecture.
+
+
model_weights
BYTEA. Weights of the model which may be used for warm start or transfer learning. Weights are stored as a PostgreSQL binary data type.
+
+
name
TEXT. Name of model (free text).
+
+
description
TEXT. Description of model (free text).
+
+
__internal_madlib_id__
TEXT. Unique id for model arch. This is an id used internally be MADlib.
Load model weights. To load weights from previous MADlib run, use UPDATE to load directly into the table. For example, if 'model_weights' are the weights in the output table 'iris_model' from a previous run of 'madlib_keras_fit()' :
+UPDATE model_arch_library SET model_weights = model_weights FROM iris_model WHERE model_id = 2;
+SELECT model_id, name, description, (model_weights IS NOT NULL) AS has_model_weights FROM model_arch_library ORDER BY model_id;
+
+ model_id | name | description | has_model_weights
+----------+--------+---------------------+-------------------
+ 1 | Sophie | A simple model | f
+ 2 | Maria | Also a simple model | t
+
+
To load weights from tensorflow.keras using a PL/Python function, we need to flatten then serialize the weights to store as a PostgreSQL binary data type. Byte format is more efficient on space and memory compared to a numeric array. The model weights will be de-serialized when passed to Keras functions.
+CREATE OR REPLACE FUNCTION load_weights() RETURNS VOID AS
+$$
+import keras
+from tensorflow.keras.layers import *
+from tensorflow.keras import Sequential
+import numpy as np
+import plpy
+#
+# create model
+model = Sequential()
+model.add(Dense(10, activation='relu', input_shape=(4,)))
+model.add(Dense(10, activation='relu'))
+model.add(Dense(3, activation='softmax'))
+#
+# get weights, flatten and serialize
+weights = model.get_weights()
+weights_flat = [w.flatten() for w in weights]
+weights1d = np.concatenate(weights_flat).ravel()
+weights_bytea = weights1d.tostring()
+#
+# load query
+load_query = plpy.prepare("""SELECT madlib.load_keras_model(
+ 'model_arch_library',
+ $1, $2)
+ """, ['json','bytea'])
+plpy.execute(load_query, [model.to_json(), weights_bytea])
+$$ language plpython3u;
+-- Call load function
+SELECT load_weights();
+SELECT model_id, name, description, (model_weights IS NOT NULL) AS has_model_weights FROM model_arch_library ORDER BY model_id;
+
+ model_id | name | description | has_model_weights
+----------+--------+---------------------+-------------------
+ 1 | Sophie | A simple model | f
+ 2 | Maria | Also a simple model | t
+ 3 | Ella | Model x | t
+
+
Load weights from tensorflow.keras using psycopg2. (Psycopg is a PostgreSQL database adapter for the Python programming language.) As above we need to flatten then serialize the weights to store as a PostgreSQL binary data type. Note that the psycopg2.Binary function used below will increase the size of the Python object for the weights, so if your model is large it might be better to use a PL/Python function as above.
+import psycopg2
+import psycopg2 as p2
+conn = p2.connect('postgresql://gpadmin@35.239.240.26:5432/madlib')
+cur = conn.cursor()
+from tensorflow.keras.layers import *
+from tensorflow.keras import Sequential
+import numpy as np
+#
+# create model
+model = Sequential()
+model.add(Dense(10, activation='relu', input_shape=(4,)))
+model.add(Dense(10, activation='relu'))
+model.add(Dense(3, activation='softmax'))
+#
+# get weights, flatten and serialize
+weights = model.get_weights()
+weights_flat = [w.flatten() for w in weights]
+weights1d = np.concatenate(weights_flat).ravel()
+weights_bytea = psycopg2.Binary(weights1d.tostring())
+#
+# load query
+query = "SELECT madlib.load_keras_model('model_arch_library', %s,%s)"
+cur.execute(query,[model.to_json(),weights_bytea])
+conn.commit()
+SELECT model_id, name, description, (model_weights IS NOT NULL) AS has_model_weights FROM model_arch_library ORDER BY model_id;
+
+ model_id | name | description | has_model_weights
+----------+--------+---------------------+-------------------
+ 1 | Sophie | A simple model | f
+ 2 | Maria | Also a simple model | t
+ 3 | Ella | Model x | t
+ 4 | Grace | Model y | t
+
+
Delete one of the models:
+SELECT madlib.delete_keras_model('model_arch_library', -- Output table
+ 1 -- Model id
+ );
+SELECT model_id, name, description, (model_weights IS NOT NULL) AS has_model_weights FROM model_arch_library ORDER BY model_id;
+
+ model_id | name | description | has_model_weights
+----------+-------+---------------------+-------------------
+ 2 | Maria | Also a simple model | t
+ 3 | Ella | Model x | t
+ 4 | Grace | Model y | t
+
This module allows you to explore network architectures and hyperparameters by training many models a time across the database cluster.
+
The aim is to support efficient empirical comparison of multiple training configurations. This process is called model selection, and the implementation here is based on a parallel execution strategy called model hopper parallelism (MOP) [1,2].
+
Models are designed in Keras [3], which is a high-level neural network API written in Python. It can run on top of different backends and the one that is currently supported by MADlib is TensorFlow [4].
+
The main use case is classification using sequential models, which are made up of a linear stack of layers. This includes multilayer perceptrons (MLPs) and convolutional neural networks (CNNs). Regression is not currently supported.
+
Before using this model selection method, you will need to Prepocess Data which prepares data for use by models that support mini-batch as an optimization option. This is a one-time operation and you would only need to re-run the preprocessor if your input data has changed. The advantage of using mini-batching is that it can perform better than stochastic gradient descent because it uses more than one training example at a time, typically resulting faster and smoother convergence [5].
+
You can set up the models and hyperparameters to try with the Define Model Configurations function to define the unique combinations of model architectures, compile and fit parameters.
+
Note
1. If 'madlib_keras_fit_multiple_model()' is running on GPDB 5 and some versions of GPDB 6, the database will keep adding to the disk space (in proportion to model size) and will only release the disk space once the fit multiple query has completed execution. This is not the case for GPDB 6.5.0+ where disk space is released during the fit multiple query.
+
+2. CUDA GPU memory cannot be released until the process holding it is terminated. When a MADlib deep learning function is called with GPUs, Greenplum internally creates a process (called a slice) which calls TensorFlow to do the computation. This process holds the GPU memory until one of the following two things happen: query finishes and user logs out of the Postgres client/session; or, query finishes and user waits for the timeout set by gp_vmem_idle_resource_timeout. The default value for this timeout is 18 sec [8]. So the recommendation is: log out/reconnect to the session after every GPU query; or wait for gp_vmem_idle_resource_timeout before you run another GPU query (you can also set it to a lower value).
+
+3. This method does not currently support multi-input or multi-output neural networks.
+
Fit
The fit (training) function has the following format:
TEXT. Name of the table containing the training data. This is the name of the output table from the data preprocessor. Independent and dependent variables are specified in the preprocessor step which is why you do not need to explictly state them here as part of the fit function.
+
+
+
model_output_table
+
TEXT. Name of the output table containing the multiple models created.
Note
pg_temp is not allowed as an output table schema for fit multiple. Details of output tables are shown below.
+
+
model_selection_table
+
TEXT. Name of the table containing model selection parameters to be tried. Here we mean both hyper-parameter tuning and model architecture search.
+
+
+
num_iterations
+
INTEGER. Number of iterations to train.
+
Note
This parameter is different than the number of passes over the dataset, which is commonly referred to as the number of epochs. Since MADlib operates in a distributed system, the number of epochs is actually equal to this parameter 'num_iterations' X 'epochs' as specified in the Keras fit parameter.
+
+
use_gpus (optional)
+
BOOLEAN, default: FALSE (i.e., CPU). Determines whether GPUs are to be used for training the neural network. Set to TRUE to use GPUs.
+
Note
This parameter must not conflict with how the distribution rules are set in the preprocessor function. For example, if you set a distribution rule to use certain segments on hosts that do not have GPUs attached, you will get an error if you set ‘use_gpus’ to TRUE. Also, we have seen some memory related issues when segments share GPU resources. For example, if you have 1 GPU per segment host and your cluster has 4 segments per segment host, it means that all 4 segments will share the same GPU on each host. The current recommended configuration is 1 GPU per segment.
+
+
validation_table (optional)
+
TEXT, default: none. Name of the table containing the validation dataset. Note that the validation dataset must be preprocessed in the same way as the training dataset, so this is the name of the output table from running the data preprocessor on the validation dataset. Using a validation dataset can mean a longer training time, depending on its size. This can be controlled using the 'metrics_compute_frequency' parameter described below.
+
+
+
metrics_compute_frequency (optional)
+
INTEGER, default: once at the end of training after 'num_iterations'. Frequency to compute per-iteration metrics for the training dataset and validation dataset (if specified). There can be considerable cost to computing metrics every iteration, especially if the training dataset is large. This parameter is a way of controlling the frequency of those computations. For example, if you specify 5, then metrics will be computed every 5 iterations as well as at the end of training after 'num_iterations'. If you use the default, metrics will be computed only once after 'num_iterations' have completed.
+
+
+
warm_start (optional)
+
BOOLEAN, default: FALSE. Initalize weights with the coefficients from the last call to the fit function. If set to TRUE, weights will be initialized from the model table generated by the previous training run.
+
Note
The warm start feature works based on the name of the model output table from a previous training run. When using warm start, do not drop the model output table or the model output summary table before calling the fit function, since these are needed to obtain the weights from the previous run. If you are not using warm start, the model output table and the model output table summary must be dropped in the usual way before calling the training function.
+
+
name (optional)
+
TEXT, default: NULL. Free text string to identify a name, if desired.
+
+
+
description (optional)
+
TEXT, default: NULL. Free text string to provide a description, if desired.
+
+
+
use_caching (optional)
+
BOOLEAN, default: FALSE. Use caching of data in memory on the segment in order to speed up processing.
+
Note
When set to TRUE, byte arrays on each segment are maintained in cache (SD). This can speed up training significantly, however the memory usage per segment increases. In effect, it requires enough available memory on a segment so that all training data residing on that segment can be read into memory.
+
+
+
Output tables
+ The model output table produced by fit contains the following columns. There is one row per model as per the rows in the 'model_selection_table':
+
+
mst_key
INTEGER. ID that defines a unique tuple for model architecture-compile parameters-fit parameters, as defined in the 'model_selection_table'.
+
+
model_weights
BYTEA8. Byte array containing the weights of the neural net.
+
+
model_arch
TEXT. A JSON representation of the model architecture used in training.
+
+
An info table named <model_output_table>_info is also created, which has the following columns. There is one row per model as per the rows in the 'model_selection_table':
+
+
mst_key
INTEGER. ID that defines a unique tuple for model architecture-compile parameters-fit parameters, as defined in the 'model_selection_table'.
+
+
model_id
INTEGER. ID that defines model in the 'model_arch_table'.
+
+
compile_params
Compile parameters passed to Keras.
+
+
fit_params
Fit parameters passed to Keras.
+
+
model_type
General identifier for type of model trained. Currently says 'madlib_keras'.
+
+
model_size
Size of the model in KB. Models are stored in 'bytea' data format which is used for binary strings in PostgreSQL type databases.
+
+
metrics_elapsed_time
Array of elapsed time for metric computations as per the 'metrics_compute_frequency' parameter. Useful for drawing a curve showing loss, accuracy or other metrics as a function of time. For example, if 'metrics_compute_frequency=5' this would be an array of elapsed time for every 5th iteration, plus the last iteration. Note that this field reports the time for training + validation if there is a validation table provided.
+
+
metrics_type
Metric specified in the 'compile_params'.
+
+
loss_type
Loss specified in the 'compile_params'.
+
+
training_metrics_final
Final value of the training metric after all iterations have completed. The metric reported is the one specified in the 'metrics_type' parameter.
+
+
training_loss_final
Final value of the training loss after all iterations have completed.
+
+
training_metrics
Array of training metrics as per the 'metrics_compute_frequency' parameter. For example, if 'metrics_compute_frequency=5' this would be an array of metrics for every 5th iteration, plus the last iteration.
+
+
training_loss
Array of training losses as per the 'metrics_compute_frequency' parameter. For example, if 'metrics_compute_frequency=5' this would be an array of losses for every 5th iteration, plus the last iteration.
+
+
validation_metrics_final
Final value of the validation metric after all iterations have completed. The metric reported is the one specified in the 'metrics_type' parameter.
+
+
validation_loss_final
Final value of the validation loss after all iterations have completed.
+
+
validation_metrics
Array of validation metrics as per the 'metrics_compute_frequency' parameter. For example, if 'metrics_compute_frequency=5' this would be an array of metrics for every 5th iteration, plus the last iteration.
+
+
validation_loss
Array of validation losses as per the 'metrics_compute_frequency' parameter. For example, if 'metrics_compute_frequency=5' this would be an array of losses for every 5th iteration, plus the last iteration.
+
+
+
+
A summary table named <model>_summary is also created, which has the following columns:
+
+
source_table
Source table used for training.
+
+
validation_table
Name of the table containing the validation dataset (if specified).
+
+
model
Name of the output table containing the model for each model selection tuple.
+
+
model_info
Name of the output table containing the model performance and other info for each model selection tuple.
+
+
dependent_varname
Dependent variable column from the original source table in the data preprocessing step.
+
+
independent_varname
Independent variables column from the original source table in the data preprocessing step.
+
+
model_arch_table
Name of the table containing the model architecture and (optionally) the initial model weights.
+
+
num_iterations
Number of iterations of training completed.
+
+
metrics_compute_frequency
Frequency that per-iteration metrics are computed for the training dataset and validation datasets.
+
+
warm_start
Indicates whether warm start used or not.
+
+
name
Name of the training run (free text).
+
+
description
Description of the training run (free text).
+
+
start_training_time
Timestamp for start of training.
+
+
end_training_time
Timestamp for end of training.
+
+
madlib_version
Version of MADlib used.
+
+
num_classes
Count of distinct classes values used.
+
+
<dependent_varname>_class_values
Array of actual class values used for a particular dependent variable. A column will be generated for each dependent variable.
+
+
dependent_vartype
Data type of the dependent variable.
+
+
normalizing_constant
Normalizing constant used from the data preprocessing step.
+
+
metrics_iters
Array indicating the iterations for which metrics are calculated, as derived from the parameters 'num_iterations' and 'metrics_compute_frequency'. For example, if 'num_iterations=5' and 'metrics_compute_frequency=2', then 'metrics_iters' value would be {2,4,5} indicating that metrics were computed at iterations 2, 4 and 5 (at the end). If 'num_iterations=5' and 'metrics_compute_frequency=1', then 'metrics_iters' value would be {1,2,3,4,5} indicating that metrics were computed at every iteration.
TEXT. Name of the table containing the model to use for validation.
+
+
+
test_table
+
TEXT. Name of the table containing the evaluation dataset. Note that test/validation data must be preprocessed in the same way as the training dataset, so this is the name of the output table from the data preprocessor. Independent and dependent variables are specified in the preprocessor step which is why you do not need to explictly state them here as part of the fit function.
+
+
+
output_table
+
TEXT. Name of table that validation output will be written to. Table contains:
+
+
loss
Loss value on evaluation dataset, where 'loss_type' below identifies the type of loss.
+
+
metric
Metric value on evaluation dataset, where 'metrics_type' below identifies the type of metric.
+
+
metrics_type
Type of metric used that was used in the training step. (It means you cannot have a different metric in evaluate compared to training.)
+
+
loss_type
Type of loss function that was used in the training step. (It means you cannot have a different loss in evaluate compared to training.)
+
+
+
+
+
use_gpus (optional)
+
BOOLEAN, default: FALSE (i.e., CPU). Determines whether GPUs are to be used for training the neural network. Set to TRUE to use GPUs.
+
Note
This parameter must not conflict with how the distribution rules are set in the preprocessor function. For example, if you set a distribution rule to use certain segments on hosts that do not have GPUs attached, you will get an error if you set ‘use_gpus’ to TRUE. Also, we have seen some memory related issues when segments share GPU resources. For example, if you have 1 GPU per segment host and your cluster has 4 segments per segment host, it means that all 4 segments will share the same GPU on each host. The current recommended configuration is 1 GPU per segment.
+
+
mst_key (optional)
+
INTEGER, default: NULL. ID that defines a unique tuple for model architecture-compile parameters-fit parameters as defined in the model selection table.
TEXT. Name of the table containing the model to use for prediction.
+
+
+
test_table
+
TEXT. Name of the table containing the dataset to predict on. Note that test data is not preprocessed (unlike fit and evaluate) so put one test image per row for prediction. Also see the comment below for the 'independent_varname' parameter regarding normalization.
+
+
+
+
id_col
+
TEXT. Name of the id column in the test data table.
+
+
+
independent_varname
+
TEXT. Column with independent variables in the test table. If a 'normalizing_const' is specified when preprocessing the training dataset, this same normalization will be applied to the independent variables used in predict.
+
+
+
output_table
+
TEXT. Name of the table that prediction output will be written to. Table contains:
+
+
id
Gives the 'id' for each prediction, corresponding to each row from the test_table.
+
+
class_name
The estimated variable.
+
+
class_value
The estimated class for classification.
+
+
prob
Probability of a given class.
+
+
rank
The rank of a given class based on the ordering of probabilities.
+
+
+
+
+
pred_type (optional)
+
TEXT or INTEGER or DOUBLE PRECISION default: 'prob'. The type and range of output desired. This parameter allows the following options.
+
'response': the actual prediction
+
'prob': the probability value for each class
+
0<value<1: the lower limit for the probability (double precision)
+
1<=value: the lower limit for the rank of the prediction (integer)
+
+
+
+
use_gpus (optional)
+
BOOLEAN, default: FALSE (i.e., CPU). Determines whether GPUs are to be used for prediction/inference. Set to TRUE to use GPUs.
+
Note
The prediction function uses the whole cluster. If you are using GPUs, it requires that GPUs are attached to all hosts, and that there are the same number of GPUs on each host (homogeneous cluster). This is different from the fit() and evaluate() functions that support GPUs on only some of the hosts (heterogeneous cluster). Therefore, if you have GPUs only on some of the hosts, or an uneven numbers of GPUs per host, then set this parameter to FALSE to use CPUs.
+
+
mst_key (optional)
+
INTEGER, default: NULL. ID that defines a unique tuple for model architecture-compile parameters-fit parameters as defined in the model selection table.
Generate model configurations using grid search. The output table for grid search contains the unique combinations of model architectures, compile and fit parameters.
+DROP TABLE IF EXISTS mst_table, mst_table_summary;
+SELECT madlib.generate_model_configs(
+ 'model_arch_library', -- model architecture table
+ 'mst_table', -- model selection table output
+ ARRAY[1,2], -- model ids from model architecture table
+ $$
+ {'loss': ['categorical_crossentropy'],
+ 'optimizer_params_list': [ {'optimizer': ['Adam'], 'lr': [0.001, 0.01, 0.1]} ],
+ 'metrics': ['accuracy']}
+ $$, -- compile_param_grid
+ $$
+ { 'batch_size': [4, 8],
+ 'epochs': [1]
+ }
+ $$, -- fit_param_grid
+ 'grid' -- search_type
+ );
+SELECT * FROM mst_table ORDER BY mst_key;
+
Predict. Now predict using one of the models we built. We will use the validation data set for prediction as well, which is not usual but serves to show the syntax. The prediction is in the estimated_class_text column:
+DROP TABLE IF EXISTS iris_predict;
+SELECT madlib.madlib_keras_predict('iris_multi_model', -- model
+ 'iris_test', -- test_table
+ 'id', -- id column
+ 'attributes', -- independent var
+ 'iris_predict', -- output table
+ 'response', -- prediction type
+ FALSE, -- use gpus
+ 3 -- mst_key to use
+ );
+SELECT * FROM iris_predict ORDER BY id;
+
+SELECT COUNT(*) FROM iris_predict JOIN iris_test USING (id)
+WHERE iris_predict.class_value != iris_test.class_text;
+
+ count
+-------+
+ 0
+
Accuracy:
+SELECT round(count(*)*100/(150*0.2),2) as test_accuracy_percent from
+ (select iris_test.class_text as actual, iris_predict.class_value as estimated
+ from iris_predict inner join iris_test
+ on iris_test.id=iris_predict.id) q
+WHERE q.actual=q.estimated;
+
Validation dataset. Now use a validation dataset and compute metrics every 3rd iteration using the 'metrics_compute_frequency' parameter. This can help reduce run time if you do not need metrics computed at every iteration. Also turn on caching.
+DROP TABLE IF EXISTS iris_predict;
+SELECT madlib.madlib_keras_predict('iris_multi_model', -- model
+ 'iris_test', -- test_table
+ 'id', -- id column
+ 'attributes', -- independent var
+ 'iris_predict', -- output table
+ 'prob', -- prediction type
+ FALSE, -- use gpus
+ 3 -- mst_key to use
+ );
+SELECT * FROM iris_predict ORDER BY id, rank;
+
Warm start. Next, use the warm_start parameter to continue learning, using the coefficients from the run above. Note that we don't drop the model table or model summary table:
Note that the loss and accuracy values pick up from where the previous run left off.
+
+
Notes
+
+
Refer to the deep learning section of the Apache MADlib wiki [6] for important information including supported libraries and versions.
+
Classification is currently supported, not regression.
+
Reminder about the distinction between warm start and transfer learning. Warm start uses model state (weights) from the model output table from a previous training run - set the 'warm_start' parameter to TRUE in the fit function. Transfer learning uses initial model state (weights) stored in the 'model_arch_table' - in this case set the 'warm_start' parameter to FALSE in the fit function.
+
Here are some more details on how warm start works. These details are mostly applicable when implementing autoML algorithms on top of MADlib's model selection. In short, the 'model_selection_table' dictates which models get trained and output to the 'model_output_table' and associated summary and info tables. When 'warm_start' is TRUE, models are built for each 'mst_key' in the 'model_selection_table'. If there are prior runs for an 'mst_key' then the weights from that run will be used. If there are no prior runs for an 'mst_key' then random initialization will be used. For example, let's say we start with 'mst_keys' of 1, 2, 3, and 4 in the 'model_selection_table'. We run fit once to get model and info tables for 1, 2, 3, and 4. Then we modify the 'model_selection_table' as part of an autoML scheme, in which we remove the 'mst_key' for 1 and add a new 'mst_key' for 5. Next we run fit with warm start. The result will be models created for 'mst_keys' of 2, 3, 4, and 5. Warm start will be used for 2, 3, and 4 (using prior run) and random initialization will be used for 5 (no prior run). The 'mst_key' of 1 will be dropped.
+
The 'num_iterations' parameter and the Keras fit parameter 'epochs' can substantially affect accuracy and run-time. In general, increasing the number of 'epochs' for a fixed 'num_iterations' will speed up training, but may result in lower accuracy. It's best to do some experimentation to find out what works for your models and dataset.
+
+
Technical Background
+
For an introduction to deep learning foundations, including MLP and CNN, refer to [7].
+
This module trains many models a time across the database cluster in order to explore network architectures and hyperparameters. It uses model hopper parallelism (MOP) and has high convergence efficiency since it does not do model averaging [2].
+
On the effect of database cluster size: as the database cluster size increases, it will be faster to train a set of models, as long as you have at least as many model selection tuples as segments. This is because model state is "hopped" from segment to segment and training takes place in parallel [1,2]. If you have fewer model selection tuples to train than segments, then some segments may not be busy 100% of the time so speedup will not necessarily increase on a larger cluster. Inference (prediction) is an embarrassingly parallel operation so inference runtimes will be proportionally faster as the number of segments increases.
[2] "Cerebro: A Data System for Optimized Deep Learning Model Selection," Supun Nakandala, Yuhao Zhang, and Arun Kumar, Proceedings of the VLDB Endowment (2020), Vol. 13, No. 11 https://adalabucsd.github.io/papers/2020_Cerebro_VLDB.pdf
This module generates model configurations using grid search or random search.
+
Once the configurations are defined, they can be used by the fit function in Train Model Configurations. By model configurations we mean both hyperparameters and model architectures. The output table from this module defines the combinations of model architectures, compile and fit parameters to be trained in parallel.
VARCHAR. Table containing model architectures and weights. For more information on this table refer to the module Define Model Architectures.
+
+
+
model_selection_table
+
VARCHAR. Model selection table created by this module. If this table already exists, it will be appended to. A summary table named <model_selection_table>_summary is also created. Contents of both output tables are described below.
+
+
+
model_id_list
+
INTEGER[]. Array of model IDs from the 'model_arch_table' to be included in the run combinations. For hyperparameter search, this will typically be one model ID. For model architecture search, this will be the different model IDs that you want to compare.
+
+
+
compile_params_grid
+
VARCHAR. String representation of a Python dictionary of compile parameters to be tested. Each entry of the dictionary should consist of keys as compile parameter names, and values as a Python list of compile parameter values to be passed to Keras. Also, optimizer parameters are a nested dictionary to allow different optimizer types to have different parameters or ranges of parameters. Here is an example:
The following types of sampling are supported: 'linear', 'log' and 'log_near_one'. The 'log_near_one' sampling is useful for exponentially weighted average types of parameters like momentum, which are very sensitive to changes near 1. It has the effect of producing more values near 1 than regular log-based sampling.
+
In the case of grid search, omit the sampling type and just put the grid points in the list.
+
Note
+
Custom loss functions and custom metrics can be used as defined in Define Custom Functions. List the custom function name and provide the name of the table where the serialized Python objects reside using the parameter 'object_table' below.
+
The following loss function is not supported: sparse_categorical_crossentropy. The following metrics are not supported: sparse_categorical_accuracy, sparse_top_k_categorical_accuracy.
+
The Keras accuracy parameter top_k_categorical_accuracy returns top 5 accuracy by default. If you want a different top k value, use the helper function Top k Accuracy Function to create a custom Python function to compute the top k accuracy that you want.
+
+
+
+
fit_params_grid
+
VARCHAR. String representation of a Python dictionary of fit parameters to be tested. Each entry of the dictionary should consist of keys as fit parameter names, and values as a Python list of fit parameter values to be passed to Keras. Here is an example:
Callbacks are not currently supported except for TensorBoard which you can specify in the usual way, e.g., 'callbacks': ['[TensorBoard(log_dir="/tmp/logs/fit")]']
+
+
search_type
+
VARCHAR, 'grid' or 'random'. Search strategy for generating model configurations. For grid search, will generate all combinations of model IDs + compile params + fit params. For random search, specify the number of configs you want to generate using the 'num_configs' parameter below. Note that you can also use short forms for the 'grid' or 'random' keywords, e.g.,'rand' or 'r' instead of writing out 'random' in full.
+
+
+
num_configs (optional)
+
INTEGER, default: NULL. Number of model configs to generate. Only applies when search_type='random'.
+
+
+
random_state (optional)
+
INTEGER, default: NULL. Pseudo random number generator state used for random uniform sampling from lists of possible values. Pass an integer for reproducible output across multiple function calls. Only applies when search_type='random'.
+
+
+
object_table (optional)
+
VARCHAR, default: NULL. Name of the table containing Python objects in the case that custom loss functions or custom metrics are specified in the 'compile_params_grid'. Note that this table has to be created by the Define Custom Functions method. It is not allowed to pass a schema name, since it will be automatically pulled from this functions associated madlib schema.
+
+
+
+
Output table
+ The model selection output table contains the following columns:
+
+
mst_key
INTEGER. ID that defines a unique tuple for model architecture-compile parameters-fit parameters.
+
+
model_id
VARCHAR. Model architecture ID from the 'model_arch_table'.
+
+
compile_params
VARCHAR. Keras compile parameters.
+
+
fit_params
VARCHAR. Keras fit parameters.
+
+
A summary table named <model_selection_table>_summary is also created, which contains the following column:
+
+
model_arch_table
VARCHAR. Name of the model architecture table containing the model architecture IDs.
+
+
object_table
VARCHAR. Name of the object table containing the serialized Python objects for custom loss functions and custom metrics. If there are none, this field will be blank.
+
+
Load Model Selection Table [Deprecated]
+
This method is deprecated and replaced by the method 'generate_model_configs' described above.
VARCHAR. Table containing model architectures and weights. For more information on this table refer to Define Model Architectures.
+
+
+
model_selection_table
+
VARCHAR. Model selection table created by this utility. A summary table named <model_selection_table>_summary is also created. Contents of both output tables are the same as described above for the method 'generate_model_configs'.
+
+
+
model_id_list
+
INTEGER[]. Array of model IDs from the 'model_arch_table' to be included in the run combinations. For hyperparameter search, this will typically be one model ID. For model architecture search, this will be the different model IDs that you want to test.
+
+
+
compile_params_list
+
VARCHAR[]. Array of compile parameters to be tested. Each element of the array should consist of a string of compile parameters exactly as it is to be passed to Keras. For custom loss functions or custom metrics, list the custom function name in the usual way, and also provide the name of the table where the serialized objects reside in the parameter 'object_table' below.
+
+
+
fit_params_list
+
VARCHAR[]. Array of fit parameters to be tested. Each element of the array should consist of a string of fit parameters exactly as it is to be passed to Keras. Callbacks are not currently supported except for TensorBoard which you can specify in the usual way, e.g., callbacks=[TensorBoard(log_dir="/tmp/logs/fit")]
+
+
+
object_table (optional)
+
VARCHAR, default: NULL. Name of the table containing Python objects in the case that custom loss functions or custom metrics are specified in the parameter 'compile_params_list'.
+
+
+
+
Examples
+
The model selection table works in conjunction with a model architecture table, so we first create a model architecture table with two different models. Use Keras to define a model architecture with 1 hidden layer:
+ model_id | name | description
+----------+--------+--------------------------
+ 1 | Sophie | MLP with 1 hidden layer
+ 2 | Maria | MLP with 2 hidden layers
+(2 rows)
+
+
Generate model configurations using grid search. The output table for grid search contains the unique combinations of model architectures, compile and fit parameters.
+DROP TABLE IF EXISTS mst_table, mst_table_summary;
+SELECT madlib.generate_model_configs(
+ 'model_arch_library', -- model architecture table
+ 'mst_table', -- model selection table output
+ ARRAY[1,2], -- model ids from model architecture table
+ $$
+ {'loss': ['categorical_crossentropy'],
+ 'optimizer_params_list': [ {'optimizer': ['Adam', 'SGD'], 'lr': [0.001, 0.01]} ],
+ 'metrics': ['accuracy']}
+ $$, -- compile_param_grid
+ $$
+ { 'batch_size': [64, 128],
+ 'epochs': [10]
+ }
+ $$, -- fit_param_grid
+ 'grid' -- search_type
+ );
+SELECT * FROM mst_table ORDER BY mst_key;
+
Note that above uses the same learning rate for the two optimizers. If you wanted to use different learning rates and different parameters for different optimizers (common):
Generate model configurations using random search. The output table for random search contains the specified number of model architectures, compile and fit parameters, sampled from the specified distributions.
Incremental loading for more complex combinations. If it is easier to generate the model configurations incrementally rather than all at once, you can do that by not dropping the model selection table and associated summary table, in which case the new model configurations will be appended to the existing table. Here we combine two of the previous examples in to a single output table:
Create model selection table manually. If you want more control over the content of the model selection table, you could use grid or random search to generate a large number of combinations, then SELECT a subset of rows for training. Alternatively, you could manually create the model selection table and the associated summary table. Both must be created since they are needed by the multiple model fit module. For example, let's say we don't want all combinations but only want batch_size=4 for model_id=1 and batch_size=8 for model_id=2:
Custom loss functions and custom metrics. Let's say we have a table 'custom_function_table' that contains a custom loss function called 'my_custom_loss' and a custom accuracy function called 'my_custom_accuracy' based on Define Custom Functions. Generate the model configurations with:
+DROP TABLE IF EXISTS mst_table, mst_table_summary;
+SELECT madlib.generate_model_configs(
+ 'model_arch_library', -- model architecture table
+ 'mst_table', -- model selection table output
+ ARRAY[1,2], -- model ids from model architecture table
+ $$
+ {'loss': ['my_custom_loss'],
+ 'optimizer_params_list': [ {'optimizer': ['Adam', 'SGD'], 'lr': [0.001, 0.01]} ],
+ 'metrics': ['my_custom_accuracy']}
+ $$, -- compile_param_grid
+ $$
+ { 'batch_size': [64, 128],
+ 'epochs': [10]
+ }
+ $$, -- fit_param_grid
+ 'grid', -- search_type
+ NULL, -- num_configs
+ NULL, -- random_state
+ 'custom_function_table' -- table with custom functions
+ );
+SELECT * FROM mst_table ORDER BY mst_key;
+
Similarly, if you created a custom top k categorical accuracy function 'top_3_accuracy' in Define Custom Functions you can generate the model configurations as:
+DROP TABLE IF EXISTS mst_table, mst_table_summary;
+SELECT madlib.generate_model_configs(
+ 'model_arch_library', -- model architecture table
+ 'mst_table', -- model selection table output
+ ARRAY[1,2], -- model ids from model architecture table
+ $$
+ {'loss': ['categorical_crossentropy'],
+ 'optimizer_params_list': [ {'optimizer': ['Adam', 'SGD'], 'lr': [0.001, 0.01]} ],
+ 'metrics': ['top_3_accuracy']}
+ $$, -- compile_param_grid
+ $$
+ { 'batch_size': [64, 128],
+ 'epochs': [10]
+ }
+ $$, -- fit_param_grid
+ 'grid', -- search_type
+ NULL, -- num_configs
+ NULL, -- random_state
+ 'custom_function_table' -- table with custom functions
+ );
+SELECT * FROM mst_table ORDER BY mst_key;
+
[Deprecated] Load model selection table. This method is replaced by the 'generate_model_configs' method described above. Select the model(s) from the model architecture table that you want to run, along with the compile and fit parameters. Unique combinations will be created:
+DROP TABLE IF EXISTS mst_table, mst_table_summary;
+SELECT madlib.load_model_selection_table('model_arch_library', -- model architecture table
+ 'mst_table', -- model selection table output
+ ARRAY[1,2], -- model ids from model architecture table
+ ARRAY[ -- compile params
+ $$loss='categorical_crossentropy',optimizer='Adam(lr=0.1)',metrics=['categorical_accuracy']$$,
+ $$loss='categorical_crossentropy', optimizer='Adam(lr=0.01)',metrics=['categorical_accuracy']$$,
+ $$loss='categorical_crossentropy',optimizer='Adam(lr=0.001)',metrics=['categorical_accuracy']$$
+ ],
+ ARRAY[ -- fit params
+ $$batch_size=4,epochs=1$$,
+ $$batch_size=8,epochs=1$$
+ ]
+ );
+SELECT * FROM mst_table ORDER BY mst_key;
+
Clustering refers to the problem of partitioning a set of objects according to some problem-dependent measure of similarity. In the k-means variant, given \( n \) points \( x_1, \dots, x_n \in \mathbb R^d \), the goal is to position \( k \) centroids \( c_1, \dots, c_k \in \mathbb R^d \) so that the sum of distances between each point and its closest centroid is minimized. Each centroid represents a cluster that consists of all points to which this centroid is closest.
+
This module can compute clusters given the number of centroids k as an input, using a variety of seeding methods. It can also automatically select the best k value from a range of suggested k values, using the simplified silhouette method or the elbow method.
+
Clustering Function
+
The k-means algorithm can be invoked in different ways, depending on the source of the initial set of centroids:
k-means++ centroid seeding method [1]. This method can speed up convergence by seeding centroids spread out over the whole range of the input points, while at the same time not being too susceptible to outliers.
Supply an initial centroid set in a relation identified by the rel_initial_centroids argument, for the case where initial centroids are stored in a table.
TEXT. The name of the table containing the input data points. Data points and predefined centroids (if used) are expected to be stored row-wise, in a column of type SVEC (or any type convertible to SVEC, like FLOAT[] or INTEGER[]). Data points with non-finite values (NULL, NaN, infinity) in any component are skipped during analysis.
+
+
+
expr_point
+
TEXT. The name of the column with point coordinates or an array expression.
+
+
+
k
+
INTEGER. The number of centroids to calculate.
+
+
+
fn_dist (optional)
+
TEXT, default: 'squared_dist_norm2'. The name of the function to use to calculate the distance from a data point vector to a centroid vector. The following distance functions can be used (computation of barycenter/mean in parentheses):
+
+dist_norm1: 1-norm/Manhattan (element-wise median). MADlib does not provide a median aggregate function for performance reasons.
INTEGER, default: 20. The maximum number of iterations to perform.
+
+
+
min_frac_reassigned (optional)
+
DOUBLE PRECISION, default: 0.001. The minimum fraction of centroids reassigned to continue iterating. When fewer than this fraction of centroids are reassigned in an iteration, the calculation completes.
+
+
+
seeding_sample_ratio (optional)
+
DOUBLE PRECISION, default: 1.0. The proportion of subsample of original dataset to use for kmeans++ centroid seeding method. Kmeans++ scans through the data sequentially 'k' times and can be too slow for big datasets. When 'seeding_sample_ratio' is greater than 0 (thresholded to be maximum value of 1.0), the seeding is run on a uniform random subsample of the data. Note: the final K-means algorithm is run on the complete dataset. This parameter only builds a subsample for the seeding and is only available for kmeans++.
+
+
+
rel_initial_centroids
+
TEXT. Table or view containing the set of initial centroids.
+
+
+
expr_centroid
+
TEXT. The name of the column (or the array expression) in the rel_initial_centroids table or view that contains the centroid coordinates.
+
+
+
initial_centroids
+
DOUBLE PRECISION[][]. Array expression with the initial centroid coordinates.
+
+
Output
+ The output of the k-means module is a composite type with the following columns:
+
+
centroids
DOUBLE PRECISION[][]. The final centroid positions.
+
+
cluster_variance
DOUBLE PRECISION[]. The value of the objective function per cluster.
+
+
objective_fn
DOUBLE PRECISION. The value of the objective function.
+
+
frac_reassigned
DOUBLE PRECISION. The fraction of points reassigned in the last iteration.
+
+
num_iterations
INTEGER. The total number of iterations executed.
+
+
Auto Clustering Function
+
The optimal number of centroids can be determined automatically, from a set of candidate values that you provide, for random seeding or k-means++ seeding. The simplified silhouette method or the elbow method are used to pick the best k value.
Arguments
+ The arguments for auto k selection are the same as described above, with the following additions:
+
output_table
+
TEXT. Name of the output table containing results for each k value. Details of the output table are shown below. A summary table called <output_table>_summary will also be created for the best k value as per the selection algorithm.
+
+
+
k
+
INTEGER[]. Array of k values to test. Does not need to be contiguous but all elements must be >1 and cannot repeat within the array. Parameter 'k_selection_algorithm' determines the evaluation method.
+
+
+
k_selection_algorithm (optional)
+
TEXT, default: 'silhouette'. Method to evaluate optimal number of centroids k. Current approaches supported: 'silhouette', 'elbow' or 'both'. The text can be any subset of the strings; for e.g., 'silh' will use the silhouette method. Note that for large data sets, the silhouette computation can be expensive.
+
Note
Note that the auto k-means algorithms require the NumPy python package to be installed on the system since the elbow method uses the NumPy gradient function [2]. For Greenplum clusters, installing NumPy only on the host machine of the master segment will be sufficient. The suggested installation method is: pip install numpy –user
+
+
+
Output Tables
+ Two output tables are created for auto k-means. The first is called 'output_table' and contains one row per k value:
+
+
k
INTEGER. Number of centroids.
+
+
centroids
DOUBLE PRECISION[][]. The final centroid positions.
+
+
cluster_variance
DOUBLE PRECISION[]. The value of the objective function per cluster.
+
+
objective_fn
DOUBLE PRECISION. The value of the objective function.
+
+
frac_reassigned
DOUBLE PRECISION. The fraction of points reassigned in the last iteration.
+
+
num_iterations
INTEGER. The total number of iterations executed.
+
+
k
INTEGER. Number of centroids as per the specified 'k_selection_algorithm'. If 'both' is specified, the best k value will correspond to the silhouette method.
+
+
silhouette
DOUBLE PRECISION. The value of the simplified silhouette score for the k value, if computed.
+
+
elbow
DOUBLE PRECISION. The value of the elbow score for the k value, if computed.
+
+
selection_algorithm
TEXT. Algorithm used to select the best k (either 'silhouette' or 'elbow')
+
+
A summary table named <output_table>_summary is also created, which has the same output as the 'output_table' above but only contains one row for best k value as per the selection algorithm. If 'both' is specified for 'k_selection_algorithm' the best k value returned will correspond to the silhouette method.
+
Cluster Assignment
+
After clustering, the cluster assignment for each data point can be computed with the help of the closest_column() utility function:
+
+closest_column( m,
+ x,
+ dist
+ )
+
Arguments
+
m
+
DOUBLE PRECISION[][]. Learned centroids from the training function.
+
+
+
x
+
DOUBLE PRECISION[]. Data points.
+
+
+
dist (optional)
+
TEXT, default: 'squared_dist_norm2'. The name of the function to use to calculate the distance from a data point vector to a centroid vector. See the fn_dist argument of the k-means training function for more details on distance functions.
+
+
Output
+ The output is a composite type with the following columns:
DOUBLE PRECISION. Distance to the cluster centroid.
+
+
Simple Silhouette
+
A common method to assess the quality of the clustering is the silhouette coefficient, a simplified version of which is implemented in MADlib [3]. There are two silhouette functions: average score across all data points, and a per-point score. The average silhouette function has the following syntax:
TEXT. The name of the table containing the input data points.
+
+
+
expr_point
+
TEXT. The name of the column with point coordinates or an array expression.
+
+
+
centroids
+
DOUBLE PRECISION[][]. An expression evaluating to an array of centroids.
+
+
+
fn_dist (optional)
+
TEXT, default: 'squared_dist_norm2'. The name of the function to use to calculate the distance from a data point vector to a centroid vector. See the fn_dist argument of the k-means training function for more details on distance functions.
+
+
+
output_table
+
TEXT. Name of the output table to contain sihouette score for each point.
+
+
+
pid
+
TEXT. Column containing point ID in the table 'rel_source' containing the data points.
+
+
+
centroids_table
+
TEXT. Name of the table containing the centroids.
+
+
+
centroids_col
+
TEXT. Name of the column in the table 'centroids_table' containing the centroids.
+
+
+
+
Output
+ For the average function, a single value for simple silhouette score is returned. For the per-point function, the output table contains the following columns:
+
+
pid
INTEGER. Point ID.
+
+
+
+
centroid_id
INTEGER. The cluster that the point is assigned to.
+
+
+
+
neighbor_centroid_id
INTEGER. The next closest cluster to the one that the point is assigned to.
+
+
+
+
simple_silhouette
DOUBLE PRECISION. Simplified silhouette score for the point.
+
+
Examples
+
Note
Your results may not be exactly the same as below due to the nature random nature of the k-means algorithm. Also, remember to be consistent in the distance functions that you use in the k-means, silhouette and helper functions.
Run k-means clustering using kmeans++ for centroid seeding. Use squared Euclidean distance which is a commonly used distance function.
+DROP TABLE IF EXISTS km_result;
+CREATE TABLE km_result AS
+SELECT * FROM madlib.kmeanspp( 'km_sample', -- Table of source data
+ 'points', -- Column containing point co-ordinates
+ 2, -- Number of centroids to calculate
+ 'madlib.squared_dist_norm2', -- Distance function
+ 'madlib.avg', -- Aggregate function
+ 20, -- Number of iterations
+ 0.001 -- Fraction of centroids reassigned to keep iterating
+ );
+\x on;
+SELECT * FROM km_result;
+
Find the cluster assignment for each point. Use the closest_column() function to map each point to the cluster that it belongs to.
+DROP TABLE IF EXISTS point_cluster_map;
+CREATE TABLE point_cluster_map AS
+SELECT data.*, (madlib.closest_column(centroids, points, 'madlib.squared_dist_norm2')).*
+FROM km_sample as data, km_result;
+ALTER TABLE point_cluster_map RENAME column_id to cluster_id; -- change column name
+SELECT * FROM point_cluster_map ORDER BY pid;
+
Display cluster ID. There are two steps to get the cluster id associated with the centroid coordinates, if you need it. First unnest the cluster centroids 2-D array to get a set of 1-D centroid arrays:
+DROP TABLE IF EXISTS km_centroids_unnest;
+-- Run unnest function
+CREATE TABLE km_centroids_unnest AS
+SELECT (madlib.array_unnest_2d_to_1d(centroids)).*
+FROM km_result;
+SELECT * FROM km_centroids_unnest ORDER BY 1;
+
Note that the ID column returned by array_unnest_2d_to_1d() is just a row ID and not the cluster ID assigned by k-means. The second step to get the cluster_id is:
+SELECT cent.*, (madlib.closest_column(centroids, unnest_result, 'madlib.squared_dist_norm2')).column_id as cluster_id
+FROM km_centroids_unnest as cent, km_result
+ORDER BY cent.unnest_row_id;
+
The best selection above is made by the silhouette algorithm by default. Note that the elbow method may select a different k value as the best. To see results for all k values:
Simplified silhouette per point. Let's say we want the simplified silhouette coefficient for each point in the data set, for the case where k=3:
+DROP TABLE IF EXISTS km_points_silh;
+SELECT * FROM madlib.simple_silhouette_points( 'km_sample', -- Input points table
+ 'km_points_silh', -- Output table
+ 'pid', -- Point ID column in input table
+ 'points', -- Points column in input table
+ (SELECT centroids FROM km_result_auto WHERE k=3), -- centroids array
+ 'madlib.squared_dist_norm2' -- Distance function
+ );
+\x off
+SELECT * FROM km_points_silh ORDER BY pid;
+
In the most common case, \( \operatorname{dist} \) is the square of the Euclidean distance, though other distance measures can be used.
+
This problem is computationally difficult (NP-hard), yet the local-search heuristic proposed by Lloyd [4] performs reasonably well in practice. In fact, it is so ubiquitous today that it is often referred to as the standard algorithm or even just the k-means algorithm. It works as follows:
+
+
Seed the \( k \) centroids, meaning specify their initial positions (see below).
+
Repeat until convergence:
+
Assign each point to its closest centroid.
+
Move each centroid to a position that minimizes the sum of distances in this cluster.
+
+
+
Convergence is achieved when no points change their assignments during step 2a.
+
+
Since the objective function decreases in every step, this algorithm is guaranteed to converge to a local optimum.
+
The quality of k-means is highly influenced by the choice of the seeding. In MADlib we support uniform-at-random sampling, kmeans++ as well as the ability for the user to enter manual seeding based on other methods.
+
k-means++ seeding [1] starts with a single centroid chosen randomly among the input points. It then iteratively chooses new centroids from the input points until there are a total of k centroids. The probability for picking a particular point is proportional to its minimum distance to any existing centroid. Intuitively, k-means++ favors seedings where centroids are spread out over the whole range of the input points, while at the same time not being too susceptible to outliers.
+
Silhouette
+
+
To evaluate the validity of clustering with different k values, the objective function is not ideal because it decreases as k value increases, so it has a bias toward selecting the largest k as the best result [6]. Therefore we use other internal measures to evaluate cluster validity.
+
One such measure is silhouette score. The original formulation [7] computes the average distance of a data point to all the other data points in its own cluster, and to all the data points in the neighbouring cluster nearest to the data point. This is expensive for a large number of points since it requires the full pairwise distance matrix over all data.
+
In the simplified silhouette [3] which is used in MADlib, the distance of a data point to a cluster is represented with the distance to the cluster centroid instead of the average distance to all (other) data points in the cluster. This has the advantage of being much faster to compute, and can be shown to be comparable in performance to the full silhouette [8].
+
Elbow Method
+ The elbow method considers the percentage of variance explained as a function of number of clusters. The idea is not to add another cluster if it doesn't model the data better. Graphically it means identifying the "elbow" in the curve of sum of squared errors vs. number of clusters (k). This was possibly originally suggested in [9]. To locate the elbow, we use the 2nd derivative of the objective function using the NumPy gradient function [2].
+
Literature
+
[1] David Arthur, Sergei Vassilvitskii: k-means++: the advantages of careful seeding, Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'07), pp. 1027-1035.
[3] E. R. Hruschka, L. N. C. Silva, R. J. G. B. Campello: Clustering Gene-Expression Data: A Hybrid Approach that Iterates Between k-Means and Evolutionary Search. In: Studies in Computational Intelligence - Hybrid Evolutionary Algorithms. pp. 313-335. Springer. 2007.
+
[4] Lloyd, Stuart: Least squares quantization in PCM. Technical Note, Bell Laboratories. Published much later in: IEEE Transactions on Information Theory 28(2), pp. 128-137. 1982.
+
[5] Leisch, Friedrich: A Toolbox for K-Centroids Cluster Analysis. In: Computational Statistics and Data Analysis, 51(2). pp. 526-544. 2006.
+
[6] Jain, A.K.: Data clustering: 50 years beyond k-means. Pattern recognition letters 31(8), 651–666 (2010)
+
[7] Peter J. Rousseeuw (1987). “Silhouettes: a Graphical Aid to the Interpretation and Validation of Cluster Analysis”. Computational and Applied Mathematics 20: 53-65
+
[8] Wang F., Franco-Penya HH., Kelleher J.D., Pugh J., Ross R. (2017) An Analysis of the Application of Simplified Silhouette to the Evaluation of k-means Clustering Validity. In: Perner P. (eds) Machine Learning and Data Mining in Pattern Recognition. MLDM 2017. Lecture Notes in Computer Science, vol 10358. Springer, Cham
+
[9] Robert L. Thorndike (December 1953). "Who Belongs in the Family?". Psychometrika. 18 (4): 267–276. doi:10.1007/BF02289263.
+
Related Topics
+
File kmeans.sql_in documenting the k-Means SQL functions
K-nearest neighbors is a method for finding the \(k\) closest points to a given data point in terms of a given metric. Its input consists of data points as features from testing examples and it looks for \(k\) closest points in the training set for each of the data points in test set. The output of KNN depends on the type of task. For classification, the output is the majority vote of the classes of the \(k\) nearest data points. For regression, the output is the average of the values of \(k\) nearest neighbors of the given test point.
+
For unsupervised nearest neighbors, set the training set to match the test set so the nearest neighbor of each point is the point itself, with zero distance.
+
Both exact and approximate methods are supported. The approximate methods can be used in the case that run-time is too long using the exact method.
TEXT. Name of the table containing the training data points. Training data points are expected to be stored row-wise in a column of type DOUBLE PRECISION[].
+
+
+
point_column_name
+
TEXT. Name of the column with training data points or expression that evaluates to a numeric array
+
+
+
point_id
+
TEXT. Name of the column in 'point_source’ containing source data ids. The ids are of type INTEGER with no duplicates. They do not need to be contiguous. This parameter must be used if the list of nearest neighbors are to be output, i.e., if the parameter 'output_neighbors' below is TRUE or if 'label_column_name' is NULL.
+
+
+
label_column_name
+
TEXT. Name of the column with labels/values of training data points. If this column is a Boolean, integer or text, it will run KNN classification, else if it is double precision values will run KNN regression. If you set this to NULL, it will only return the set of neighbors without actually doing classification or regression.
+
+
+
test_source
+
TEXT. Name of the table containing the test data points. Testing data points are expected to be stored row-wise in a column of type DOUBLE PRECISION[].
+
+
+
test_column_name
+
TEXT. Name of the column with testing data points or expression that evaluates to a numeric array
+
Note
For unsupervised nearest neighbors, make the test dataset the same as the source dataset, so the nearest neighbor of each point is the point itself, with a zero distance.
+
+
test_id
+
TEXT. Name of the column having ids of data points in test data table.
+
+
+
output_table
+
TEXT. Name of the table to store final results.
+
+
+
k (optional)
+
INTEGER. default: 1. Number of nearest neighbors to consider. For classification, should be an odd number to break ties, otherwise the result may depend on ordering of the input data.
+
+
+
output_neighbors (optional)
+
BOOLEAN default: TRUE. Outputs the list of k-nearest neighbors (and their respective distances to the target point) that were used in the voting/averaging, sorted from closest to furthest.
+
+
+
fn_dist (optional)
+
TEXT, default: 'squared_dist_norm2'. The name of the function used to calculate the distance between data points.
+user defined function with signature DOUBLE PRECISION[] x, DOUBLE PRECISION[] y -> DOUBLE PRECISION. Must return a value greater than or equal to zero.
+
+
Note
Always qualify the distance function with the schema name. For example, if you install MADlib in a schema called 'madlib' then the 'fn_dist' parameter would be 'madlib.dist_norm2', etc.
+
+
weighted_avg (optional)
+
BOOLEAN, default: FALSE. Calculates classification or regression values using a weighted average. The idea is to weigh the contribution of each of the k neighbors according to their distance to the test point, giving greater influence to closer neighbors. The distance function 'fn_dist' specified above is used. For classification, majority voting weighs a neighbor according to inverse distance. For regression, the inverse distance weighting approach is used from Shepard [4].
+
+
+
algorithm (optional)
+
TEXT, default: 'brute_force'. The name of the algorithm used to compute nearest neighbors. The following options are supported:
+
+brute_force: Produces an exact result by searching all points in the search space. You can also use a short form "b" or "brute" etc. to select brute force.
+
+kd_tree: Produces an approximate result by searching a subset of the search space, that is, only certain leaf nodes in the kd-tree as specified by "algorithm_params" below. You can also use a short form "k" or "kd" etc. to select kd-tree.
+
+
+
+
algorithm_params (optional)
+
TEXT, default: 'depth=3, leaf_nodes=2'. These parameters apply to the kd-tree algorithm only.
+
+depth: Depth of the kd-tree. Increasing this value will decrease run-time but reduce the accuracy.
+
+leaf_nodes: Number of leaf nodes (regions) to search for each test point. Inceasing this value will improve the accuracy but increase run-time.
+
+
Note
Please note that the kd-tree accuracy will be lower for datasets with a high number of features. It is advised to use at least two leaf nodes. Refer to the Technical Background for more information on how the kd-tree is implemented.
+
+
+
Output Format
+
The output of the KNN module is a table with the following columns:
+
+
id
INTEGER. The ids of test data points.
+
+
test_column_name
DOUBLE PRECISION[]. The test data points.
+
+
prediction
INTEGER. Label in case of classification, average value in case of regression.
+
+
k_nearest_neighbours
INTEGER[]. List of nearest neighbors, sorted closest to furthest from the corresponding test point.
+
+
distance
DOUBLE PRECISION[]. List of distance to nearest neighbors, sorted closest to furthest from the corresponding test point.
+
+
Examples
+
+
Prepare some training data for classification:
+DROP TABLE IF EXISTS knn_train_data;
+CREATE TABLE knn_train_data (
+ id integer,
+ data integer[],
+ label integer -- Integer label means for classification
+ );
+INSERT INTO knn_train_data VALUES
+(1, '{1,1}', 1),
+(2, '{2,2}', 1),
+(3, '{3,3}', 1),
+(4, '{4,4}', 1),
+(5, '{4,5}', 1),
+(6, '{20,50}', 0),
+(7, '{10,31}', 0),
+(8, '{81,13}', 0),
+(9, '{1,111}', 0);
+
+
Prepare some training data for regression:
+DROP TABLE IF EXISTS knn_train_data_reg;
+CREATE TABLE knn_train_data_reg (
+ id integer,
+ data integer[],
+ label float -- Float label means for regression
+ );
+INSERT INTO knn_train_data_reg VALUES
+(1, '{1,1}', 1.0),
+(2, '{2,2}', 1.0),
+(3, '{3,3}', 1.0),
+(4, '{4,4}', 1.0),
+(5, '{4,5}', 1.0),
+(6, '{20,50}', 0.0),
+(7, '{10,31}', 0.0),
+(8, '{81,13}', 0.0),
+(9, '{1,111}', 0.0);
+
+
Prepare some testing data:
+DROP TABLE IF EXISTS knn_test_data CASCADE;
+CREATE TABLE knn_test_data (
+ id integer,
+ data integer[]
+ );
+INSERT INTO knn_test_data VALUES
+(1, '{2,1}'),
+(2, '{2,6}'),
+(3, '{15,40}'),
+(4, '{12,1}'),
+(5, '{2,90}'),
+(6, '{50,45}');
+
+
Run KNN for classification. Prepend the distance function parameter with the schema where MADlib is installed (in this example 'madlib.squared_dist_norm2'):
+DROP TABLE IF EXISTS knn_result_classification;
+SELECT * FROM madlib.knn(
+ 'knn_train_data', -- Table of training data
+ 'data', -- Col name of training data
+ 'id', -- Col name of id in train data
+ 'label', -- Training labels
+ 'knn_test_data', -- Table of test data
+ 'data', -- Col name of test data
+ 'id', -- Col name of id in test data
+ 'knn_result_classification', -- Output table
+ 3, -- Number of nearest neighbors
+ True, -- True to list nearest-neighbors by id
+ 'madlib.squared_dist_norm2' -- Distance function
+ );
+SELECT * from knn_result_classification ORDER BY id;
+
Note that the nearest neighbors are sorted from closest to furthest from the corresponding test point.
+
Run KNN for regression:
+DROP TABLE IF EXISTS knn_result_regression;
+SELECT * FROM madlib.knn(
+ 'knn_train_data_reg', -- Table of training data
+ 'data', -- Col name of training data
+ 'id', -- Col Name of id in train data
+ 'label', -- Training labels
+ 'knn_test_data', -- Table of test data
+ 'data', -- Col name of test data
+ 'id', -- Col name of id in test data
+ 'knn_result_regression', -- Output table
+ 3, -- Number of nearest neighbors
+ True, -- True to list nearest-neighbors by id
+ 'madlib.dist_norm2' -- Distance function
+ );
+SELECT * FROM knn_result_regression ORDER BY id;
+
List nearest neighbors only, without doing classification or regression:
+DROP TABLE IF EXISTS knn_result_list_neighbors;
+SELECT * FROM madlib.knn(
+ 'knn_train_data_reg', -- Table of training data
+ 'data', -- Col name of training data
+ 'id', -- Col Name of id in train data
+ NULL, -- NULL training labels means just list neighbors
+ 'knn_test_data', -- Table of test data
+ 'data', -- Col name of test data
+ 'id', -- Col name of id in test data
+ 'knn_result_list_neighbors', -- Output table
+ 3 -- Number of nearest neighbors
+ );
+SELECT * FROM knn_result_list_neighbors ORDER BY id;
+
Result, with neighbors sorted from closest to furthest:
Run KNN for classification using the weighted average:
+DROP TABLE IF EXISTS knn_result_classification;
+SELECT * FROM madlib.knn(
+ 'knn_train_data', -- Table of training data
+ 'data', -- Col name of training data
+ 'id', -- Col name of id in train data
+ 'label', -- Training labels
+ 'knn_test_data', -- Table of test data
+ 'data', -- Col name of test data
+ 'id', -- Col name of id in test data
+ 'knn_result_classification', -- Output table
+ 3, -- Number of nearest neighbors
+ True, -- True to list nearest-neighbors by id
+ 'madlib.squared_dist_norm2', -- Distance function
+ True -- For weighted average
+ );
+SELECT * FROM knn_result_classification ORDER BY id;
+
Use kd-tree option. First we build a kd-tree to depth 4 and search half (8) of the 16 leaf nodes (i.e., 2^4 total leaf nodes):
+DROP TABLE IF EXISTS knn_result_classification_kd;
+SELECT madlib.knn(
+ 'knn_train_data', -- Table of training data
+ 'data', -- Col name of training data
+ 'id', -- Col name of id in train data
+ NULL, -- Training labels
+ 'knn_test_data', -- Table of test data
+ 'data', -- Col name of test data
+ 'id', -- Col name of id in test data
+ 'knn_result_classification_kd', -- Output table
+ 3, -- Number of nearest neighbors
+ True, -- True to list nearest-neighbors by id
+ 'madlib.squared_dist_norm2', -- Distance function
+ False, -- For weighted average
+ 'kd_tree', -- Use kd-tree
+ 'depth=4, leaf_nodes=8' -- Kd-tree options
+ );
+SELECT * FROM knn_result_classification_kd ORDER BY id;
+
The result above is the same as brute force. If we search just 1 leaf node, run-time will be faster but accuracy will be lower. This shows up in this very small data set by not being able to find 3 nearest neighbors for all test points:
+DROP TABLE IF EXISTS knn_result_classification_kd;
+SELECT madlib.knn(
+ 'knn_train_data', -- Table of training data
+ 'data', -- Col name of training data
+ 'id', -- Col name of id in train data
+ NULL, -- Training labels
+ 'knn_test_data', -- Table of test data
+ 'data', -- Col name of test data
+ 'id', -- Col name of id in test data
+ 'knn_result_classification_kd', -- Output table
+ 3, -- Number of nearest neighbors
+ True, -- True to list nearest-neighbors by id
+ 'madlib.squared_dist_norm2', -- Distance function
+ False, -- For weighted average
+ 'kd_tree', -- Use kd-tree
+ 'depth=4, leaf_nodes=1' -- Kd-tree options
+ );
+SELECT * FROM knn_result_classification_kd ORDER BY id;
+
Unsupervised nearest neighbors. Here the training set matches the test set so the nearest neighbor of each point is the point itself, with a zero distance:
+DROP TABLE IF EXISTS knn_result_classification_unsup;
+SELECT * FROM madlib.knn(
+ 'knn_train_data', -- Table of training data
+ 'data', -- Col name of training data
+ 'id', -- Col name of id in train data
+ NULL, -- NULL training labels means just list neighbors
+ 'knn_train_data', -- Table of test data (same as training data)
+ 'data', -- Col name of test data
+ 'id', -- Col name of id in test data
+ 'knn_result_classification_unsup', -- Output table
+ 3, -- Number of nearest neighbors
+ True, -- True to list nearest-neighbors by id
+ 'madlib.squared_dist_norm2' -- Distance function
+ );
+SELECT * from knn_result_classification_unsup ORDER BY id;
+
Result, with point and neighbors sorted from closest to furthest:
User defined distance function. There are several built-in distance functions, but you can create your own using a UDF if desired. For example, to create a Chebyshev distance function [6], first create the function:
+CREATE OR REPLACE FUNCTION chebychev_distance (x double precision[], y double precision[])
+ RETURNS double precision
+AS $$
+ from scipy.spatial import distance
+ return distance.chebyshev(x, y)
+$$ LANGUAGE plpython3u;
+
Then pass the function as an argument:
+DROP TABLE IF EXISTS knn_result_classification_udf;
+SELECT * FROM madlib.knn(
+ 'knn_train_data', -- Table of training data
+ 'data', -- Col name of training data
+ 'id', -- Col name of id in train data
+ 'label', -- Training labels
+ 'knn_test_data', -- Table of test data
+ 'data', -- Col name of test data
+ 'id', -- Col name of id in test data
+ 'knn_result_classification_udf', -- Output table
+ 3, -- Number of nearest neighbors
+ True, -- True to list nearest-neighbors by id
+ 'chebychev_distance' -- Distance function
+ );
+SELECT * from knn_result_classification_udf ORDER BY id;
+
Result, with point and neighbors sorted from closest to furthest:
The training data points are vectors in a multidimensional feature space, each with a class label. The training phase of the algorithm consists only of storing the feature vectors and class labels of the training points.
+
In the prediction phase, \(k\) is a user-defined constant, and an unlabeled vector (a test point) is predicted by using the label from the the \(k\) training samples nearest to that test point.
+
Since distances between points are used to find the nearest neighbors, the data should be standardized across features. This ensures that all features are given equal weightage in the distance computation.
+
An approximation method can be used to speed the prediction phase by building appropriate data structures in the training phase. An example of such a data structure is kd-trees [5]. Using the kd-tree algorithm can improve the execution time of the \(k\)-NN operation, but at expense of sacrificing some accuracy. The kd-tree implementation divides the training dataset into multiple regions that correspond to the leaf nodes of a tree. For example, a tree of depth \(3\) will have a total of \(2^3 = 8\) regions. The algorithm will look for the nearest neighbors in a subset of all regions instead of searching the whole dataset. For a given test point, the first (home) region is found by traversing the tree and finding its associated node. If the user requests additional leaf nodes to be searched, we look at the distance between the point and the centroids of other regions and expand the search to the specified number of closest regions.
+
It's important to note that the nodes that each level of the kd-tree search over a single feature and the features are explored in the same order as that in the data.
+
The kd-tree accuracy might suffer on datasets with a high number of features (dimensions). For example, let's say we are using a dataset with 20 features and kd-tree depth of only 3. This means the kd-tree is constructed based on the first 3 features. Therefore, it is possible to miss nearest neighbors that are closer in those 17 dimensions because they got assigned to a further region (the distance computation would still uses all 20 features).
[4] Shepard, Donald (1968). "A two-dimensional interpolation function for
+irregularly-spaced data". Proceedings of the 1968 ACM National Conference. pp. 517–524.
+
[5] Bentley, J. L. (1975). "Multidimensional binary search trees used for
+associative searching". Communications of the ACM. 18 (9): 509. doi:10.1145/361002.361007.
Latent Dirichlet Allocation (LDA) is a generative probabilistic model for natural texts. It is used in problems such as automated topic discovery, collaborative filtering, and document classification.
+
In addition to an implementation of LDA, this MADlib module also provides a number of additional helper functions to interpret results of the LDA output.
+
Note
Topic modeling is often used as part of a larger text processing pipeline, which may include operations such as term frequency, stemming and stop word removal. You can use the function Term Frequency to generate the required vocabulary format from raw documents for the LDA training function. See the examples later on this page for more details.
+
Background
+
The LDA model posits that each document is associated with a mixture of various topics (e.g., a document is related to Topic 1 with probability 0.7, and Topic 2 with probability 0.3), and that each word in the document is attributable to one of the document's topics. There is a (symmetric) Dirichlet prior with parameter \( \alpha \) on each document's topic mixture. In addition, there is another (symmetric) Dirichlet prior with parameter \( \beta \) on the distribution of words for each topic.
+
The following generative process then defines a distribution over a corpus of documents:
+
+
Sample for each topic \( i \), a per-topic word distribution \( \phi_i \) from the Dirichlet ( \(\beta\)) prior.
+
For each document:
+
Sample a document length N from a suitable distribution, say, Poisson.
+
Sample a topic mixture \( \theta \) for the document from the Dirichlet ( \(\alpha\)) distribution.
+
For each of the N words:
+
Sample a topic \( z_n \) from the multinomial topic distribution \( \theta \).
+
Sample a word \( w_n \) from the multinomial word distribution \( \phi_{z_n} \) associated with topic \( z_n \).
+
+
+
+
+
+
In practice, only the words in each document are observable. The topic mixture of each document and the topic for each word in each document are latent unobservable variables that need to be inferred from the observables, and this is referred to as the inference problem for LDA. Exact inference is intractable, but several approximate inference algorithms for LDA have been developed. The simple and effective Gibbs sampling algorithm described in Griffiths and Steyvers [2] appears to be the current algorithm of choice.
+
This implementation provides a parallel and scalable in-database solution for LDA based on Gibbs sampling. It takes advantage of the shared-nothing MPP architecture and is a different implementation than one would find for MPI or map/reduce.
+
Training Function
The LDA training function has the following syntax:
TEXT. Name of the table storing the training dataset. Each row is in the form <docid, wordid, count> where docid, wordid, and count are non-negative integers. The docid column refers to the document ID, the wordid column is the word ID (the index of a word in the vocabulary), and count is the number of occurrences of the word in the document. Please note:
+
+
wordid must be contiguous integers going from from 0 to voc_size − 1.
+
column names for docid, wordid, and count are currently fixed, so you must use these exact names in the data_table.
+
+
The function Term Frequency can be used to generate vocabulary in the required format from raw documents.
+
+
+
model_table
+
TEXT. This is an output table generated by LDA which contains the learned model. It has one row with the following columns:
+
+
voc_size
INTEGER. Size of the vocabulary. As mentioned above for the input table, wordid consists of contiguous integers going from 0 to voc_size − 1.
+
+
topic_num
INTEGER. Number of topics.
+
+
alpha
DOUBLE PRECISION. Dirichlet prior for the per-document topic multinomial.
+
+
beta
DOUBLE PRECISION. Dirichlet prior for the per-topic word multinomial.
+
+
model
BIGINT[]. The encoded model description (not human readable).
+
+
num_iterations
INTEGER. Number of iterations that training ran for, which may be less than the maximum value specified in the parameter 'iter_num' if the perplexity tolerance was reached.
+
+
perplexity
DOUBLE PRECISION[]. Array of perplexity values as per the 'evaluate_every' parameter. For example, if 'evaluate_every=5' this would be an array of perplexity values for every 5th iteration, plus the last iteration.
+
+
perplexity_iters
INTEGER[]. Array indicating the iterations for which perplexity is calculated, as derived from the parameters 'iter_num' and 'evaluate_every'. For example, if 'iter_num=5' and 'evaluate_every=2', then 'perplexity_iters' value would be {2,4,5} indicating that perplexity is computed at iterations 2, 4 and 5 (at the end), unless of course it terminated earlier due to 'perplexity_tol'. If 'iter_num=5' and 'evaluate_every=1', then 'perplexity_iters' value would be {1,2,3,4,5} indicating that perplexity is computed at every iteration, again assuming it ran the full number of iterations.
+
+
+
output_data_table
+
TEXT. The name of the table generated by LDA that stores the output data. It has the following columns:
+
+
docid
INTEGER. Document id from input 'data_table'.
+
+
wordcount
INTEGER. Count of number of words in the document, including repeats. For example, if a word appears 3 times in the document, it is counted 3 times.
+
+
words
INTEGER[]. Array of wordid in the document, not including repeats. For example, if a word appears 3 times in the document, it appears only once in the words array.
+
+
counts
INTEGER[]. Frequency of occurance of a word in the document, indexed the same as the words array above. For example, if the 2nd element of the counts array is 4, it means that the word in the 2nd element of the words array occurs 4 times in the document.
+
+
topic_count
INTEGER[]. Array of the count of words in the document that correspond to each topic. This array is of length topic_num. Topic ids are continuous integers going from 0 to topic_num − 1.
+
+
topic_assignment
INTEGER[]. Array indicating which topic each word in the document corresponds to. This array is of length wordcount. Words that are repeated n times in the document will show up consecutively n times in this array.
+
+
+
voc_size
+
INTEGER. Size of the vocabulary. As mentioned above for the input 'data_table', wordid consists of continuous integers going from 0 to voc_size − 1.
+
topic_num
+
INTEGER. Desired number of topics.
+
iter_num
+
INTEGER. Maximum number of iterations. If a 'perplexity_tol' is set, LDA may train for less than the maximum number of iterations if the tolerance is reached.
+
alpha
+
DOUBLE PRECISION. Dirichlet prior for the per-document topic multinomial (e.g., 50/topic_num is a reasonable value to start with as per Griffiths and Steyvers [2]).
+
beta
+
DOUBLE PRECISION. Dirichlet prior for the per-topic word multinomial (e.g., 0.01 is a reasonable value to start with).
+
evaluate_every (optional)
+
INTEGER, default: 0. How often to evaluate perplexity. Set it to 0 or a negative number to not evaluate perplexity in training at all. Evaluating perplexity can help you check convergence during the training process, but it will also increase total training time. For example, evaluating perplexity in every iteration might increase training time up to two-fold.
+
perplexity_tol (optional)
+
DOUBLE PRECISION, default: 0.1. Perplexity tolerance to stop iteration. Only used when the parameter 'evaluate_every' is greater than 0.
+
+
+
Prediction Function
+
Prediction involves labelling test documents using a learned LDA model:
TEXT. Name of the table storing the test dataset (new document to be labeled).
+
model_table
+
TEXT. The model table generated by the training process.
+
output_predict_table
+
TEXT. The prediction output table. Each row in the table stores the topic distribution and the topic assignments for a document in the dataset. This table has the exact same columns and interpretation as the 'output_data_table' from the training function above.
+
+
Perplexity
Perplexity describes how well the model fits the data by computing word likelihoods averaged over the documents. This function returns a single perplexity value.
TEXT. The model table generated by the training process.
+
vocab_table
+
TEXT. The vocabulary table in the form <wordid, word>. Reminder that this table can be created using the term_frequency function (Term Frequency) with the parameter compute_vocab set to TRUE.
+
output_table
+
TEXT. The output table with per-topic description generated by this helper function. It has the following columns:
+
+
topicid
INTEGER. Topic id.
+
+
wordid
INTEGER. Word id.
+
+
prob
DOUBLE PRECISION. Probability that this topic will generate the word.
+
+
word
TEXT. Word in text form.
+
+
+
top_k
+
TEXT. The desired number of top words to show for each topic.
TEXT. The model table generated by the training process.
+
output_table
+
TEXT. The output table with per-word topic counts generated by this helper function. It has the following columns:
+
+
wordid
INTEGER. Word id.
+
+
topic_count
INTEGER[]. Count of word association with each topic, i.e., shows how many times a given word is assigned to a topic. Array is of length number of topics.
TEXT. The model table generated by the training process.
+
output_table
+
TEXT. The output table with per-topic word counts generated by this helper function. It has the following columns:
+
+
topicid
INTEGER. Topic id.
+
+
word_count
INTEGER[]. Array showing which words are associated with the topic by frequency. Array is of length number of words.
+
+
+
+
Per-document word to topic mapping
+
Applies to both LDA training and LDA prediction.
+
+lda_get_word_topic_mapping( output_data_table, -- From training or prediction
+ output_table
+ )
+
Arguments
+
output_data_table
+
TEXT. The output data table generated by either LDA training or LDA prediction.
+
output_table
+
TEXT. The output table with word to topic mappings generated by this helper function. It has the following columns:
+
+
docid
INTEGER. Document id.
+
+
wordid
INTEGER. Word id.
+
+
topicid
INTEGER. Topic id.
+
+
+
+
Examples
+
+
Prepare a training dataset for LDA. The examples below are small strings extracted from various Wikipedia documents:
+DROP TABLE IF EXISTS documents;
+CREATE TABLE documents(docid INT4, contents TEXT);
+INSERT INTO documents VALUES
+(0, 'Statistical topic models are a class of Bayesian latent variable models, originally developed for analyzing the semantic content of large document corpora.'),
+(1, 'By the late 1960s, the balance between pitching and hitting had swung in favor of the pitchers. In 1968 Carl Yastrzemski won the American League batting title with an average of just .301, the lowest in history.'),
+(2, 'Machine learning is closely related to and often overlaps with computational statistics; a discipline that also specializes in prediction-making. It has strong ties to mathematical optimization, which deliver methods, theory and application domains to the field.'),
+(3, 'California''s diverse geography ranges from the Sierra Nevada in the east to the Pacific Coast in the west, from the Redwood–Douglas fir forests of the northwest, to the Mojave Desert areas in the southeast. The center of the state is dominated by the Central Valley, a major agricultural area.');
+
You can apply stemming, stop word removal and tokenization at this point in order to prepare the documents for text processing. Depending upon your database version, various tools are available. Databases based on more recent versions of PostgreSQL may do something like:
+SELECT tsvector_to_array(to_tsvector('english',contents)) from documents;
+
In this example, we assume a database based on an older version of PostgreSQL and just perform basic punctuation removal and tokenization. The array of words is added as a new column to the documents table:
+ALTER TABLE documents ADD COLUMN words TEXT[];
+UPDATE documents SET words =
+ regexp_split_to_array(lower(
+ regexp_replace(contents, E'[,.;\']','', 'g')
+ ), E'[\\s+]');
+\x on
+SELECT * FROM documents ORDER BY docid;
+
+-[ RECORD 1 ]---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
+docid | 0
+contents | Statistical topic models are a class of Bayesian latent variable models, originally developed for analyzing the semantic content of large document corpora.
+words | {statistical,topic,models,are,a,class,of,bayesian,latent,variable,models,originally,developed,for,analyzing,the,semantic,content,of,large,document,corpora}
+-[ RECORD 2 ]---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
+docid | 1
+contents | By the late 1960s, the balance between pitching and hitting had swung in favor of the pitchers. In 1968 Carl Yastrzemski won the American League batting title with an average of just .301, the lowest in history.
+words | {by,the,late,1960s,the,balance,between,pitching,and,hitting,had,swung,in,favor,of,the,pitchers,in,1968,carl,yastrzemski,won,the,american,league,batting,title,with,an,average,of,just,301,the,lowest,in,history}
+-[ RECORD 3 ]---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
+docid | 2
+contents | Machine learning is closely related to and often overlaps with computational statistics; a discipline that also specializes in prediction-making. It has strong ties to mathematical optimization, which deliver methods, theory and application domains to the field.
+words | {machine,learning,is,closely,related,to,and,often,overlaps,with,computational,statistics,a,discipline,that,also,specializes,in,prediction-making,it,has,strong,ties,to,mathematical,optimization,which,deliver,methods,theory,and,application,domains,to,the,field}
+-[ RECORD 4 ]---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
+docid | 3
+contents | California's diverse geography ranges from the Sierra Nevada in the east to the Pacific Coast in the west, from the Redwood–Douglas fir forests of the northwest, to the Mojave Desert areas in the southeast. The center of the state is dominated by the Central Valley, a major agricultural area.
+words | {californias,diverse,geography,ranges,from,the,sierra,nevada,in,the,east,to,the,pacific,coast,in,the,west,from,the,redwood–douglas,fir,forests,of,the,northwest,to,the,mojave,desert,areas,in,the,southeast,the,center,of,the,state,is,dominated,by,the,central,valley,a,major,agricultural,area}
+
+
Build a word count table by extracting the words and building a histogram for each document using the term_frequency function (Term Frequency).
+DROP TABLE IF EXISTS documents_tf, documents_tf_vocabulary;
+SELECT madlib.term_frequency('documents', -- input table
+ 'docid', -- document id column
+ 'words', -- vector of words in document
+ 'documents_tf', -- output documents table with term frequency
+ TRUE); -- TRUE to created vocabulary table
+\x off
+SELECT * FROM documents_tf ORDER BY docid LIMIT 20;
+
Here is the associated vocabulary table. Note that wordid starts at 0:
+SELECT * FROM documents_tf_vocabulary ORDER BY wordid LIMIT 20;
+
+ wordid | word
+--------+--------------
+ 0 | 1960s
+ 1 | 1968
+ 2 | 301
+ 3 | a
+ 4 | agricultural
+ 5 | also
+ 6 | american
+ 7 | an
+ 8 | analyzing
+ 9 | and
+ 10 | application
+ 11 | are
+ 12 | area
+ 13 | areas
+ 14 | average
+ 15 | balance
+ 16 | batting
+ 17 | bayesian
+ 18 | between
+ 19 | by
+(20 rows)
+
The total number of words in the vocabulary across all documents is:
+SELECT COUNT(*) FROM documents_tf_vocabulary;
+
+ count
++------
+ 103
+(1 row)
+
+
Train LDA model. For Dirichlet priors we use initial rule-of-thumb values of 50/(number of topics) for alpha and 0.01 for beta. Reminder that column names for docid, wordid, and count are currently fixed, so you must use these exact names in the input table. After a successful run of the LDA training function two tables are generated, one for storing the learned model and the other for storing the output data table.
+DROP TABLE IF EXISTS lda_model, lda_output_data;
+SELECT madlib.lda_train( 'documents_tf', -- documents table in the form of term frequency
+ 'lda_model', -- model table created by LDA training (not human readable)
+ 'lda_output_data', -- readable output data table
+ 103, -- vocabulary size
+ 5, -- number of topics
+ 10, -- number of iterations
+ 5, -- Dirichlet prior for the per-doc topic multinomial (alpha)
+ 0.01 -- Dirichlet prior for the per-topic word multinomial (beta)
+ );
+\x on
+SELECT * FROM lda_output_data ORDER BY docid;
+
Review learned model using helper functions. First, we get topic description by top-k words. These are the k words with the highest probability for the topic. Note that if there are ties in probability, more than k words may actually be reported for each topic. Also note that topicid starts at 0:
+DROP TABLE IF EXISTS helper_output_table;
+SELECT madlib.lda_get_topic_desc( 'lda_model', -- LDA model generated in training
+ 'documents_tf_vocabulary', -- vocabulary table that maps wordid to word
+ 'helper_output_table', -- output table for per-topic descriptions
+ 5); -- k: number of top words for each topic
+\x off
+SELECT * FROM helper_output_table ORDER BY topicid, prob DESC LIMIT 40;
+
Get the per-word topic counts. This mapping shows how many times a given word is assigned to a topic. E.g., wordid 3 is assigned to topicid 0 three times:
+DROP TABLE IF EXISTS helper_output_table;
+SELECT madlib.lda_get_word_topic_count( 'lda_model', -- LDA model generated in training
+ 'helper_output_table'); -- output table for per-word topic counts
+SELECT * FROM helper_output_table ORDER BY wordid LIMIT 20;
+
Get the per-topic word counts. This mapping shows which words are associated with each topic by frequency:
+DROP TABLE IF EXISTS topic_word_count;
+SELECT madlib.lda_get_topic_word_count( 'lda_model',
+ 'topic_word_count');
+\x on
+SELECT * FROM topic_word_count ORDER BY topicid;
+
+DROP TABLE IF EXISTS helper_output_table;
+SELECT madlib.lda_get_word_topic_mapping('lda_output_data', -- Output table from training
+ 'helper_output_table');
+\x off
+SELECT * FROM helper_output_table ORDER BY docid LIMIT 40;
+
Use a learned LDA model for prediction (that is, to label new documents). In this example, we use the same input table as we used to train, just for demonstration purpose. Normally, the test document is a new one that we want to predict on.
+DROP TABLE IF EXISTS outdata_predict;
+SELECT madlib.lda_predict( 'documents_tf', -- Document to predict
+ 'lda_model', -- LDA model from training
+ 'outdata_predict' -- Output table for predict results
+ );
+\x on
+SELECT * FROM outdata_predict;
+
The test table is expected to be in the same form as the training table and can be created with the same process. The LDA prediction results have the same format as the output table generated by the LDA training function.
+
Review prediction using helper function. (This is the same per-document word to topic mapping that we used on the learned model.)
+DROP TABLE IF EXISTS helper_output_table;
+SELECT madlib.lda_get_word_topic_mapping('outdata_predict', -- Output table from prediction
+ 'helper_output_table');
+\x off
+SELECT * FROM helper_output_table ORDER BY docid LIMIT 40;
+
Perplexity by iteration. Now let's look at how perplexity changes from one iteration to the next:
+DROP TABLE IF EXISTS lda_model_perp, lda_output_data_perp;
+SELECT madlib.lda_train( 'documents_tf', -- documents table in the form of term frequency
+ 'lda_model_perp', -- model table created by LDA training (not human readable)
+ 'lda_output_data_perp', -- readable output data table
+ 103, -- vocabulary size
+ 5, -- number of topics
+ 30, -- number of iterations
+ 5, -- Dirichlet prior for the per-doc topic multinomial (alpha)
+ 0.01, -- Dirichlet prior for the per-topic word multinomial (beta)
+ 2, -- Evaluate perplexity every n iterations
+ 0.3 -- Tolerance to stop iteration
+ );
+\x on
+SELECT voc_size, topic_num, alpha, beta, num_iterations, perplexity, perplexity_iters from lda_model_perp;
+
Iterating stops at 14 since the tolerance is reached. There are 7 perplexity values because we computed it only every 2nd iteration to save time. As expected, the perplexity on the training data is that same as the final iteration value:
+\x off
+SELECT madlib.lda_get_perplexity( 'lda_model_perp',
+ 'lda_output_data_perp'
+ );
+
The linalg module consists of useful utility functions for basic linear algebra operations. Several of these functions can be used while implementing new algorithms. These functions operate on vectors (1-D FLOAT8 array) and matrices (2-D FLOAT8 array). Note that other array types may need to be casted into FLOAT8[] before calling the functions.
+
Refer to the linalg.sql_in file for documentation on each of the utility functions.
Linear regression models a linear relationship of a scalar dependent variable \( y \) to one or more explanatory independent variables \( x \) and builds a model of coefficients.
+
Training Function
+
The linear regression training function has the following syntax.
TEXT. Name of the table containing the training data.
+
+
+
out_table
+
TEXT. Name of the generated table containing the output model.
+
The output table contains the following columns:
+
+
<...>
Any grouping columns provided during training. Present only if the grouping option is used.
+
+
coef
FLOAT8[]. Vector of the coefficients of the regression.
+
+
r2
FLOAT8. R-squared coefficient of determination of the model.
+
+
std_err
FLOAT8[]. Vector of the standard error of the coefficients.
+
+
t_stats
FLOAT8[]. Vector of the t-statistics of the coefficients.
+
+
p_values
FLOAT8[]. Vector of the p-values of the coefficients.
+
+
condition_no
FLOAT8 array. The condition number of the \(X^{*}X\) matrix. A high condition number is usually an indication that there may be some numeric instability in the result yielding a less reliable model. A high condition number often results when there is a significant amount of colinearity in the underlying design matrix, in which case other regression techniques, such as elastic net regression, may be more appropriate.
+
+
bp_stats
FLOAT8. The Breush-Pagan statistic of heteroskedacity. Present only if the heteroskedacity argument was set to True when the model was trained.
+
+
bp_p_value
FLOAT8. The Breush-Pagan calculated p-value. Present only if the heteroskedacity parameter was set to True when the model was trained.
+
+
num_rows_processed
BIGINT. The number of rows that are actually used in each group.
+
+
num_missing_rows_skipped
INTEGER. The number of rows that have NULL values in the dependent and independent variables, and were skipped in the computation for each group.
+
+
variance_covariance
FLOAT[]. Variance/covariance matrix.
+
+
A summary table named <out_table>_summary is created together with the output table. It has the following columns:
+
+
method
'linregr' for linear regression.
+
+
source_table
The data source table name
+
+
out_table
The output table name
+
+
dependent_varname
The dependent variable
+
+
independent_varname
The independent variables
+
+
num_rows_processed
The total number of rows that were used in the computation.
+
+
num_missing_rows_skipped
The total number of rows that were skipped because of NULL values in them.
+
+
grouping_cols
Names of the grouping columns.
+
+
+
+
Note
For p-values, we just return the computation result directly. Other statistical packages like 'R' produce the same result, but on printing the result to screen, another format function is used and any p-value that is smaller than the machine epsilon (the smallest positive floating-point number 'x' such that '1 + x != 1') will be printed on screen as "< xxx" (xxx is the value of the machine epsilon). Although the result may look different, they are in fact the same.
+
+
dependent_varname
+
TEXT. Expression to evaluate for the dependent variable.
+
+
+
independent_varname
+
TEXT. Expression list to evaluate for the independent variables. An intercept variable is not assumed. It is common to provide an explicit intercept term by including a single constant 1 term in the independent variable list.
+
+
+
grouping_cols (optional)
+
TEXT, default: NULL. An expression list used to group the input dataset into discrete groups, running one regression per group. Similar to the SQL GROUP BY clause. When this value is null, no grouping is used and a single result model is generated for the whole data set.
+
+
+
heteroskedasticity_option (optional)
+
BOOLEAN, default: FALSE. When TRUE, the heteroskedasticity of the model is also calculated and returned with the results.
+
+
Warning
The aggregate 'linregr' has been deprecated in favor of the function 'linregr_train'. If the aggregate 'linregr' is used to output the results of linear regression to a table, it is recommended to follow the general pattern shown below (replace text within '<...>' with the appropriate variable names).
+CREATE TABLE <output table> AS
+SELECT (r).*
+FROM (
+ SELECT linregr(<dependent variable>, <independent variable>) as r
+ FROM <source table>
+ ) q;
+
+
Prediction Function
The prediction function is as follows:
+linregr_predict(coef, col_ind)
+
Arguments
+
coef
+
FLOAT8[]. Vector of the coefficients of regression from training.
+
col_ind
+
FLOAT8[]. An array containing the independent variable column names, as was used for the training.
(Note that in this example we are dynamically creating the array of independent variables from column names. If you have large numbers of independent variables beyond the PostgreSQL limit of maximum columns per table, you would pre-build the arrays and store them in a single column.)
+
Next we generate three output models, one for each value of "bedroom".
Alternatively you can unnest the results for easier reading of output.
+\x OFF
+SELECT unnest(ARRAY['intercept','tax','bath','size']) as attribute,
+ unnest(coef) as coefficient,
+ unnest(std_err) as standard_error,
+ unnest(t_stats) as t_stat,
+ unnest(p_values) as pvalue
+FROM houses_linregr;
+
Compare predicted price with actual. (This example uses the original data table to perform the prediction. Typically a different test dataset with the same features as the original training dataset would be used for prediction.)
+\x OFF
+SELECT houses.*,
+ madlib.linregr_predict( m.coef,
+ ARRAY[1,tax,bath,size]
+ ) as predict,
+ price -
+ madlib.linregr_predict( m.coef,
+ ARRAY[1,tax,bath,size]
+ ) as residual
+FROM houses, houses_linregr m ORDER BY id;
+
All table names can be optionally schema qualified (current_schemas() would be searched if a schema name is not provided) and all table and column names should follow case-sensitivity and quoting rules per the database. (For instance, 'mytable' and 'MyTable' both resolve to the same entity, i.e. 'mytable'. If mixed-case or multi-byte characters are desired for entity names then the string should be double-quoted; in this case the input would be '"MyTable"').
+
Technical Background
+
Ordinary least-squares (OLS) linear regression refers to a stochastic model in which the conditional mean of the dependent variable (usually denoted \( Y \)) is an affine function of the vector of independent variables (usually denoted \( \boldsymbol x \)). That is,
for some unknown vector of coefficients \( \boldsymbol c \). The assumption is that the residuals are i.i.d. distributed Gaussians. That is, the (conditional) probability density of \( Y \) is given by
OLS linear regression finds the vector of coefficients \( \boldsymbol c \) that maximizes the likelihood of the observations.
+
Let
+
\( \boldsymbol y \in \mathbf R^n \) denote the vector of observed dependent variables, with \( n \) rows, containing the observed values of the dependent variable,
+
\( X \in \mathbf R^{n \times k} \) denote the design matrix with \( k \) columns and \( n \) rows, containing all observed vectors of independent variables. \( \boldsymbol x_i \) as rows,
+
\( X^T \) denote the transpose of \( X \),
+
\( X^+ \) denote the pseudo-inverse of \( X \).
+
+
Maximizing the likelihood is equivalent to maximizing the log-likelihood \( \sum_{i=1}^n \log f(y_i \mid \boldsymbol x_i) \), which simplifies to minimizing the residual sum of squares \( RSS \) (also called sum of squared residuals or sum of squared errors of prediction),
+\[ RSS = \sum_{i=1}^n ( y_i - \boldsymbol c^T \boldsymbol x_i )^2 = (\boldsymbol y - X \boldsymbol c)^T (\boldsymbol y - X \boldsymbol c) \,. \]
+
+
The first-order conditions yield that the \( RSS \) is minimized at
+\[ \boldsymbol c = (X^T X)^+ X^T \boldsymbol y \,. \]
+
+
Computing the total sum of squares \( TSS \), the explained sum of squares \( ESS \) (also called the regression sum of squares), and the coefficient of determination \( R^2 \) is done according to the following formulas:
+\begin{align*} ESS & = \boldsymbol y^T X \boldsymbol c - \frac{ \| y \|_1^2 }{n} \\ TSS & = \sum_{i=1}^n y_i^2 - \frac{ \| y \|_1^2 }{n} \\ R^2 & = \frac{ESS}{TSS} \end{align*}
+
+
Note: The last equality follows from the definition \( R^2 = 1 - \frac{RSS}{TSS} \) and the fact that for linear regression \( TSS = RSS + ESS \). A proof of the latter can be found, e.g., at: http://en.wikipedia.org/wiki/Sum_of_squares
+
We estimate the variance \( Var[Y - \boldsymbol c^T \boldsymbol x \mid \boldsymbol x] \) as
+\[ \sigma^2 = \frac{RSS}{n - k} \]
+
+
and compute the t-statistic for coefficient \( i \) as
The \( p \)-value for coefficient \( i \) gives the probability of seeing a value at least as extreme as the one observed, provided that the null hypothesis ( \( c_i = 0 \)) is true. Letting \( F_\nu \) denote the cumulative density function of student-t with \( \nu \) degrees of freedom, the \( p \)-value for coefficient \( i \) is therefore
where \( T \) is a student-t distributed random variable with mean 0.
+
The condition number [2] \( \kappa(X) = \|X\|_2\cdot\|X^{-1}\|_2\) is computed as the product of two spectral norms [3]. The spectral norm of a matrix \(X\) is the largest singular value of \(X\) i.e. the square root of the largest eigenvalue of the positive-semidefinite matrix \(X^{*}X\):
where \(X^{*}\) is the conjugate transpose of \(X\). The condition number of a linear regression problem is a worst-case measure of how sensitive the result is to small perturbations of the input. A large condition number (say, more than 1000) indicates the presence of significant multicollinearity.
This module implements "factor model" for representing an incomplete matrix using a low-rank approximation [1]. Mathematically, this model seeks to find matrices U and V (also referred as factors) that, for any given incomplete matrix A, minimizes:
+
+\[ \|\boldsymbol A - \boldsymbol UV^{T} \|_2 \]
+
+
subject to \(rank(\boldsymbol UV^{T}) \leq r\), where \(\|\cdot\|_2\) denotes the Frobenius norm. Let \(A\) be a \(m \times n\) matrix, then \(U\) will be \(m \times r\) and \(V\) will be \(n \times r\), in dimension, and \(1 \leq r \ll \min(m, n)\). This model is not intended to do the full decomposition, or to be used as part of inverse procedure. This model has been widely used in recommendation systems (e.g., Netflix [2]) and feature selection (e.g., image processing [3]).
+
Function Syntax
+
Low-rank matrix factorization of an incomplete matrix into two factors.
TEXT. The name of the table to receive the output.
+
Output factors matrix U and V are in a flattened format.
RESULT AS (
+ matrix_u DOUBLE PRECISION[],
+ matrix_v DOUBLE PRECISION[],
+ rmse DOUBLE PRECISION
+);
Features correspond to row i is matrix_u[i:i][1:r]. Features correspond to column j is matrix_v[j:j][1:r].
+
+
rel_source
+
TEXT. The name of the table containing the input data.
+
The input matrix> is expected to be of the following form:
{TABLE|VIEW} input_table (
+ row INTEGER,
+ col INTEGER,
+ value DOUBLE PRECISION
+)
Input is contained in a table that describes an incomplete matrix, with available entries specified as (row, column, value). The input matrix should be 1-based, which means row >= 1, and col >= 1. NULL values are not expected.
+
+
col_row
+
TEXT. The name of the column containing the row number.
+
col_column
+
TEXT. The name of the column containing the column number.
+
col_value
+
DOUBLE PRECISION. The value at (row, col).
+
row_dim (optional)
+
INTEGER, default: "SELECT max(col_row) FROM rel_source". The number of columns in the matrix.
+
column_dim (optional)
+
INTEGER, default: "SELECT max(col_col) FROM rel_source". The number of rows in the matrix.
+
max_rank
+
INTEGER, default: 20. The rank of desired approximation.
+
stepsize (optional)
+
DOUBLE PRECISION, default: 0.01. Hyper-parameter that decides how aggressive the gradient steps are.
+
scale_factor (optional)
+
DOUBLE PRECISION, default: 0.1. Hyper-parameter that decides scale of initial factors.
+
num_iterations (optional)
+
INTEGER, default: 10. Maximum number if iterations to perform regardless of convergence.
+
tolerance (optional)
+
DOUBLE PRECISION, default: 0.0001. Acceptable level of error in convergence.
+
+
Examples
+
+
Prepare an input table/view:
+DROP TABLE IF EXISTS lmf_data;
+CREATE TABLE lmf_data (
+ row INT,
+ col INT,
+ val FLOAT8
+);
+
+
Populate the input table with some data.
+INSERT INTO lmf_data VALUES (1, 1, 5.0);
+INSERT INTO lmf_data VALUES (3, 100, 1.0);
+INSERT INTO lmf_data VALUES (999, 10000, 2.0);
+
[1] N. Srebro and T. Jaakkola. “Weighted Low-Rank Approximations.” In: ICML. Ed. by T. Fawcett and N. Mishra. AAAI Press, 2003, pp. 720–727. isbn: 1-57735-189-4.
[3] J. Wright, A. Ganesh, S. Rao, Y. Peng, and Y. Ma. “Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Matrices via Convex Optimization.” In: NIPS. Ed. by Y. Bengio, D. Schuurmans, J. D. Lafferty, C. K. I. Williams, and A. Culotta. Curran Associates, Inc., 2009, pp. 2080–2088. isbn: 9781615679119.
+
+
+
+
+
+
Generated on Mon Aug 28 2023 08:03:55 for MADlib by
+
+ 1.8.13
Binomial logistic regression models the relationship between a dichotomous dependent variable and one or more predictor variables. The dependent variable may be a Boolean value or a categorial variable that can be represented with a Boolean expression. The probabilities describing the possible outcomes of a single trial are modeled, as a function of the predictor variables, using a logistic function.
+
Training Function
The logistic regression training function has the following format:
TEXT. Name of the table containing the training data.
+
+
+
out_table
+
TEXT. Name of the generated table containing the output model.
+
The output table produced by the logistic regression training function contains the following columns:
+
+
+
<...>
TEXT. Grouping columns, if provided in input. This could be multiple columns depending on the grouping_cols input.
+
+
+
+
coef
FLOAT8. Vector of the coefficients of the regression.
+
+
+
+
log_likelihood
FLOAT8. The log-likelihood \( l(\boldsymbol c) \).
+
+
+
+
std_err
FLOAT8[]. Vector of the standard error of the coefficients.
+
+
+
+
z_stats
FLOAT8[]. Vector of the z-statistics of the coefficients.
+
+
+
+
p_values
FLOAT8[]. Vector of the p-values of the coefficients.
+
+
+
+
odds_ratios
FLOAT8[]. The odds ratio, \( \exp(c_i) \).
+
+
+
+
condition_no
FLOAT8[]. The condition number of the \(X^{*}X\) matrix. A high condition number is usually an indication that there may be some numeric instability in the result yielding a less reliable model. A high condition number often results when there is a significant amount of colinearity in the underlying design matrix, in which case other regression techniques may be more appropriate.
+
+
+
+
num_rows_processed
INTEGER. The number of rows actually processed, which is equal to the total number of rows in the source table minus the number of skipped rows.
+
+
+
+
num_missing_rows_skipped
INTEGER. The number of rows skipped during the training. A row will be skipped if the independent_varname is NULL or contains NULL values.
+
+
+
+
num_iterations
INTEGER. The number of iterations actually completed. This would be different from the nIterations argument if a tolerance parameter is provided and the algorithm converges before all iterations are completed.
+
+
+
+
variance_covariance
FLOAT[]. Variance/covariance matrix.
+
+
A summary table named <out_table>_summary is also created at the same time, which has the following columns:
+
+
method
'logregr' for logistic regression.
+
+
+
+
source_table
The data source table name.
+
+
+
+
out_table
The output table name.
+
+
+
+
dependent_varname
The dependent variable name.
+
+
+
+
independent_varname
The independent variable names.
+
+
+
+
optimizer_params
A string that contains all the optimizer parameters, and has the form of 'optimizer=..., max_iter=..., tolerance=...'
+
+
+
+
num_all_groups
How many groups of data were fit by the logistic model.
+
+
+
+
num_failed_groups
How many groups failed in training.
+
+
+
+
num_rows_processed
The total number of rows used in the computation.
+
+
+
+
num_missing_rows_skipped
The total number of rows skipped.
+
+
+
+
grouping_cols
Names of the grouping columns.
+
+
+
+
dependent_varname
+
TEXT. Name of the dependent variable column (of type BOOLEAN) in the training data, or an expression evaluating to a BOOLEAN.
+
+
+
independent_varname
+
TEXT. Expression list to evaluate for the independent variables. An intercept variable is not assumed so it is common to provide an explicit intercept term by including a single constant 1 term in the independent variable list.
+
+
+
grouping_cols (optional)
+
TEXT, default: NULL. An expression list used to group the input dataset into discrete groups, running one regression per group. Similar to the SQL "GROUP BY" clause. When this value is NULL, no grouping is used and a single model is generated for the whole data set.
+
+
+
max_iter (optional)
+
INTEGER, default: 20. The maximum number of iterations allowed.
+
+
+
optimizer (optional)
+
TEXT, default: 'irls'. The name of the optimizer to use:
+
+
'newton' or 'irls'
Iteratively reweighted least squares
+
+
'cg'
conjugate gradient
+
+
'igd'
incremental gradient descent.
+
+
+
+
tolerance (optional)
+
FLOAT8, default: 0.0001. The difference between log-likelihood values in successive iterations that indicate convergence. A zero disables the convergence criterion, so that execution stops after the maximum iterations have completed, as set in the 'max_iter' parameter above.
+
+
+
verbose (optional)
+
BOOLEAN, default: FALSE. Provides verbose output of the results of training.
+
+
+
Note
For p-values, we just return the computation result directly. Other statistical packages like 'R' produce the same result, but on printing the result to screen, another format function is used and any p-value that is smaller than the machine epsilon (the smallest positive floating-point number 'x' such that '1 + x != 1') will be printed on screen as "< xxx" (xxx is the value of the machine epsilon). Although the result may look different, they are in fact the same.
+
Prediction Function
Two prediction functions are provided. One predicts the boolean value of the dependent variable, and the other predicts the probability of the value of the dependent variable being 'True'. Syntax is the same for both functions.
+
The function to predict the boolean value (True/False) of the dependent variable has the following syntax:
+logregr_predict(coefficients,
+ ind_var
+ )
+
The function to predict the probability of the dependent variable being 'True' has the following syntax:
DOUBLE PRECISION[]. Model coefficients obtained from training logregr_train().
+
+
+
ind_var
+
Independent variables expressed as a DOUBLE array. This should be the same length as the array obtained by evaluation of the 'independent_varname' argument in logregr_train().
+
+
Examples
+
Create the training data table. This data set is related to predicting a second heart attack given treatment and health factors.
Note that in the example above we are dynamically creating the array of independent variables from column names. If you have large numbers of independent variables beyond the PostgreSQL limit of maximum columns per table, you would typically pre-build the arrays and store them in a single column.
+
View the regression results.
+-- Set extended display on for easier reading of output
+\x on
+SELECT * from patients_logregr;
+
Alternatively, unnest the arrays in the results for easier reading of output:
+\x off
+SELECT unnest(array['intercept', 'treatment', 'trait_anxiety']) as attribute,
+ unnest(coef) as coefficient,
+ unnest(std_err) as standard_error,
+ unnest(z_stats) as z_stat,
+ unnest(p_values) as pvalue,
+ unnest(odds_ratios) as odds_ratio
+ FROM patients_logregr;
+
Predict the dependent variable using the logistic regression model. (This example uses the original data table to perform the prediction. Typically a different test dataset with the same features as the original training dataset would be used for prediction.)
+\x off
+-- Display prediction value along with the original value
+SELECT p.id, madlib.logregr_predict(coef, ARRAY[1, treatment, trait_anxiety]),
+ p.second_attack::BOOLEAN
+FROM patients p, patients_logregr m
+ORDER BY p.id;
+
Result:
+ id | logregr_predict | second_attack
+----+-----------------+---------------
+ 1 | t | t
+ 2 | t | t
+ 3 | f | t
+ 4 | t | t
+ 5 | f | t
+ 6 | t | t
+ 7 | t | t
+ 8 | t | t
+ 9 | t | t
+ 10 | t | t
+ 11 | t | f
+ 12 | f | f
+ 13 | f | f
+ 14 | f | f
+ 15 | f | f
+ 16 | f | f
+ 17 | t | f
+ 18 | f | f
+ 19 | f | f
+ 20 | t | f
+(20 rows)
+
+
Predict the probability of the dependent variable being TRUE.
+\x off
+-- Display prediction value along with the original value
+SELECT p.id, madlib.logregr_predict_prob(coef, ARRAY[1, treatment, trait_anxiety]),
+ p.second_attack::BOOLEAN
+FROM patients p, patients_logregr m
+ORDER BY p.id;
+
Result:
+ id | logregr_predict_prob | second_attack
+----+----------------------+---------------
+ 1 | 0.720223028941527 | t
+ 2 | 0.894354902502048 | t
+ 3 | 0.192269541755171 | t
+ 4 | 0.685513072239347 | t
+ 5 | 0.167747881508857 | t
+ 6 | 0.79809810891514 | t
+ 7 | 0.928568075752503 | t
+ 8 | 0.959305763693571 | t
+ 9 | 0.877576117431452 | t
+ 10 | 0.685513072239347 | t
+ 11 | 0.586700895943317 | f
+ 12 | 0.192269541755171 | f
+ 13 | 0.116032010632994 | f
+ 14 | 0.0383829143134982 | f
+ 15 | 0.0674976224147597 | f
+ 16 | 0.192269541755171 | f
+ 17 | 0.545870774302621 | f
+ 18 | 0.267675422387132 | f
+ 19 | 0.398618639285111 | f
+ 20 | 0.685513072239347 | f
+(20 rows)
+
+
+
+
Notes
All table names can be optionally schema qualified (current_schemas() would be searched if a schema name is not provided) and all table and column names should follow case-sensitivity and quoting rules per the database. (For instance, 'mytable' and 'MyTable' both resolve to the same entity, i.e. 'mytable'. If mixed-case or multi-byte characters are desired for entity names then the string should be double-quoted; in this case the input would be '"MyTable"').
+
Technical Background
+
(Binomial) logistic regression refers to a stochastic model in which the conditional mean of the dependent dichotomous variable (usually denoted \( Y \in \{ 0,1 \} \)) is the logistic function of an affine function of the vector of independent variables (usually denoted \( \boldsymbol x \)). That is,
for some unknown vector of coefficients \( \boldsymbol c \) and where \( \sigma(x) = \frac{1}{1 + \exp(-x)} \) is the logistic function. Logistic regression finds the vector of coefficients \( \boldsymbol c \) that maximizes the likelihood of the observations.
+
Let
+
\( \boldsymbol y \in \{ 0,1 \}^n \) denote the vector of observed dependent variables, with \( n \) rows, containing the observed values of the dependent variable,
+
\( X \in \mathbf R^{n \times k} \) denote the design matrix with \( k \) columns and \( n \) rows, containing all observed vectors of independent variables \( \boldsymbol x_i \) as rows.
The Hessian of this objective is \( H = -X^T A X \) where \( A = \text{diag}(a_1, \dots, a_n) \) is the diagonal matrix with \( a_i = \sigma(\boldsymbol c^T \boldsymbol x) \cdot \sigma(-\boldsymbol c^T \boldsymbol x) \,. \) Since \( H \) is non-positive definite, \( l(\boldsymbol c) \) is convex. There are many techniques for solving convex optimization problems. Currently, logistic regression in MADlib can use one of three algorithms:
+
Iteratively Reweighted Least Squares
+
A conjugate-gradient approach, also known as Fletcher-Reeves method in the literature, where we use the Hestenes-Stiefel rule for calculating the step size.
+
Incremental gradient descent, also known as incremental gradient methods or stochastic gradient descent in the literature.
+
+
We estimate the standard error for coefficient \( i \) as
The Wald \( p \)-value for coefficient \( i \) gives the probability (under the assumptions inherent in the Wald test) of seeing a value at least as extreme as the one observed, provided that the null hypothesis ( \( c_i = 0 \)) is true. Letting \( F \) denote the cumulative density function of a standard normal distribution, the Wald \( p \)-value for coefficient \( i \) is therefore
where \( Z \) is a standard normally distributed random variable.
+
The odds ratio for coefficient \( i \) is estimated as \( \exp(c_i) \).
+
The condition number is computed as \( \kappa(X^T A X) \) during the iteration immediately preceding convergence (i.e., \( A \) is computed using the coefficients of the previous iteration). A large condition number (say, more than 1000) indicates the presence of significant multicollinearity.
+
Literature
+
A selection of references pertaining to logistic regression, with some good pointers to other literature.
[3] Paul Komarek, Andrew W. Moore: Making Logistic Regression A Core Data Mining Tool With TR-IRLS, IEEE International Conference on Data Mining 2005, pp. 685-688, http://komarix.org/ac/papers/tr-irls.short.pdf
+
[4] D. P. Bertsekas: Incremental gradient, subgradient, and proximal methods for convex optimization: a survey, Technical report, Laboratory for Information and Decision Systems, 2010, http://web.mit.edu/dimitrib/www/Incremental_Survey_LIDS.pdf
A marginal effect (ME) or partial effect measures the effect on the conditional mean of \( y \) for a change in one of the regressors, say \(X_k\). In the linear regression model, the ME equals the relevant slope coefficient, greatly simplifying analysis. For nonlinear models, specialized algorithms are required for calculating ME. The marginal effect computed is the average of the marginal effect at every data point present in the source table.
+
MADlib provides marginal effects regression functions for linear, logistic and multinomial logistic regressions. The implementation is similar to reference [1].
VARCHAR. The name of the result table. The output table has the following columns.
+
+
variables
INTEGER[]. The indices of the basis variables.
+
+
margins
DOUBLE PRECISION[]. The marginal effects.
+
+
std_err
DOUBLE PRECISION[]. An array of the standard errors, computed using the delta method.
+
+
z_stats
DOUBLE PRECISION[]. An array of the z-stats of the marginal effects.
+
+
p_values
DOUBLE PRECISION[]. An array of the Wald p-values of the marginal effects.
+
+
+
x_design (optional)
+
VARCHAR, default: NULL. The design of independent variables, necessary only if interaction term or indicator (categorical) terms are present. This parameter is necessary since the independent variables in the underlying regression is not parsed to extract the relationship between variables.
+
Example: The independent_varname in the regression method can be specified in either of the following ways:
In the second version, the column x is an array containing data identical to that expressed in the first version, computed in a prior data preparation step. Supply an x_design argument to the margins() function in the following way:
The variable names ('gpa', 'weight', ...), referred to here as identifiers, should be unique for each basis variable and need not be the same as the original variable name in independent_varname. They should, however, be in the same order as the corresponding variables in independent_varname. The length of x_design is expected to be the same as the length of independent_varname. Each identifier name can contain only alphanumeric characters and the underscore.
+
Indicator (dummy) variables are prefixed with an 'i.' (This is only necessary for the basis term; it is not needed in the interaction terms.) Indicator variables that are obtained from the same categorical variable (for example, 'color_blue' and 'color_green') need to have a common and unique suffix (for example, '.color'). The '.' is used to add the prefix and suffix. If a reference indicator variable is present, it should contain the prefix 'ir.'.
+
An identifier may contain alphanumeric characters and underscores. To include other characters, the string must be double-quoted. Escape-characters are not currently supported.
+
+
+
source_table (optional)
+
VARCHAR, default: NULL. Name of the data table to apply marginal effects on. If not provided or NULL then the marginal effects are computed on the training table.
+
+
+
marginal_vars (optional)
+
VARCHAR, default: NULL. Comma-separated string containing specific variable identifiers to calculate marginal effects on. When NULL, marginal effects for all variables are returned.
+
+
+
Note
No output will be provided for the reference indicator variable, since the marginal effect for that variable is undefined. If a reference variable is included in the independent variables and marginal_vars, the margins() function will ignore that variable for the output. The variable can still be included in the regression and margins, since it will affect the values for other related indicator variables.
+
Marginal Effects for Logistic Regression
+
Warning
This function has been deprecated in favor of the margins() function.
VARCHAR. The name of the result table. The output table has the following columns.
+
+
margins
DOUBLE PRECISION[]. The marginal effects.
+
+
std_err
DOUBLE PRECISION[]. An array of the standard errors, using the delta method.
+
+
z_stats
DOUBLE PRECISION[]. An array of the z-stats of the marginal effects.
+
+
p_values
DOUBLE PRECISION[]. An array of the Wald p-values of the marginal effects.
+
+
A summary table named <output_table>_summary is also created, which is the same as the summary table created by logregr_train() function. Refer to the documentation for logistic regression for details.
+
+
+
dependent_variable
+
VARCHAR. The name of the column for dependent variables.
+
independent_variable
+
VARCHAR. The name of the column for independent variables. Can be any SQL expression that evaluates to an array.
+
grouping_cols (optional)
+
VARCHAR, default: NULL. Not currently implemented. Any non-NULL value is ignored. An expression list used to group the input dataset into discrete groups, running one regression per group. Similar to the SQL "GROUP BY" clause. When this value is NULL, no grouping is used and a single result model is generated.
+
marginal_vars (optional)
+
INTEGER[], default: NULL. An index list (base 1) representing the independent variables to compute marginal effects on. When NULL, computes marginal effects on all variables.
+
max_iter (optional)
+
INTEGER, default: 20. The maximum number of iterations for the logistic regression.
+
optimizer (optional)
+
VARCHAR, default: 'irls'. The optimizer to use for the logistic regression: newton/irls, cg, or igd.
+
tolerance (optional)
+
DOUBLE PRECISION, default: 1e-4. Termination criterion for logistic regression (relative).
+
verbose_mode (optional)
+
BOOLEAN, default FALSE. When TRUE, provides verbose output of the results of training.
+
+
Marginal Effects for Multinomial Logistic Regression
+
Warning
This function has been deprecated in favor of the margins() function.
VARCHAR. The name of result table. The output table has the following columns.
+
+
category
The category.
+
+
ref_category
The refererence category used for modeling.
+
+
margins
DOUBLE PRECISION[]. The marginal effects.
+
+
std_err
DOUBLE PRECISION[]. An array of the standard errors, using the delta method.
+
+
z_stats
DOUBLE PRECISION[]. An array of the z-stats of the marginal effects.
+
+
p_values
DOUBLE PRECISION[]. An array of the Wald p-values of the marginal effects.
+
+
A summary table named <out_table>_summary is also created, which is the same as the summary table created by mlogregr_train() function. Refer to the documentation for multinomial logistic regression for details.
+
+
+
dependent_varname
+
VARCHAR. The name of the column for dependent variables.
+
independent_varname
+
VARCHAR. The name of the column for independent variables. Can be any SQL expression that evaluates to an array.
+
ref_category (optional)
+
INTEGER, default: 0. Reference category for the multinomial logistic regression.
+
grouping_cols (optional)
+
VARCHAR, default: NULL. Not currently implemented. Any non-NULL value is ignored. An expression list used to group the input dataset into discrete groups, running one regression per group. Similar to the SQL "GROUP BY" clause. When this value is NULL, no grouping is used and a single result model is generated.
+
marginal_vars(optional)
+
INTEGER[], default: NULL. An index list (base 1) representing the independent variables to compute marginal effects on. When NULL, computes marginal effects on all variables.
+
optimizer_params (optional)
+
TEXT, default: NULL, which uses the default values of optimizer parameters: max_iter=20, optimizer='newton', tolerance=1e-4. It should be a string that contains 'key=value' pairs separated by commas.
+
verbose_mode (optional)
+
BOOLEAN, default FALSE. When TRUE, provides verbose output of the results of training.
Create a sample data set for multinomial logistic regression. (The full dataset has three categories.) Use the dataset from the Multinomial Regression example.
The marginal_vars argument is a list with the names matching those in 'x_design'. If no 'x_design' is present (i.e. no interaction and no indicator variables), then marginal_vars must be the indices (base 1) of variables in 'independent_varname'. Use NULL to use all independent variables. It is important to note that the independent_varname array in the underlying regression is assumed to start with a lower bound index of 1. Arrays that don't follow this would result in an incorrect solution.
+
Technical Background
+
The standard approach to modeling dichotomous/binary variables (so \(y \in \{0, 1\} \)) is to estimate a generalized linear model under the assumption that \( y \) follows some form of Bernoulli distribution. Thus the expected value of \( y \) becomes,
+\[ y = G(X' \beta), \]
+
+
where G is the specified binomial distribution. For logistic regression, the function \( G \) represents the inverse logit function.
There are several methods for calculating the marginal effects for dichotomous dependent variables. This package uses the average of the marginal effects at every sample observation.
A 'row' column (called row_id above) provides the row number of each row and a 'val' column (called row_vec above) provides each row as an array. The row column should contain a series of integers from 1 to N with no duplicates, where N is the row dimensionality.
+
+
Sparse: The matrix is represented using the row and column indices for each non-zero entry of the matrix. This representation is useful for sparse matrices, containing multiple zero elements. Given below is an example of a sparse 4x7 matrix with just 6 out of 28 entries being non-zero. The dimensionality of the matrix is inferred using the max value in row and col columns. Note the last entry is included (even though it is 0) to provide the dimensionality of the matrix, indicating that the 4th row and 7th column contain all zeros.
For sparse matrices, the row and col columns together should not contain a duplicate entry and the val column should be of scalar (non-array) data type.
+ For comparison, the dense representation of this matrix is shown below. Note the dimensionality of the dense matrix is 4 x 7 since the max value of row and col is 4 and 7 respectively, leading to all zeros in the last row and last column.
The functions below support several numeric types (unless otherwise noted) including SMALLINT, INTEGER, BIGINT, DOUBLE PRECISION (FLOAT8), NUMERIC (internally casted into FLOAT8, so loss of precision can happen).
+
Matrix Operations
+
Below are the supported matrix operations. The meaning of the arguments and other terms are common to all functions and are provided at the end of the list in the glossary of arguments.
+
+
Representation
+-- Convert to sparse representation
+ matrix_sparsify( matrix_in, in_args, matrix_out, out_args)
+
+-- Convert to dense representation
+ matrix_densify( matrix_in, in_args, matrix_out, out_args)
+
+-- Get dimensions of matrix
+ matrix_ndims( matrix_in, in_args )
+
+-- Extract row from matrix given row index
+ matrix_extract_row( matrix_in, in_args, index)
+
+-- Extract column from matrix given column index
+ matrix_extract_col( matrix_in, in_args, index)
+
+-- Extract main (principal) diagonal elements
+ matrix_extract_diag( matrix_in, in_args)
+
+
Reduction operations (aggregate across specific dimension)
+-- Get max value along dim. Returns corresponding index if fetch_index = True
+ matrix_max( matrix_in, in_args, dim, matrix_out, fetch_index)
+
+-- Get min value along dim. Returns corresponding index if fetch_index = True
+ matrix_min( matrix_in, in_args, dim, matrix_out, fetch_index)
+
+-- Get sum value along dimension
+ matrix_sum( matrix_in, in_args, dim)
+
+-- Get mean value along dimension
+ matrix_mean( matrix_in, in_args, dim)
+
+-- Get matrix norm
+ matrix_norm( matrix_in, in_args, norm_type)
+
+
Creation methods
+-- Create a matrix initialized with ones of given row and column dimension
+ matrix_ones( row_dim, col_dim, matrix_out, out_args)
+
+-- Create a matrix initialized with zeros of given row and column dimension
+ matrix_zeros( row_dim, col_dim, matrix_out, out_args)
+
+-- Create an square identity matrix of size dim x dim
+ matrix_identity( dim, matrix_out, out_args)
+
+-- Create a diag matrix initialized with given diagonal elements
+ matrix_diag( diag_elements, matrix_out, out_args)
+-- Create a matrix initialized with values sampled from a distribution
+-- Supported distributions: normal, uniform, bernoulli
+ matrix_random( distribution, row_dim, col_dim, in_args, matrix_out, out_args )
+
+
Decomposition operations
Note
Please note that the decomposition operations below are only implemented for in-memory operations. The matrix data is collected on a single node and the decomposition is computed. This makes the operations applicable to smaller matrices since the calculation is not distributed across multiple nodes.
The table below provides a glossary of the arguments used in the matrix operations.
+
+
matrix_in, matrix_a, matrix_b
+
TEXT. Name of the table containing the input matrix.
+
For functions accepting one matrix, matrix_in denotes the input matrix.
+
For functions accepting two matrices, matrix_a denotes the first matrix and matrix_b denotes the second matrix. These two matrices can independently be in either dense or sparse format.
+
+
+
+
in_args, a_args, b_args
+
TEXT. A comma-delimited string containing multiple named arguments of the form "name=value". This argument is used as a container for multiple parameters related to a single matrix.
+
The following parameters are supported for this string argument:
+
+
row
(Default: 'row_num') Name of the column containing row index of the matrix.
+
+
col
(Default: 'col_num') Name of the column containing column index of the matrix.
+
+
val
(Default: 'val') Name of the column containing the entries of the matrix.
+
+
trans
(Default: False) Boolean flag to indicate if the matrix should be transposed before the operation. This is currently functional only for matrix_mult.
+
+
For example, the string argument with default values will be 'row=row_num, col=col_num, val=val, trans=False'. Alternatively, the string argument can be set to NULL or be blank ('') if default values are to be used.
+
+
+
matrix_out
+
TEXT. Name of the table to store the result matrix.
+
For Cholesky, QR and LU decompositions, a prefix (matrix_out_prefix) is used as a basis to build the names of the various output tables.
+
For Cholesky decomposition ( \( PA = LDL* \)), the following suffixes are added to matrix_out_prefix:
+
_p for row permutation matrix P
+
_l for lower triangular factor L
+
_d for diagonal matrix D
+
+
For QR decomposition ( \( A = QR \)) the following suffixes are added to matrix_out_prefix:
+
_q for orthogonal matrix Q
+
_r for upper triangular factor R
+
+
For LU decomposition with full pivoting ( \( PAQ = LU \)), the following suffixes are added to matrix_out_prefix:
+
_p for row permutation matrix P
+
_q for column permutation matrix Q
+
_l for lower triangular factor L
+
_u for upper triangular factor U
+
+
+
+
out_args
+
TEXT. A comma-delimited string containing named arguments of the form "name=value". This is an optional parameter and the default value is set as follows:
+
For functions with one input matrix, default out_args will the be same as specified in in_args.
+
For functions with two input matrices, default out_args will be the same as specified in a_args.
+
+
The following parameters are supported for this string argument:
+
+
row
Name of the column containing row index of the matrix.
+
+
col
Name of the column containing column index of the matrix.
+
+
val
Name of the column containing the entries of the matrix.
+
+
fmt
Format of the output matrix. It could be either 'dense' or 'sparse'. When 'fmt' is not provided, the output fmt is inferred from the input matrices.
+
+
Note
One exception is for matrix_eigen: the default output column name is ’eigen_values’ in the format (real, imaginary).
+
+
index
+
INTEGER. An integer representing a row or column index of the matrix. Should be a number from 1 to N, where N is the maximum size of the dimension.
+
+
+
dim
+
INTEGER. Should either be 1 or 2. This value indicates the dimension to operate along for the reduction/aggregation operations. The value of dim should be interpreted as the dimension to be flattened i.e. whose length reduces to 1 in the result.
+
For any reduction function applied on an N x M matrix:
+
+
if dim=1, operation is applied on successive elements in each column; return value is a single vector with M elements (i.e. matrix with 1 row and M columns).
+
+
if dim=2, operation is applied on successive elements in each row; return value is a single vector with N elements (i.e. matrix with N rows and 1 column).
+
+
+
+
row_dim
+
INTEGER. Should be greater than 0. This value indicates the row dimension of result matrix.
+
+
+
col_dim
+
INTEGER. Should be greater than 0. This value indicates the column dimension of result matrix.
+
+
+
diag_elements
+
ARRAY OF FLOAT. Should not be empty. This value indicates the float array which is used to generate diag elements of result output matrix.
+
+
+
norm_type
+
TEXT. Optional parameter. Defaults to Frobenius norm. Other supported norms for this string argument:
+
+
+
'one' or 'o'
1 norm
+
+
float > 0
Element-wise norm
+
+
'inf' or 'i'
Infinite norm
+
+
'max' or 'm'
Max absolute value norm
+
+
'fro' or 'f'
Frobenius norm (default)
+
+
+
+
+
Examples (Dense Format)
+
Here are some examples of matrix operations in dense format. Later on this page we will show examples of matrix operations in sparse format.
+
+
First let’s create example data tables in dense format:
+CREATE TABLE "mat_A" (
+ row_id integer,
+ row_vec integer[]
+);
+INSERT INTO "mat_A" (row_id, row_vec) VALUES (1, '{9,6,5,8,5,6,6,3,10,8}');
+INSERT INTO "mat_A" (row_id, row_vec) VALUES (2, '{8,2,2,6,6,10,2,1,9,9}');
+INSERT INTO "mat_A" (row_id, row_vec) VALUES (3, '{3,9,9,9,8,6,3,9,5,6}');
+INSERT INTO "mat_A" (row_id, row_vec) VALUES (4, '{6,4,2,2,2,7,8,8,0,7}');
+INSERT INTO "mat_A" (row_id, row_vec) VALUES (5, '{6,8,9,9,4,6,9,5,7,7}');
+INSERT INTO "mat_A" (row_id, row_vec) VALUES (6, '{4,10,7,3,9,5,9,2,3,4}');
+INSERT INTO "mat_A" (row_id, row_vec) VALUES (7, '{8,10,7,10,1,9,7,9,8,7}');
+INSERT INTO "mat_A" (row_id, row_vec) VALUES (8, '{7,4,5,6,2,8,1,1,4,8}');
+INSERT INTO "mat_A" (row_id, row_vec) VALUES (9, '{8,8,8,5,2,6,9,1,8,3}');
+INSERT INTO "mat_A" (row_id, row_vec) VALUES (10, '{4,6,3,2,6,4,1,2,3,8}');
+
+CREATE TABLE "mat_B" (
+ row_id integer,
+ vector integer[]
+);
+INSERT INTO "mat_B" (row_id, vector) VALUES (1, '{9,10,2,4,6,5,3,7,5,6}');
+INSERT INTO "mat_B" (row_id, vector) VALUES (2, '{5,3,5,2,8,6,9,7,7,6}');
+INSERT INTO "mat_B" (row_id, vector) VALUES (3, '{0,1,2,3,2,7,7,3,10,1}');
+INSERT INTO "mat_B" (row_id, vector) VALUES (4, '{2,9,0,4,3,6,8,6,3,4}');
+INSERT INTO "mat_B" (row_id, vector) VALUES (5, '{3,8,7,7,0,5,3,9,2,10}');
+INSERT INTO "mat_B" (row_id, vector) VALUES (6, '{5,3,1,7,6,3,5,3,6,4}');
+INSERT INTO "mat_B" (row_id, vector) VALUES (7, '{4,8,4,4,2,7,10,0,3,3}');
+INSERT INTO "mat_B" (row_id, vector) VALUES (8, '{4,6,0,1,3,1,6,6,9,8}');
+INSERT INTO "mat_B" (row_id, vector) VALUES (9, '{6,5,1,7,2,7,10,6,0,6}');
+INSERT INTO "mat_B" (row_id, vector) VALUES (10, '{1,4,4,4,8,5,2,8,5,5}');
+
+
Transpose a matrix
+SELECT madlib.matrix_trans('"mat_B"', 'row=row_id, val=vector',
+ 'mat_r');
+SELECT * FROM mat_r ORDER BY row_id;
+
+-- Note that the result matrix has inherited 'vector' as the name of the value column by default
+ row_id | vector
+--------+-------------------------
+ 1 | {9,5,0,2,3,5,4,4,6,1}
+ 2 | {10,3,1,9,8,3,8,6,5,4}
+ 3 | {2,5,2,0,7,1,4,0,1,4}
+ 4 | {4,2,3,4,7,7,4,1,7,4}
+ 5 | {6,8,2,3,0,6,2,3,2,8}
+ 6 | {5,6,7,6,5,3,7,1,7,5}
+ 7 | {3,9,7,8,3,5,10,6,10,2}
+ 8 | {7,7,3,6,9,3,0,6,6,8}
+ 9 | {5,7,10,3,2,6,3,9,0,5}
+ 10 | {6,6,1,4,10,4,3,8,6,5}
+(10 rows)
+
Get min and max values along a specific dimension, as well as the corresponding indices. Note that in this example dim=2 implies that the min and max is computed on each row, returning a column vector i.e. the column (dim=2) is flattened.
+SELECT madlib.matrix_inverse('"mat_A"', 'row=row_id, val=row_vec', 'mat_r');
+SELECT row_vec FROM mat_r ORDER BY row_id;
+
+
Generic inverse of matrix
+SELECT madlib.matrix_pinv('"mat_A"', 'row=row_id, val=row_vec', 'mat_r');
+SELECT row_vec FROM mat_r ORDER BY row_id;
+
+
Eigenvalues of matrix (note default column name of eigenvalues)
+SELECT madlib.matrix_eigen('"mat_A"', 'row=row_id, val=row_vec', 'mat_r');
+SELECT eigen_values FROM mat_r ORDER BY row_id;
+
+
Cholesky decomposition of matrix
+SELECT madlib.matrix_cholesky('"mat_A"', 'row=row_id, val=row_vec', 'matrix_out_prefix');
+SELECT row_vec FROM matrix_out_prefix_p ORDER BY row_id;
+SELECT row_vec FROM matrix_out_prefix_l ORDER BY row_id;
+SELECT row_vec FROM matrix_out_prefix_d ORDER BY row_id;
+
+
QR decomposition of matrix
+SELECT madlib.matrix_qr('"mat_A"', 'row=row_id, val=row_vec', 'matrix_out_prefix');
+SELECT row_vec FROM matrix_out_prefix_q ORDER BY row_id;
+SELECT row_vec FROM matrix_out_prefix_r ORDER BY row_id;
+
+
LU decomposition of matrix
+SELECT madlib.matrix_lu('"mat_A"', 'row=row_id, val=row_vec', 'matrix_out_prefix');
+SELECT row_vec FROM matrix_out_prefix_l ORDER BY row_id;
+SELECT row_vec FROM matrix_out_prefix_u ORDER BY row_id;
+SELECT row_vec FROM matrix_out_prefix_p ORDER BY row_id;
+SELECT row_vec FROM matrix_out_prefix_q ORDER BY row_id;
+
+-- Note the double quotes for "rowNum" are required as per PostgreSQL rules since “N” is capitalized
+SELECT madlib.matrix_trans('"mat_A_sparse"', 'row="rowNum", val=entry',
+ 'matrix_r_sparse');
+SELECT "rowNum", col_num, entry FROM matrix_r_sparse ORDER BY col_num;
+
MFVSketch: Most Frequent Values variant of CountMin sketch, implemented as a UDA.
+
Produces an n-bucket histogram for a column where each bucket counts one of the most frequent values in the column. The output is an array of doubles {value, count} in descending order of frequency; counts are approximated via CountMin sketches. Ties are handled arbitrarily.
+
The MFV frequent-value UDA comes in two different versions:
+
a faithful implementation that preserves the approximation guarantees of Cormode/Muthukrishnan,
+mfvsketch_top_histogram( col_name,
+ n )
+
+
and a "quick and dirty" version that can do parallel aggregation in Greenplum at the expense of missing some of the most frequent values.
+mfvsketch_quick_histogram( col_name,
+ n )
+
+
+
In PostgreSQL the two UDAs are identical. In Greenplum, the quick version should produce good results unless the number of values requested is small, or the distribution is flat.
+
Note
This is a User Defined Aggregate which returns the results when used in a query. Use "CREATE TABLE AS ", with the UDA as subquery if the results are to be stored. This is unlike the usual MADlib stored procedure interface which places the results in a table instead of returning it.
+
Examples
+
+
Generate some data.
+CREATE TABLE data(class INT, a1 INT);
+INSERT INTO data SELECT 1,1 FROM generate_series(1,10000);
+INSERT INTO data SELECT 1,2 FROM generate_series(1,15000);
+INSERT INTO data SELECT 1,3 FROM generate_series(1,10000);
+INSERT INTO data SELECT 2,5 FROM generate_series(1,1000);
+INSERT INTO data SELECT 2,6 FROM generate_series(1,1000);
+
+
Produce a histogram of 5 bins and return the most frequent value and associated count in each bin.
The mini-batch preprocessor is a utility that prepares input data for use by models that support mini-batch as an optimization option. (This is currently only the case for Neural Networks.) It is effectively a packing operation that builds arrays of dependent and independent variables from the source data table.
+
The advantage of using mini-batching is that it can perform better than stochastic gradient descent (default MADlib optimizer) because it uses more than one training example at a time, typically resulting in faster and smoother convergence [1].
+
Note
This preprocessor should not be used for deep learning methods. Please refer to the section on Deep Learning for more information.
+
Mini-Batch Preprocessor
The mini-batch preprocessor has the following format:
TEXT. Name of the table containing input data. Can also be a view.
+
+
+
output_table
+
TEXT. Name of the output table from the preprocessor which will be used as input to algorithms that support mini-batching. Note that the arrays packed into the output table are randomized and normalized, so they will not match up in an obvious way with the rows in the source table.
+
+
+
dependent_varname
+
TEXT. Name of the dependent variable column.
+
+
+
independent_varname
+
TEXT. Column name or expression list to evaluate for the independent variable. Please note that independent variables are cast to double precision by the preprocessor, so categorical variables should be one-hot or dummy encoded as appropriate. See Encoding Categorical Variables for more details on this.
Note
Supported expressions for independent variables include:
+
‘ARRAY[x1,x2,x3]’, where x1, x2, and x3 are columns in the source table containing scalar values.
+
Single column in the source table containing an array like ARRAY[1,2,3] or {1,2,3}.
+
+
+
+The following forms are not currently supported:
+
‘x1,x2,x3’, where x1,x2,x3 are columns in source table with scalar values
+
ARRAY[x1,x2] where x1 is scalar and x2 is array
+
ARRAY[x1,x2] where both x1 and x2 are arrays
+
ARRAY[x1] where x1 is array
+
+
+
+
grouping_col (optional)
+
TEXT, default: NULL. An expression list used to group the input dataset into discrete groups, which runs the preprocessing separately for each group. When this value is NULL, no grouping is used and a single preprocessor step is run for the whole data set.
Note
If you plan to use grouping in model training, then you must set up the groups in the preprocessor exactly as you want to use them in training.
+
+
buffer_size (optional)
+
INTEGER, default: computed. Buffer size is the number of rows from the source table that are packed into one row of the preprocessor output table. The default value is computed considering size of the source table, number of independent variables, number of groups, and number of segments in the database cluster. For larger data sets, the computed buffer size will typically be a value in the millions.
+
+
+
one_hot_encode_int_dep_var (optional)
+
BOOLEAN. default: FALSE. Flag to one-hot encode dependent variables that are scalar integers. This parameter is ignored if the dependent variable is not a scalar integer.
+
Note
The mini-batch preprocessor automatically encodes dependent variables that are boolean and character types such as text, char and varchar. However, scalar integers are a special case because they can be used in both classification and regression problems, so you must tell the mini-batch preprocessor whether you want to encode them or not. In the case that you have already encoded the dependent variable yourself, you can ignore this parameter. Also, if you want to encode float values for some reason, cast them to text first.
+
+
+
Output tables
+ The output table produced by the mini-batch preprocessor contains the following columns:
+
+
__id__
INTEGER. Unique id for packed table.
+
+
dependent_varname
FLOAT8[]. Packed array of dependent variables. If the dependent variable in the source table is categorical, the preprocessor will one-hot encode it.
+
+
independent_varname
FLOAT8[]. Packed array of independent variables.
+
+
grouping_cols
TEXT. Name of grouping columns.
+
+
A summary table named <output_table>_summary is also created, which has the following columns:
+
+
source_table
Name of the source table.
+
+
output_table
Name of output table generated by preprocessor.
+
+
dependent_varname
Dependent variable from the source table.
+
+
independent_varname
Independent variable from the source table.
+
+
buffer_size
Buffer size used in preprocessing step.
+
+
class_values
Class values (i.e., levels) of the dependent variable if categorical. If the dependent variable is not categorical, this will be NULL./td>
+
+
num_rows_processed
The total number of rows that were used in the preprocessing operation.
+
+
num_missing_rows_skipped
The total number of rows that were skipped because of NULL values in either the dependent or independent variables.
+
+
grouping_col
Comma separated list of grouping column names if grouping is used. If no grouping, will be NULL.
+
+
A standardization table named <output_table>_standardization is also created. This is needed by the models that will use the preprocessed data so is likely not of much interest to users. It has the following columns:
+
+
grouping columns
If 'grouping_col' is specified, a column for each grouping column is created.
+
+
mean
Mean of independent variables.
+
+
std
Population standard deviation of independent variables.
+
+
Examples
+
Create an input data set based on the well known iris data set:
For small datasets like in this example, buffer size is mainly determined by the number of segments in the database. This example is run on a Greenplum database with 2 segments, so there are 2 rows with a buffer size of 26. For PostgresSQL, there would be only one row with a buffer size of 52 since it is a single node database. For larger data sets, other factors go into computing buffers size besides number of segments. Also, note that the dependent variable has been one-hot encoded since it is categorical. Here is a sample of the packed output table:
This is an old implementation of multinomial logistic regression. Replacement of this function is available as the Multinomial regression module Multinomial Regression
Multinomial logistic regression is a widely used regression analysis tool that models the outcomes of categorical dependent random variables. The model assumes that the conditional mean of the dependent categorical variables is the logistic function of an affine combination of independent variables. Multinomial logistic regression finds the vector of coefficients that maximizes the likelihood of the observations.
+
Training Function
The multinomial logistic regression training function has the following syntax:
TEXT. The name of the table containing the input data.
+
+
+
+
output_table
+
TEXT. The name of the generated table containing the output model. The output table produced by the multinomial logistic regression training function contains the following columns:
+
+
category
INTEGER. The category. Categories are encoded as integers with values from {0, 1, 2,..., numCategories – 1}
+
+
ref_category
INTEGER. The reference category. Categories are encoded as integers with values from {0, 1, 2,..., numCategories – 1}
+
+
coef
FLOAT8[]. An array of coefficients, \( \boldsymbol c \).
+
+
log_likelihood
FLOAT8. The log-likelihood, \( l(\boldsymbol c) \).
+
+
std_err
FLOAT8[]. An array of the standard errors.
+
+
z_stats
FLOAT8[]. An array of the Wald z-statistics.
+
+
p_values
FLOAT8[]. An array of the Wald p-values.
+
+
odds_ratios
FLOAT8[]. An array of the odds ratios.
+
+
condition_no
FLOAT8. The condition number of the matrix, computed using the coefficients of the iteration immediately preceding convergence.
+
+
num_iterations
INTEGER. The number of iterations executed before the algorithm completed.
+
+
A summary table named <out_table>_summary is also created at the same time, and it contains the following columns:
+
+
+
source_table
The data source table name.
+
+
out_table
The output table name.
+
+
dependent_varname
The dependent variable.
+
+
independent_varname
The independent variables.
+
+
optimizer_params
The optimizer parameters. It is a copy of the optimizer_params in the training function's arguments.
+
+
ref_category
An integer, the value of reference category used.
+
+
num_rows_processed
INTEGER. The number of rows actually processed, which is equal to the total number of rows in the source table minus the number of skipped rows.
+
+
num_missing_rows_skipped
INTEGER. The number of rows skipped during the training. A row will be skipped if the ind_col is NULL or contains NULL values.
+
+
+
+
dependent_varname
+
TEXT. The name of the column containing the dependent variable.
+
+
+
independent_varname
+
TEXT. Expression list to evaluate for the independent variables. An intercept variable is not assumed. The number of independent variables cannot exceed 65535.
+
+
+
ref_category (optional)
+
INTEGER, default: 0. The reference category ranges from [0, numCategories – 1].
+
+
+
optimizer_params (optional)
+
VARCHAR, default: NULL, which uses the default values of optimizer parameters. It should be a string that contains pairs of 'key=value' separated by commas. Supported parameters with their default values: max_iter=20, optimizer='irls', precision=1e-4. Currently, only 'irls' and 'newton' are allowed for 'optimizer'.
+
+
+
Note
Table names can be optionally schema qualified and table and column names should follow the same case-sensitivity and quoting rules as in the database.
+
Prediction Function
The prediction function is provided to estimate the conditional mean given a new predictor. It has the following syntax:
TEXT. Name of the table containing the multilogistic model. This should be the output table returned from mlogregr_train.
+
+
+
new_data_table
+
TEXT. Name of the table containing prediction data. This table is expected to contain the same features that were used during training. The table should also contain id_col_name used for identifying each row.
+
+
+
id_col_name
+
TEXT. Name of the column containing id information in the source data. This is a mandatory argument and is used for correlating prediction table rows with the source. The values of this column are expected to be unique for each tuple.
+
+
+
output_table
+
TEXT. Name of the table to output prediction results to. If this table already exists then an error is returned. This output table contains the id_col_name column giving the 'id' for each prediction.
+
If type = 'response', then the table has a single additional column with the prediction value of the response. The type of this column depends on the type of the response variable used during training.
+
If type = 'prob', then the table has multiple additional columns, one for each possible category. The columns are labeled as 'estimated_prob_category_value', where category_value represents the values of categories (0 to K-1).
+
+
+
type
+
TEXT, optional, default: 'response'.
+
When type = 'prob', the probabilities of each category (including the reference category) is given.
+
When type = 'response', a single output is provided which represents the prediction category for each tuple. This represents the category with the highest probability.
Multinomial logistic regression models the outcomes of categorical dependent random variables (denoted \( Y \in \{ 0,1,2 \ldots k \} \)). The model assumes that the conditional mean of the dependent categorical variables is the logistic function of an affine combination of independent variables (usually denoted \( \boldsymbol x \)). That is,
+ for some unknown vector of coefficients \( \boldsymbol c \) and where \( \sigma(x) = \frac{1}{1 + \exp(-x)} \) is the logistic function. Multinomial logistic regression finds the vector of coefficients \( \boldsymbol c \) that maximizes the likelihood of the observations.
+
Let
+
\( \boldsymbol y \in \{ 0,1 \}^{n \times k} \) denote the vector of observed dependent variables, with \( n \) rows and \( k \) columns, containing the observed values of the dependent variable,
+
\( X \in \mathbf R^{n \times k} \) denote the design matrix with \( k \) columns and \( n \) rows, containing all observed vectors of independent variables \( \boldsymbol x_i \) as rows.
The Hessian of this objective is \( H = -X^T A X \) where \( A = \text{diag}(a_1, \dots, a_n) \) is the diagonal matrix with \( a_i = \sigma(\boldsymbol c^T \boldsymbol x) \cdot \sigma(-\boldsymbol c^T \boldsymbol x) \,. \) Since \( H \) is non-positive definite, \( l(\boldsymbol c) \) is convex. There are many techniques for solving convex optimization problems. Currently, logistic regression in MADlib can use:
+
Iteratively Reweighted Least Squares
+
+
We estimate the standard error for coefficient \( i \) as
The Wald \( p \)-value for coefficient \( i \) gives the probability (under the assumptions inherent in the Wald test) of seeing a value at least as extreme as the one observed, provided that the null hypothesis ( \( c_i = 0 \)) is true. Letting \( F \) denote the cumulative density function of a standard normal distribution, the Wald \( p \)-value for coefficient \( i \) is therefore
where \( Z \) is a standard normally distributed random variable.
+
The odds ratio for coefficient \( i \) is estimated as \( \exp(c_i) \).
+
The condition number is computed as \( \kappa(X^T A X) \) during the iteration immediately preceding convergence (i.e., \( A \) is computed using the coefficients of the previous iteration). A large condition number (say, more than 1000) indicates the presence of significant multicollinearity.
+
The multinomial logistic regression uses a default reference category of zero, and the regression coefficients in the output are in the order described below. For a problem with \( K \) dependent variables \( (1, ..., K) \) and \( J \) categories \( (0, ..., J-1) \), let \( {m_{k,j}} \) denote the coefficient for dependent variable \( k \) and category \( j \). The output is \( {m_{k_1, j_0}, m_{k_1, j_1} \ldots m_{k_1, j_{J-1}}, m_{k_2, j_0}, m_{k_2, j_1}, \ldots m_{k_2, j_{J-1}} \ldots m_{k_K, j_{J-1}}} \). The order is NOT CONSISTENT with the multinomial regression marginal effect calculation with function marginal_mlogregr. This is deliberate because the interfaces of all multinomial regressions (robust, clustered, ...) will be moved to match that used in marginal.
+
Literature
+
A collection of nice write-ups, with valuable pointers into further literature:
+
[1] Annette J. Dobson: An Introduction to Generalized Linear Models, Second Edition. Nov 2001
[3] Scott A. Czepiel: Maximum Likelihood Estimation of Logistic Regression Models: Theory and Implementation, Retrieved Jul 12 2012, http://czep.net/stat/mlelr.pdf
+
Related Topics
+
File multilogistic.sql_in documenting the multinomial logistic regression functions
In statistics, multinomial regression is a classification method that generalizes binomial regression to multiclass problems, i.e. with more than two possible discrete outcomes. That is, it is a model that is used to predict the probabilities of the different possible outcomes of a categorically distributed dependent variable, given a set of independent variables (which may be real-valued, binary-valued, categorical-valued, etc.).
+
Training Function
The multinomial regression training function has the following syntax:
VARCHAR. Name of the table containing the training data.
+
+
+
model_table
+
VARCHAR. Name of the generated table containing the model.
+
The model table produced by multinom() contains the following columns:
+
+
+
<...>
Grouping columns, if provided in input. This could be multiple columns depending on the grouping_col input.
+
+
+
+
category
VARCHAR. String representation of category value.
+
+
+
+
coef
FLOAT8[]. Vector of the coefficients in linear predictor.
+
+
+
+
log_likelihood
FLOAT8. The log-likelihood \( l(\boldsymbol \beta) \). The value will be the same across categories within the same group.
+
+
+
+
std_err
FLOAT8[]. Vector of the standard errors of the coefficients.
+
+
+
+
z_stats
FLOAT8[]. Vector of the z-statistics of the coefficients.
+
+
+
+
p_values
FLOAT8[]. Vector of the p-values of the coefficients.
+
+
+
+
num_rows_processed
BIGINT. Number of rows processed.
+
+
+
+
num_rows_skipped
BIGINT. Number of rows skipped due to missing values or failures.
+
+
+
+
num_iterations
INTEGER. Number of iterations actually completed. This would be different from the nIterations argument if a tolerance parameter is provided and the algorithm converges before all iterations are completed.
+
+
A summary table named <model_table>_summary is also created at the same time, which has the following columns:
+
+
method
VARCHAR. String describes the model: 'multinom'.
+
+
+
+
source_table
VARCHAR. Data source table name.
+
+
+
+
model_table
VARCHAR. Model table name.
+
+
+
+
dependent_varname
VARCHAR. Expression for dependent variable.
+
+
+
+
independent_varname
VARCHAR. Expression for independent variables.
+
+
+
+
ref_category
VARCHAR. String representation of reference category.
+
+
+
+
link_func
VARCHAR. String that contains link function parameters: only 'logit' is implemented now
+
+
+
+
grouping_col
VARCHAR. String representation of grouping columns.
+
+
+
+
optimizer_params
VARCHAR. String that contains optimizer parameters, and has the form of 'optimizer=..., max_iter=..., tolerance=...'.
+
+
+
+
num_all_groups
INTEGER. Number of groups in glm training.
+
+
+
+
num_failed_groups
INTEGER. Number of failed groups in glm training.
+
+
+
+
total_rows_processed
BIGINT. Total number of rows processed in all groups.
+
+
+
+
total_rows_skipped
BIGINT. Total number of rows skipped in all groups due to missing values or failures.
+
+
+
+
+
+
dependent_varname
+
VARCHAR. Name of the dependent variable column.
+
+
+
independent_varname
+
VARCHAR. Expression list to evaluate for the independent variables. An intercept variable is not assumed. It is common to provide an explicit intercept term by including a single constant 1 term in the independent variable list.
+
+
+
link_function (optional)
+
VARCHAR, default: 'logit'. Parameters for link function. Currently, we support logit.
+
+
+
ref_category (optional)
+
VARCHAR, default: '0'. Parameters to specify the reference category.
+
+
+
grouping_col (optional)
+
VARCHAR, default: NULL. An expression list used to group the input dataset into discrete groups, running one regression per group. Similar to the SQL "GROUP BY" clause. When this value is NULL, no grouping is used and a single model is generated.
+
+
+
optim_params (optional)
+
VARCHAR, default: 'max_iter=100,optimizer=irls,tolerance=1e-6'. Parameters for optimizer. Currently, we support tolerance=[tolerance for relative error between log-likelihoods], max_iter=[maximum iterations to run], optimizer=irls.
+
+
+
verbose (optional)
+
BOOLEAN, default: FALSE. Provides verbose output of the results of training.
+
+
Note
For p-values, we just return the computation result directly. Other statistical packages, like 'R', produce the same result, but on printing the result to screen, another format function is used and any p-value that is smaller than the machine epsilon (the smallest positive floating-point number 'x' such that '1 + x != 1') will be printed on screen as "< xxx" (xxx is the value of the machine epsilon). Although the result may look different, they are in fact the same.
+
Prediction Function
Multinomial regression prediction function has the following format:
TEXT. Name of the generated table containing the model, which is the output table from multinom().
+
+
+
predict_table_input
+
TEXT. The name of the table containing the data to predict on. The table must contain id column as the primary key.
+
+
+
output_table
+
TEXT. Name of the generated table containing the predicted values.
+
The model table produced by multinom_predict contains the following columns:
+
+
+
id
SERIAL. Column to identify the predicted value.
+
+
+
+
category
TEXT. Available if the predicted type = 'response'. Column contains the predicted categories
+
+
+
+
category_value
FLOAT8. The predicted probability for the specific category_value.
+
+
+
+
predict_type
+
TEXT. Either 'response' or 'probability'. Using 'response' will give the predicted category with the largest probability. Using probability will give the predicted probabilities for all categories
+
verbose
+
BOOLEAN. Control whether verbose is displayed. The default is FALSE.
Predicting dependent variable using multinomial model. (This example uses the original data table to perform the prediction. Typically a different test dataset with the same features as the original training dataset would be used for prediction.)
+\x off
+-- Add the id column for prediction function
+ALTER TABLE test3 ADD COLUMN id SERIAL;
+-- Predict probabilities for all categories using the original data
+SELECT madlib.multinom_predict('test3_output','test3', 'test3_prd_prob', 'probability');
+-- Display the predicted value
+SELECT * FROM test3_prd_prob;
+
+
+
Technical Background
When link = 'logit', multinomial logistic regression models the outcomes of categorical dependent random variables (denoted \( Y \in \{ 0,1,2 \ldots k \} \)). The model assumes that the conditional mean of the dependent categorical variables is the logistic function of an affine combination of independent variables (usually denoted \( \boldsymbol x \)). That is,
+ for some unknown vector of coefficients \( \boldsymbol c \) and where \( \sigma(x) = \frac{1}{1 + \exp(-x)} \) is the logistic function. Multinomial logistic regression finds the vector of coefficients \( \boldsymbol c \) that maximizes the likelihood of the observations.
+
Let
+
\( \boldsymbol y \in \{ 0,1 \}^{n \times k} \) denote the vector of observed dependent variables, with \( n \) rows and \( k \) columns, containing the observed values of the dependent variable,
+
\( X \in \mathbf R^{n \times k} \) denote the design matrix with \( k \) columns and \( n \) rows, containing all observed vectors of independent variables \( \boldsymbol x_i \) as rows.
The Hessian of this objective is \( H = -X^T A X \) where \( A = \text{diag}(a_1, \dots, a_n) \) is the diagonal matrix with \( a_i = \sigma(\boldsymbol c^T \boldsymbol x) \cdot \sigma(-\boldsymbol c^T \boldsymbol x) \,. \) Since \( H \) is non-positive definite, \( l(\boldsymbol c) \) is convex. There are many techniques for solving convex optimization problems. Currently, logistic regression in MADlib can use:
+
Iteratively Reweighted Least Squares
+
+
We estimate the standard error for coefficient \( i \) as
The Wald \( p \)-value for coefficient \( i \) gives the probability (under the assumptions inherent in the Wald test) of seeing a value at least as extreme as the one observed, provided that the null hypothesis ( \( c_i = 0 \)) is true. Letting \( F \) denote the cumulative density function of a standard normal distribution, the Wald \( p \)-value for coefficient \( i \) is therefore
where \( Z \) is a standard normally distributed random variable.
+
The odds ratio for coefficient \( i \) is estimated as \( \exp(c_i) \).
+
The condition number is computed as \( \kappa(X^T A X) \) during the iteration immediately preceding convergence (i.e., \( A \) is computed using the coefficients of the previous iteration). A large condition number (say, more than 1000) indicates the presence of significant multicollinearity.
+
The multinomial logistic regression uses a default reference category of zero, and the regression coefficients in the output are in the order described below. For a problem with \( K \) dependent variables \( (1, ..., K) \) and \( J \) categories \( (0, ..., J-1) \), let \( {m_{k,j}} \) denote the coefficient for dependent variable \( k \) and category \( j \). The output is \( {m_{k_1, j_0}, m_{k_1, j_1} \ldots m_{k_1, j_{J-1}}, m_{k_2, j_0}, m_{k_2, j_1}, \ldots m_{k_2, j_{J-1}} \ldots m_{k_K, j_{J-1}}} \). The order is NOT CONSISTENT with the multinomial regression marginal effect calculation with function marginal_mlogregr. This is deliberate because the interfaces of all multinomial regressions (robust, clustered, ...) will be moved to match that used in marginal.
+
Literature
+
A collection of nice write-ups, with valuable pointers into further literature:
+
[1] Annette J. Dobson: An Introduction to Generalized Linear Models, Second Edition. Nov 2001
[3] Scott A. Czepiel: Maximum Likelihood Estimation of Logistic Regression Models: Theory and Implementation, Retrieved Jul 12 2012, http://czep.net/stat/mlelr.pdf
Multilayer Perceptron (MLP) is a type of neural network that can be used for regression and classification.
+
MLPs consist of several fully connected hidden layers with non-linear activation functions. In the case of classification, the final layer of the neural net has as many nodes as classes, and the output of the neural net can be interpreted as the probability that a given input feature belongs to a specific class.
+
MLP can be used with or without mini-batching. The advantage of using mini-batching is that it can perform better than the default MADlib optimizer, because it uses more than one training example at a time, typically resulting faster and smoother convergence [3].
+
Note
In order to use mini-batching, you must first run the Mini-Batch Preprocessor, which is a utility that prepares input data for use by models that support mini-batch as an optimization option, such as MLP. This is a one-time operation and you would only need to re-run the preprocessor if your input data has changed, or if you change the grouping parameter.
+
Classification Training Function
The MLP classification training function has the following format:
TEXT. Name of the table containing the training data. If you are using mini-batching, this is the name of the output table from the mini-batch preprocessor.
+
+
+
output_table
+
TEXT. Name of the output table containing the model. Details of the output table are shown below.
+
+
+
independent_varname
+
TEXT. Expression list to evaluate for the independent variables. It should be a numeric array expression. If you are using mini-batching, set this parameter to 'independent_varname' which is the hardcoded name of the column from the mini-batch preprocessor containing the packed independent variables.
+
Note
If you are not using mini-batching, please note that an intercept variable should not be included as part of this expression - this is different from other MADlib modules. Also please note that independent variables should be encoded properly. All values are cast to DOUBLE PRECISION, so categorical variables should be one-hot or dummy encoded as appropriate. See Encoding Categorical Variables for more details.
+
+
dependent_varname
+
TEXT. Name of the dependent variable column. For classification, supported types are: text, varchar, character varying, char, character integer, smallint, bigint, and boolean. If you are using mini-batching, set this parameter to 'dependent_varname' which is the hardcoded name of the column from the mini-batch preprocessor containing the packed dependent variables.
+
+
+
hidden_layer_sizes (optional)
+
INTEGER[], default: ARRAY[100]. The number of neurons in each hidden layer. The length of this array will determine the number of hidden layers. For example, ARRAY[5,10] means 2 hidden layers, one with 5 neurons and the other with 10 neurons. Use ARRAY[]::INTEGER[] for no hidden layers.
+
+
+
optimizer_params (optional)
+
TEXT, default: NULL. Parameters for optimization in a comma-separated string of key-value pairs. See the description below for details.
+
+
+
activation (optional)
+
TEXT, default: 'sigmoid'. Activation function. Currently three functions are supported: 'sigmoid' (default), 'relu', and 'tanh'. The text can be any prefix of the three strings; for e.g., specifying 's' will use sigmoid activation.
+
+
+
weights (optional)
+
TEXT, default: 1. Column name for giving different weights to different rows during training. E.g., a weight of two for a specific row is equivalent to dupicating that row. This weight is incorporated into the update during stochastic gradient descent (SGD), but is not be used for loss calculations. If not specified, weight for each row will default to 1 (equal weights). Column should be a numeric type.
Note
The 'weights' parameter cannot be used if you use mini-batching of the source dataset.
+
+
warm_start (optional)
+
BOOLEAN, default: FALSE. Initalize neural network weights with the coefficients from the last call of the training function. If set to true, neural network weights will be initialized from the output_table generated by the previous run. Note that all parameters other than optimizer_params and verbose must remain constant between calls when warm_start is used.
+
Note
The warm start feature works based on the name of the output_table. When using warm start, do not drop the output table or the output table summary before calling the training function, since these are needed to obtain the neural network weights from the previous run. If you are not using warm start, the output table and the output table summary must be dropped in the usual way before calling the training function.
+
+
+
verbose (optional)
+
BOOLEAN, default: FALSE. Provides verbose output of the results of training, including the value of loss at each iteration.
Note
There are some subtleties on the reported per-iteration loss values because we are working in a distributed system. When mini-batching is used (i.e., batch gradient descent), loss per iteration is an average of losses across all mini-batches and epochs on a segment. Losses across all segments then get averaged to give overall loss for the model for the iteration. This will tend to be a pessimistic estimate of loss. When mini-batching is not used (i.e., stochastic gradient descent), we use the model state from the previous iteration to compute the loss at the start of the current iteration on the whole data set. This is an accurate computation of loss for the iteration.
+
+
grouping_col (optional)
+
TEXT, default: NULL. A single column or a list of comma-separated columns that divides the input data into discrete groups, resulting in one model per group. When this value is NULL, no grouping is used and a single model is generated for all data. If you are using mini-batching, you must have run the mini-batch preprocessor with exactly the same groups that you specify here for MLP training. If you change the groups, or remove the groups, then you must re- run the mini-batch preprocessor.
+
+
Output tables
+ The model table produced by MLP contains the following columns:
+
+
coeffs
FLOAT8[]. Flat array containing the weights of the neural net.
+
+
n_iterations
INTEGER. Number of iterations completed by the stochastic gradient descent algorithm. The algorithm either converged in this number of iterations or hit the maximum number specified in the optimization parameters.
+
+
loss
FLOAT8. The cross entropy loss over the training data. See Technical Background section below for more details.
+
+
grouping columns
If grouping_col is specified during training, a column for each grouping column is created.
+
+
A summary table named <output_table>_summary is also created, which has the following columns:
+
+
source_table
The source table.
+
+
independent_varname
The independent variables.
+
+
dependent_varname
The dependent variable.
+
+
tolerance
The tolerance as given in optimizer_params.
+
+
learning_rate_init
The initial learning rate as given in optimizer_params.
+
+
learning_rate_policy
The learning rate policy as given in optimizer_params.
+
+
momentum
Momentum value as given in optimizer_params.
+
+
nesterov
Nesterov value as given in optimizer_params.
+
+
n_iterations
The number of iterations run.
+
+
n_tries
The number of tries as given in optimizer_params.
+
+
layer_sizes
The number of units in each layer including the input and output layers.
+
+
activation
The activation function.
+
+
is_classification
True if the model was trained for classification, False if it was trained for regression.
+
+
classes
The classes which were trained against (empty for regression).
+
+
weights
The weight column used during training for giving different weights to different rows.
+
+
grouping_col
NULL if no grouping_col was specified during training, and a comma-separated list of grouping column names if not.
+
+
+
+
A standardization table named <output_table>_standardization is also create, that has the following columns:
+
+
mean
The mean for all input features (used for normalization).
+
+
std
The standard deviation for all input features (used for normalization).
+
+
grouping columns
If grouping_col is specified during training, a column for each grouping column is created.
+
+
Regression Training Function
The MLP regression training function has the following format:
Parameters for regression are largely the same as for classification. In the model table, the loss refers to mean square error instead of cross entropy loss. In the summary table, there is no classes column. The following arguments have specifications which differ from mlp_classification:
+
dependent_varname
+
TEXT. Name of the dependent variable column. For regression, supported types are any numeric type, or array of numeric types (for multiple regression).
+
+
Optimizer Parameters
Parameters in this section are supplied in the optimizer_params argument as a string containing a comma-delimited list of name-value pairs. All of these named parameters are optional and their order does not matter. You must use the format "<param_name> = <value>" to specify the value of a parameter, otherwise the parameter is ignored.
Default: sgd. One of 'sgd', 'rmsprop' or 'adam' or any prefix of these (e.g., 'rmsp' means 'rmsprop'). These are defined below:
+
'sgd': stochastic gradient descent
+
'rmsprop': RMSprop algorithm proposed by Hinton et al. [3]
+
'adam': Adam algorithm proposed by Kingma and Ba [4]
+
+
Note
rmsprop and adam solvers only work with minibatched data. In addition, they do not support momentum. Please see the references for the explanations on these constraints.
+
+
learning_rate_init
+
Default: 0.001. Also known as the learning rate. A small value is usually desirable to ensure convergence, while a large value provides more room for progress during training. Since the best value depends on the condition number of the data, in practice one often tunes this parameter.
+
+
+
learning_rate_policy
+
Default: constant. One of 'constant', 'exp', 'inv' or 'step' or any prefix of these (e.g., 's' means 'step'). These are defined below, where 'iter' is the current iteration:
Default: 0.1. Decay rate for learning rate when learning_rate_policy is 'exp' or 'step'.
+
+
+
power
+
Default: 0.5. Exponent for learning_rate_policy = 'inv'.
+
+
+
iterations_per_step
+
Default: 100. Number of iterations to run before decreasing the learning rate by a factor of gamma. Valid for learning rate policy = 'step'.
+
+
+
n_iterations
+
Default: 100. The maximum number of iterations allowed.
+
+
+
n_tries
+
Default: 1. Number of times to retrain the network with randomly initialized neural network weights.
+
+
+
lambda
+
Default: 0. The regularization coefficient for L2 regularization.
+
+
+
tolerance
+
Default: 0.001. The criterion to end iterations. The training stops whenever the difference between the training models of two consecutive iterations is smaller than tolerance or the iteration number is larger than n_iterations. If you want to run the full number of iterations specified in n_interations, set tolerance=0.0
+
+
+
batch_size
+
Default: min(200, buffer_size) where buffer_size is set in the mini-batch preprocessor. The 'batch_size' is the size of the mini-batch used in the optimizer. This parameter is only used in the case of mini-batching.
+
+
+
n_epochs
+
Default: 1. Represents the number of times each batch is used by the optimizer. This parameter is only used in the case of mini-batching.
+
+
+
momentum
+
Default: 0.9. Momentum can help accelerate learning and avoid local minima when using gradient descent. Value must be in the range 0 to 1, where 0 means no momentum.
+
+
+
nesterov
+
Default: TRUE. Only used when the 'momentum' parameter is > 0. Nesterov momentum can provide better results than using classical momentum alone, due to its look-ahead characteristics. In classical momentum we correct the velocity and then update the model with that velocity, whereas in Nesterov Accelerated Gradient method, we first move the model in the direction of velocity, compute the gradient using this updated model, and then add this gradient back into the model. The main difference being that in classical momentum, we compute the gradient before updating the model whereas in nesterov we first update the model and then compute the gradient from the updated position.
+
+
+
rho
+
Default: 0.9. Moving average parameter for the RMSprop solver.
+
+
+
beta1
+
Default: 0.9. The exponential decay rate for the first moment estimates.
+
+
+
beta2
+
Default: 0.999. The exponential decay rate for the second moment estimates.
+
+
+
eps
+
Default: 1e-7. Constant for numerical stability in adam and rmsprop solvers.
+
+
Prediction Function
Used to generate predictions on novel data given a previously trained model. The same syntax is used for classification and regression.
TEXT. Model table produced by the training function.
+
+
+
data_table
+
TEXT. Name of the table containing the data for prediction. This table is expected to contain the same input features that were used during training. The table should also contain id_col_name used for identifying each row.
+
+
+
id_col_name
+
TEXT. The name of the id column in data_table.
+
+
+
output_table
+
TEXT. Name of the table where output predictions are written. If this table name is already in use, an error is returned. Table contains:
+
+
id
Gives the 'id' for each prediction, corresponding to each row from the data_table.
+
+
estimated_COL_NAME
(For pred_type='response') The estimated class for classification or value for regression, where COL_NAME is the name of the column to be predicted from training data.
+
+
prob_CLASS
(For pred_type='prob' for classification) The probability of a given class CLASS as given by softmax. There will be one column for each class in the training data.
+
+
+
+
+
pred_type
+
TEXT. The type of output requested: 'response' gives the actual prediction, 'prob' gives the probability of each class. For regression, only type='response' is defined.
Generate a multilayer perceptron with a single hidden layer of 5 units. Use the attributes column as the independent variables, and use the class column as the classification. Set the tolerance to 0 so that 500 iterations will be run. Use a hyperbolic tangent activation function. The model will be written to mlp_model.
+DROP TABLE IF EXISTS mlp_model, mlp_model_summary, mlp_model_standardization;
+-- Set seed so results are reproducible
+SELECT setseed(0);
+SELECT madlib.mlp_classification(
+ 'iris_data', -- Source table
+ 'mlp_model', -- Destination table
+ 'attributes', -- Input features
+ 'class_text', -- Label
+ ARRAY[5], -- Number of units per layer
+ 'learning_rate_init=0.003,
+ n_iterations=500,
+ tolerance=0', -- Optimizer params
+ 'tanh', -- Activation function
+ NULL, -- Default weight (1)
+ FALSE, -- No warm start
+ FALSE -- Not verbose
+);
+
Now let's use the model to predict. In the following example we will use the training data set for prediction as well, which is not usual but serves to show the syntax. The prediction is in the estimated_class_text column with the actual value in the class_text column.
+DROP TABLE IF EXISTS mlp_prediction;
+\x off
+SELECT madlib.mlp_predict(
+ 'mlp_model', -- Model table
+ 'iris_data', -- Test data table
+ 'id', -- Id column in test table
+ 'mlp_prediction', -- Output table for predictions
+ 'response' -- Output classes, not probabilities
+ );
+SELECT * FROM mlp_prediction JOIN iris_data USING (id) ORDER BY id;
+
Now let's use the model to predict. As before we will use the training data set for prediction as well, which is not usual but serves to show the syntax. The prediction is in the estimated_class_text column with the actual value in the class_text column.
+DROP TABLE IF EXISTS mlp_prediction;
+\x off
+SELECT madlib.mlp_predict(
+ 'mlp_model', -- Model table
+ 'iris_data', -- Test data table
+ 'id', -- Id column in test table
+ 'mlp_prediction', -- Output table for predictions
+ 'response' -- Output classes, not probabilities
+ );
+SELECT * FROM mlp_prediction JOIN iris_data USING (id) ORDER BY id;
+
+SELECT COUNT(*) FROM mlp_prediction JOIN iris_data USING (id)
+WHERE mlp_prediction.estimated_class_text != iris_data.class_text;
+
+ count
+-------+
+ 0
+
+
+
Classification with Other Parameters
+
+
Now, use the n_tries optimizer parameter to learn and choose the best model among n_tries number of models learnt by the algorithm. Run only for 50 iterations and choose the best model from this short run. Note we are not using mini-batching here.
+DROP TABLE IF EXISTS mlp_model, mlp_model_summary, mlp_model_standardization;
+-- Set seed so results are reproducible
+SELECT setseed(0);
+SELECT madlib.mlp_classification(
+ 'iris_data', -- Source table
+ 'mlp_model', -- Destination table
+ 'attributes', -- Input features
+ 'class_text', -- Label
+ ARRAY[5], -- Number of units per layer
+ 'learning_rate_init=0.003,
+ n_iterations=50,
+ tolerance=0,
+ n_tries=3', -- Optimizer params, with n_tries
+ 'tanh', -- Activation function
+ NULL, -- Default weight (1)
+ FALSE, -- No warm start
+ FALSE -- Not verbose
+);
+
Next, use the warm_start parameter to start learning a new model, using the coefficients already present in mlp_model. Note that we must not drop the mlp_model table, and cannot use the n_tries parameter if warm_start is used.
+SELECT madlib.mlp_classification(
+ 'iris_data', -- Source table
+ 'mlp_model', -- Destination table
+ 'attributes', -- Input features
+ 'class_text', -- Label
+ ARRAY[5], -- Number of units per layer
+ 'learning_rate_init=0.003,
+ n_iterations=450,
+ tolerance=0', -- Optimizer params
+ 'tanh', -- Activation function
+ NULL, -- Default weight (1)
+ TRUE, -- Warm start
+ FALSE -- Not verbose
+);
+
Notice that the loss is lower compared to the previous example, despite having the same values for every other parameter. This is because the algorithm learned three different models starting with a different set of initial weights for the coefficients, and chose the best model among them as the initial weights for the coefficients when run with warm start.
+
Next, test adam solver for adaptive learning rates. Note that we are using the minibatched dataset.
+DROP TABLE IF EXISTS mlp_model, mlp_model_summary, mlp_model_standardization;
+SELECT madlib.mlp_classification(
+ 'iris_data_packed', -- Output table from mini-batch preprocessor
+ 'mlp_model', -- Destination table
+ 'independent_varname', -- Hardcode to this, from table iris_data_packed
+ 'dependent_varname', -- Hardcode to this, from table iris_data_packed
+ ARRAY[5], -- Number of units per layer
+ 'learning_rate_init=0.1,
+ n_iterations=500,
+ tolerance=0.0001,
+ solver=adam', -- Optimizer params
+ 'tanh', -- Activation function
+ NULL, -- Default weight (1)
+ FALSE, -- No warm start
+ FALSE -- Not verbose
+);
+
A separate model is learnt for each state, and the result table displays the name of the state (grouping column) associated with the model.
+
Prediction based on grouping using the state column:
+\x off
+DROP TABLE IF EXISTS mlp_prediction;
+SELECT madlib.mlp_predict(
+ 'mlp_model_group', -- Model table
+ 'iris_data', -- Test data table
+ 'id', -- Id column in test table
+ 'mlp_prediction', -- Output table for predictions
+ 'response' -- Output classes, not probabilities
+ );
+SELECT * FROM mlp_prediction JOIN iris_data USING (state,id) ORDER BY state, id;
+
+DROP TABLE IF EXISTS mlp_regress_prediction;
+SELECT madlib.mlp_predict(
+ 'mlp_regress', -- Model table
+ 'lin_housing', -- Test data table
+ 'id', -- Id column in test table
+ 'mlp_regress_prediction', -- Output table for predictions
+ 'response' -- Output values, not probabilities
+ );
+
View results:
+\x off
+SELECT * FROM lin_housing JOIN mlp_regress_prediction USING (id) ORDER BY id;
+
+DROP TABLE IF EXISTS mlp_regress_prediction;
+SELECT madlib.mlp_predict(
+ 'mlp_regress', -- Model table
+ 'lin_housing', -- Test data table
+ 'id', -- Id column in test table
+ 'mlp_regress_prediction', -- Output table for predictions
+ 'response' -- Output values, not probabilities
+ );
+\x off
+SELECT *, ABS(y-estimated_y) as abs_diff FROM lin_housing JOIN mlp_regress_prediction USING (id) ORDER BY id;
+
Note that the results you get for all examples may vary with the database you are using.
+
Technical Background
+
To train a neural net, the loss function is minimized using stochastic gradient descent. In the case of classification, the loss function is cross entropy. For regression, mean square error is used. Weights in the neural net are updated via the backpropogation process, which uses dynamic programming to compute the partial derivative of each weight with respect to the overall loss. This partial derivative incorporates the activation function used, which requires that the activation function be differentiable.
+
For an overview of multilayer perceptrons, see [1].
+
For details on backpropogation, see [2].
+
On the effect of database cluster size: as the database cluster size increases, the per iteration loss will be higher since the model only sees 1/n of the data, where n is the number of segments. However, each iteration runs faster than single node because it is only traversing 1/n of the data. For large data sets, all else being equal, a bigger cluster will achieve a given accuracy faster than a single node although it may take more iterations to achieve that accuracy.
In statistics, ordinal regression is a type of regression analysis used for predicting an ordinal variable, i.e. a variable whose value exists on an arbitrary scale where only the relative ordering between different values is significant. The two most common types of ordinal regression models are ordered logit, which applies to data that meet the proportional odds assumption, and ordered probit.
+
Training Function
The ordinal regression training function has the following syntax:
VARCHAR. Name of the table containing the training data.
+
+
+
model_table
+
VARCHAR. Name of the generated table containing the model.
+
The model table produced by ordinal() contains the following columns:
+
+
+
<...>
Grouping columns, if provided in input. This could be multiple columns depending on the grouping_col input.
+
+
+
+
coef_threshold
FLOAT8[]. Vector of the threshold coefficients in linear predictor. The threshold coefficients are the intercepts specific to each categorical levels
+
+
+
+
std_err_threshold
FLOAT8[]. Vector of the threshold standard errors of the threshold coefficients.
+
+
+
+
z_stats_threshold
FLOAT8[]. Vector of the threshold z-statistics of the thresholdcoefficients.
+
+
+
+
p_values_threshold
FLOAT8[]. Vector of the threshold p-values of the threshold coefficients.
+
+
+
+
log_likelihood
FLOAT8. The log-likelihood \( l(\boldsymbol \beta) \). The value will be the same across categories within the same group.
+
+
+
+
coef_feature
FLOAT8[]. Vector of the feature coefficients in linear predictor. The feature coefficients are the coefficients for the independent variables. They are the same across categories.
+
+
+
+
std_err_feature
FLOAT8[]. Vector of the feature standard errors of the feature coefficients.
+
+
+
+
z_stats_feature
FLOAT8[]. Vector of the feature z-statistics of the feature coefficients.
+
+
+
+
p_values_feature
FLOAT8[]. Vector of the feature p-values of the feature coefficients.
+
+
+
+
num_rows_processed
BIGINT. Number of rows processed.
+
+
+
+
num_rows_skipped
BIGINT. Number of rows skipped due to missing values or failures.
+
+
+
+
num_iterations
INTEGER. Number of iterations actually completed. This would be different from the nIterations argument if a tolerance parameter is provided and the algorithm converges before all iterations are completed.
+
+
A summary table named <model_table>_summary is also created at the same time, which has the following columns:
+
+
method
VARCHAR. String describes the model: 'ordinal'.
+
+
+
+
source_table
VARCHAR. Data source table name.
+
+
+
+
model_table
VARCHAR. Model table name.
+
+
+
+
dependent_varname
VARCHAR. Expression for dependent variable.
+
+
+
+
independent_varname
VARCHAR. Expression for independent variables. The independent variables should not include intercept term. Otherwise there will be an error message indicating Hessian matrix is not finite. In that case, the user should drop the intercept and rerun the function agian.
+
+
+
+
cat_order
VARCHAR. String representation of category order. Default is the sorted categories in data using python sort
+
+
+
+
link_func
VARCHAR. String that contains link function parameters: 'logit' and 'probit' links are implemented now
+
+
+
+
grouping_col
VARCHAR. String representation of grouping columns.
+
+
+
+
optimizer_params
VARCHAR. String that contains optimizer parameters, and has the form of 'optimizer=..., max_iter=..., tolerance=...'.
+
+
+
+
num_all_groups
INTEGER. Number of groups in ordinal regression training.
+
+
+
+
num_failed_groups
INTEGER. Number of failed groups in ordinal regression training.
+
+
+
+
total_rows_processed
BIGINT. Total number of rows processed in all groups.
+
+
+
+
total_rows_skipped
BIGINT. Total number of rows skipped in all groups due to missing values or failures.
+
+
+
+
+
+
dependent_varname
+
VARCHAR. Name of the dependent variable column.
+
+
+
independent_varname
+
VARCHAR. Expression list to evaluate for the independent variables. The intercept should not be included here since the cumulative probability force to have intercepts for each category level.
+
+
+
cat_order
+
VARCHAR, String that represents the order of category. The order is specified by charactor '<'.
+
+
+
link_function (optional)
+
VARCHAR, default: 'logit'. Parameters for link function. Currently, we support logit and probit.
+
+
+
grouping_col (optional)
+
VARCHAR, default: NULL. An expression list used to group the input dataset into discrete groups, running one regression per group. Similar to the SQL "GROUP BY" clause. When this value is NULL, no grouping is used and a single model is generated.
+
+
+
optim_params (optional)
+
VARCHAR, default: 'max_iter=100,optimizer=irls,tolerance=1e-6'. Parameters for optimizer. Currently, we support tolerance=[tolerance for relative error between log-likelihoods], max_iter=[maximum iterations to run], optimizer=irls.
+
+
+
verbose (optional)
+
BOOLEAN, default: FALSE. Provides verbose output of the results of training.
+
+
Note
To calculate the standard error the coefficient, we are using the square root of the diagnal elements of the expected Fisher information matrix, which is a by-product of iteratively reweighted least square. This method is used in the original ordinal regression paper by McCullagh(1980). In some software like Stata, the standard error is calculated by the observed information matrix, which is supported by Efron and Hinkley (1978). In R, polr() uses the approximated observed information matrix while the optimization is achieved by first order optimization method. Therefore, there will be some difference on standard error, z-stats and p-value from other software.
+
Prediction Function
Ordinal regression prediction function has the following format:
TEXT. Name of the generated table containing the model, which is the output table from ordinal().
+
+
+
predict_table_input
+
TEXT. The name of the table containing the data to predict on. The table must contain id column as the primary key.
+
+
+
output_table
+
TEXT. Name of the generated table containing the predicted values.
+
The model table produced by ordinal_predict contains the following columns:
+
+
+
id
SERIAL. Column to identify the predicted value.
+
+
+
+
category
TEXT. Available if the predicted type = 'response'. Column contains the predicted categories
+
+
+
+
category_value
FLOAT8. The predicted probability for the specific category_value.
+
+
+
+
predict_type
+
TEXT. Either 'response' or 'probability'. Using 'response' will give the predicted category with the largest probability. Using probability will give the predicted probabilities for all categories
Predicting dependent variable using ordinal model. (This example uses the original data table to perform the prediction. Typically a different test dataset with the same features as the original training dataset would be used for prediction.)
+\x off
+-- Add the id column for prediction function
+ALTER TABLE test3 ADD COLUMN id SERIAL;
+-- Predict probabilities for all categories using the original data
+SELECT ordinal_predict('test3_output','test3', 'test3_prd_prob', 'probability');
+-- Display the predicted value
+SELECT * FROM test3_prd_prob;
+
+
+
Model Details
+
The function ordinal() fit the ordinal response model using a cumulative link model. The ordinal reponse variable, denoted by \( Y_i \), can fall in \( j = 1,.. , J\) categories. Then \( Y_i \) follows a multinomial distribution with parameter \(\pi\) where \(\pi_{ij}\) denote the probability that the \(i\)th observation falls in response category \(j\). We define the cumulative probabilities as
Next we will consider the logit link for illustration purpose. The logit function is defined as \( \mbox{logit}(\pi) = \log[\pi/(1-\pi)] \) and cumulative logits are defined as:
where \(x_i\) is a vector of explanatory variables for the \(i\)th observation and \(\beta\) is the corresponding set of regression parameters. The \(\{\theta_j\}\) parameters provide each cumulative logit (for each \(j\)) with its own intercept. A key point is that the regression part \(x^T_i\beta\) is independent of \(j\), so \(\beta\) has the same effect for each of the J − 1 cumulative logits. Note that \(x^T_i\beta\) does not contain an intercept, since the \(\{\theta_j\}\) act as intercepts. For small values of \(x^T_i\beta\) the response is likely to fall in the first category and for large values of \(x^T_i\beta\) the response is likely to fall in the last category. The horizontal displacements of the curves are given by the values of \(\{\theta_j\}\).
+
Literature
+
A collection of nice write-ups, with valuable pointers into further literature:
+
[1] Peter McCullagh: Regression Models for Ordinal Data, Journal of the Royal Statistical Society. Series B (Methodological), Volume 42, Issue 2 (1980), 109-142
+
[2] Rune Haubo B Christensen: Analysis of ordinal data with cumulative link models – estimation with the R-package ordinal. cran.r-project.org/web/packages/ordinal/vignettes/clm_intro.pdf
+
Related Topics
+
File ordinal.sql_in documenting the ordinal regression functions
Given a graph, the PageRank algorithm outputs a probability distribution representing the likelihood that a person randomly traversing the graph will arrive at any particular vertex. This algorithm was originally used by Google to rank websites where the World Wide Web was modeled as a directed graph with the vertices representing the websites. The PageRank algorithm initially proposed by Larry Page and Sergey Brin is implemented here [1].
+
We also implement personalized PageRank, in which a notion of importance provides personalization to a query. For example, importance scores can be biased according to a specified set of vertices in the graph that are of interest or special in some way [2].
TEXT. Name of the table containing the vertex data for the graph. Must contain the column specified in the 'vertex_id' parameter below.
+
TEXT, default = 'id'. Name of the column(s) in 'vertex_table' containing vertex ids. The vertex ids can be of type INTEGER or BIGINT with no duplicates. They do not need to be contiguous. If multiple columns are used as vertex ids, they are passed in the following format: [<vertex_id1>,<vertex_id2>,...]
+
+
+
edge_table
+
TEXT. Name of the table containing the edge data. The edge table must contain columns for source vertex and destination vertex.
+
+
+
edge_args
+
TEXT. A comma-delimited string containing multiple named arguments of the form "name=value". The following parameters are supported for this string argument:
+
src (INTEGER or BIGINT): Name of the column(s) containing the source vertex ids in the edge table. Default column name is 'src'.
+
dest (INTEGER or BIGINT): Name of the column(s) containing the destination vertex ids in the edge table. Default column name is 'dest'.
+
+
+
+
out_table
+
TEXT. Name of the table to store the result of PageRank. It will contain a row for every vertex from 'vertex_table' with the following columns:
+
vertex_id : The id of a vertex. Will use the input parameter 'vertex_id' for column naming.
+
pagerank : The vertex's PageRank.
+
grouping_cols : Grouping column (if any) values associated with the vertex_id.
+
+
A summary table is also created that contains information regarding the number of iterations required for convergence. It is named by adding the suffix '_summary' to the 'out_table' parameter.
+
+
+
damping_factor (optional)
+
FLOAT8, default 0.85. The probability, at any step, that a user will continue following the links in a random surfer model.
+
+
+
max_iter (optional)
+
INTEGER, default: 100. The maximum number of iterations allowed.
+
+
+
threshold (optional)
+
FLOAT8, default: (1/number of vertices * 1000). If the difference between the PageRank of every vertex of two consecutive iterations is smaller than 'threshold', or the iteration number is larger than 'max_iter', the computation stops. If you set the threshold to zero, then you will force the algorithm to run for the full number of iterations specified in 'max_iter'. It is advisable to set threshold to a value lower than 1/(number of vertices in the graph) since the PageRank value of nodes is initialized to that value.
+
+
+
grouping_cols (optional)
+
TEXT, default: NULL. A single column or a list of comma-separated columns that divides the input data into discrete groups, resulting in one distribution per group. When this value is NULL, no grouping is used and a single model is generated for all data.
Note
Expressions are not currently supported for 'grouping_cols'.
+
+
personalization_vertices (optional)
+
INTEGER[] or BIGINT[], default: NULL. A comma separated list of vertices or nodes for personalized PageRank. When this parameter is provided, personalized PageRank will run. In the absence of this parameter, regular PageRank will run. If multiple columns are used for identifying vertices, a 2D array will be required for this parameter.
[2] Jeh, Glen and Widom, Jennifer. "Scaling Personalized Web Search", Proceedings of the 12th international conference on World Wide Web, Pages 271-279 Budapest, Hungary, May 20-24, 2003, http://ilpubs.stanford.edu:8090/530/1/2002-12.pdf
+
+
+
+
+
+
Generated on Mon Aug 28 2023 08:03:56 for MADlib by
+
+ 1.8.13
The goal of the MADlib path function is to perform regular pattern matching over a sequence of rows, and to extract useful information about the pattern matches. The useful information could be a simple count of matches or something more involved like aggregations or window functions.
+
Symbols are used to identify particular rows of interest. Then, standard PostgreSQL pattern matching using symbols can be applied to identify patterns across the rows of interest. (This is similar in concept to regular expressions which match patterns within strings of text.)
+
For example, a symbol can be defined for purchase events by on-line shoppers. Then, preceding events that led to the purchase can be identified and operated on, perhaps to find the common actions that resulted in a purchase. Or conversely, to find actions that resulted in an exit without a purchase having been made.
+
Steps on how to use path functions:
+
+
Partition input rows.
+
Order the partitions.
+
Define symbols to match rows of interest.
+
Define regular expression of symbols and operators to define patterns to match in your ordered partitions.
+
Define an aggregate function to compute for each pattern match.
+
If desired, output the pattern matches for inspection or to operate on them with subsequent queries.
VARCHAR. Name of the source table, containing data for path analysis.
+
+
+
output_table
+
VARCHAR. Name of the result table.
+
+
+
partition_expr
+
VARCHAR. The 'partition_expr' can be a single column or a list of comma-separated columns/expressions to divide all rows into groups, or partitions. Matching is applied across the rows that fall into the same partition. This can be NULL or '' to indicate the matching is to be applied to the whole table.
+
+
+
order_expr
+
VARCHAR. This expression controls the order in which rows are processed or matched in a partition. For example, time is a common way to order partitions.
+
+
+
symbol
+
VARCHAR. Symbols enable you to express patterns of interest in a simple way (see definition of ‘pattern’ argument below). A symbol identifies a row of a particular type that you’re searching for as part of a pattern match. Symbol definition uses the standard PostgreSQL assignment statement 'identifier := expression;' [1]. A given row can only match one symbol. If a row matches multiple symbols, the symbol that comes first in the symbol definition list will take precedence.
+
+
+
pattern
+
VARCHAR. The 'pattern' clause defines the pattern that the path algorithm searches for. You express the pattern using symbols and operators following regular PostgreSQL pattern matching syntax and rules [2].
+
Note
Symbols defined using more than one (1) character need to be enclosed in parentheses '()' when referenced in the 'pattern' argument. For example:
+
a symbol defined as 'a' in the 'symbol' argument can be used directly in the 'pattern' argument
+
a symbol defined as 'abc' in the 'symbol' argument must be written as '(abc)' in the 'pattern' argument
+
+
+
The following pattern matching metacharacters are supported:
+
+| denotes alternation (either of two alternatives).
+
+? denotes repetition of the previous item zero or one time.
+
+* denotes repetition of the previous item zero or more times.
+
++ denotes repetition of the previous item one or more times.
+
+{m} denotes repetition of the previous item exactly m times.
+
+{m,} denotes repetition of the previous item m or more times.
+
+{m,n} denotes repetition of the previous item at least m and not more than n times.
+
+Parentheses () can be used to group items into a single logical item.
+
+
+
+
aggregate_func (optional)
+
VARCHAR, default NULL. A comma-separated list of aggregates to be applied to the pattern matches [3]. You can think of this input parameter as being like a SELECT clause. Please note that window functions cannot currently be used in the parameter 'aggregate_func'. If you want to use a window function [4], output the pattern matches and write a SQL query with a window function over the output tuples (see 'persist_rows' parameter below).
+
If you just want to output the pattern matched rows and not compute any aggregates, you can put NULL or '' in the 'aggregate_func' parameter.
+
+
+
persist_rows (optional)
+
BOOLEAN, default FALSE. If TRUE the matched rows are persisted in a separate output table. This table is named as <output_table>_tuples (the string "_tuples" is added as suffix to the value of output_table).
+
+
+
overlapping_patterns (optional)
+
BOOLEAN, default FALSE. If TRUE find every occurrence of the pattern in the partition, regardless of whether it might have been part of a previously found match.
+
+
+
+
Examples
+
The data set describes shopper behavior on a notional web site that sells beer and wine. A beacon fires an event to a log file when the shopper visits different pages on the site: landing page, beer selection page, wine selection page, and checkout. Other pages on the site like help pages show up in the logs as well. Let’s assume that the log has been sessionized.
+DROP TABLE IF EXISTS path_output, path_output_tuples;
+SELECT madlib.path(
+ 'eventlog', -- Name of input table
+ 'path_output', -- Table name to store path results
+ 'session_id', -- Partition input table by session
+ 'event_timestamp ASC', -- Order partitions in input table by time
+ 'buy:=page=''CHECKOUT''', -- Define a symbol for checkout events
+ '(buy)', -- Pattern search: purchase
+ 'sum(revenue) as checkout_rev', -- Aggregate: sum revenue by checkout
+ TRUE -- Persist matches
+ );
+SELECT * FROM path_output ORDER BY session_id, match_id;
+
Note that there are 2 checkouts within session 102, which is apparent from the 'match_id' column. This serves to illustrate that the 'aggregate_func' operates on a per pattern match basis, not on a per partition basis. If in fact we wanted revenue by partition ('session_id' in this example), then we could do:
+SELECT session_id, sum(checkout_rev) FROM path_output GROUP BY session_id ORDER BY session_id;
+
Notice that the 'symbol' and 'match_id' columns are added to the right of the matched rows.
+
We are interested in sessions with an order placed within 4 pages of entering the shopping site via the landing page. We represent this by the regular expression: '(land)[^(land)(buy)]{0,2}(buy)'. In other words, visit to the landing page followed by from 0 to 2 non-entry, non-sale pages, followed by a purchase. The SQL is as follows:
+DROP TABLE IF EXISTS path_output, path_output_tuples;
+SELECT madlib.path(
+ 'eventlog', -- Name of input table
+ 'path_output', -- Table name to store path results
+ 'session_id', -- Partition input table by session
+ 'event_timestamp ASC', -- Order partitions in input table by time
+ 'land:=page=''LANDING'',
+ wine:=page=''WINE'',
+ beer:=page=''BEER'',
+ buy:=page=''CHECKOUT'',
+ other:=page<>''LANDING'' AND page<>''WINE'' AND page<>''BEER'' AND page<>''CHECKOUT''', -- Symbols for page types
+ '(land)[^(land)(buy)]{0,2}(buy)', -- Purchase within 4 pages entering site
+ 'sum(revenue) as checkout_rev', -- Aggregate: sum revenue by checkout
+ TRUE -- Persist matches
+ );
+SELECT * FROM path_output ORDER BY session_id, match_id;
+
We may want to use a window function instead of an aggregate. Currently, only aggregates are supported in the core path function in the parameter 'aggregate_func'. However, you can write window functions on the output tuples to achieve the desired result. Continuing the previous example, let’s say we want to compute average revenue for checkouts within 4 pages of entering the shopping site via the landing page:
+SELECT DATE(event_timestamp), user_id, session_id, revenue,
+ avg(revenue) OVER (PARTITION BY DATE(event_timestamp)) as avg_checkout_rev
+ FROM path_output_tuples
+ WHERE page='CHECKOUT'
+ ORDER BY user_id, session_id;
+
Here we are partitioning the window function by day because we want daily averages, although our sample data set only has a single day.
+
Now we want to do a golden path analysis to find the most successful shopper paths through the site. Since our data set is small, we decide this means the most frequently viewed page just before a checkout is made:
+DROP TABLE IF EXISTS path_output, path_output_tuples;
+SELECT madlib.path(
+ 'eventlog', -- Name of input table
+ 'path_output', -- Table name to store path results
+ 'session_id', -- Partition input table by session
+ 'event_timestamp ASC', -- Order partitions in input table by time
+ 'land:=page=''LANDING'',
+ wine:=page=''WINE'',
+ beer:=page=''BEER'',
+ buy:=page=''CHECKOUT'',
+ other:=page<>''LANDING'' AND page<>''WINE'' AND page<>''BEER'' AND page<>''CHECKOUT''', -- Symbols for page types
+ '[^(buy)](buy)', -- Pattern to match
+ 'array_agg(page ORDER BY session_id ASC, event_timestamp ASC) as page_path', -- Build array with shopper paths
+ FALSE -- Don't persist matches
+ );
+
Now count the common paths and print the most frequent:
+SELECT count(*), page_path from
+ (SELECT * FROM path_output) q
+GROUP BY page_path
+ORDER BY count(*) DESC
+LIMIT 10;
+
There are only 2 different paths. The wine page is viewed more frequently than the beer page just before checkout.
+
To demonstrate the use of 'overlapping_patterns', consider a pattern with at least one page followed by and ending with a checkout:
+DROP TABLE IF EXISTS path_output, path_output_tuples;
+SELECT madlib.path(
+ 'eventlog', -- Name of the table
+ 'path_output', -- Table name to store the path results
+ 'session_id', -- Partition by session
+ 'event_timestamp ASC', -- Order partitions in input table by time
+ $$ nobuy:=page<>'CHECKOUT',
+ buy:=page='CHECKOUT'
+ $$, -- Definition of symbols used in the pattern definition
+ '(nobuy)+(buy)', -- At least one page followed by and ending with a CHECKOUT.
+ 'array_agg(page ORDER BY session_id ASC, event_timestamp ASC) as page_path',
+ FALSE, -- Don't persist matches
+ TRUE -- Turn on overlapping patterns
+ );
+SELECT * FROM path_output ORDER BY session_id, match_id;
+
Please note some current limitations of the path algorithm.
+
Window functions cannot currently be used in the parameter 'aggregate_func'. Instead, output the pattern matches and write a SQL query with a window function over the output tuples.
+
A given row can only match one symbol. If a row matches multiple symbols, the symbol that comes first in the symbol definition list will take precedence.
+
Maximum number of symbols that can be defined is 35.
+
The columns 'match_id' and 'symbol' are generated by the path algorithm. If coincidently you have columns in your input data named 'match_id' or 'symbol', the system generated column names will be changed to "__madlib_path_match_id__" and "__madlib_path_symbol__"
+
+
+
Nomenclature
+
Partition
+
scope of rows to be searched for pattern match
+
typical examples: user id, session id, portfolio id
+
+
Order
+
sort order of input rows in partition
+
typical example: time
+
+
Symbol
+
a row of a particular type that you’re searching for, that you want to include in a pattern
+
+
Pattern
+
regular PostgreSQL pattern match expression of symbols and operators that you want to match across rows
+
+
Pattern match
+
rows that result from a pattern match expression of symbols
+
can be multiple matches per partition
+
+
Literature
+
NOTE: The following links refer to documentation resources for the current PostgreSQL database version. Depending upon your database platform version, you may need to change "current" references in the links to your database version.
+
If your database platform uses the Greenplum Database (or related variants), please check with the project community and/or your database vendor to identify the PostgreSQL version it is based on.
Produces a model that transforms a number of (possibly) correlated variables into a (smaller) number of uncorrelated variables called principal components.
Principal component projection is a mathematical procedure that projects high dimensional data onto a lower dimensional space. This lower dimensional space is defined by the \( k \) principal components with the highest variance in the training data.
+
More details on the mathematics of PCA can be found in Principal Component Analysis and some details about principal component projection calculations can be found in the Technical Background.
+
Projection Function
The projection functions are slightly different for dense and sparse matrices. For dense matrices:
+madlib.pca_sparse_project( source_table,
+ pc_table,
+ out_table,
+ row_id,
+ col_id, -- Sparse matrices only
+ val_id, -- Sparse matrices only
+ row_dim, -- Sparse matrices only
+ col_dim, -- Sparse matrices only
+ residual_table,
+ result_summary_table
+ )
+
+
Arguments
+
source_table
+
TEXT. Source table name. Identical to pca_train, the input data matrix should have \( N \) rows and \( M \) columns, where \( N \) is the number of data points, and \( M \) is the number of features for each data point.
+
The input table for pca_project is expected to be in the one of the two standard MADlib dense matrix formats, and the sparse input table for pca_sparse_project should be in the standard MADlib sparse matrix format. These formats are described in the documentation for Principal Component Analysis.
+
+
+
pc_table
+
TEXT. Table name for the table containing principal components.
+
+
+
out_table
+
TEXT. Name of the table that will contain the low-dimensional representation of the input data.
+
The out_table encodes a dense matrix with the projection onto the principal components. The table has the following columns:
+
+
+
row_id
Row id of the output matrix.
+
+
row_vec
A vector containing elements in the row of the matrix.
+
+
+
+
row_id
+
TEXT. Column name containing the row IDs in the input source table. The column should be of type INT (or a type that can be cast to INT) and should only contain values between 1 and N. For dense matrix format, it should contain all continguous integers from 1 to N describing the full matrix.
+
+
+
col_id
+
TEXT. Column name containing the column IDs in sparse matrix representation. The column should be of type INT (or a type that can be cast to INT) and should only contain values between 1 and M. This parameter applies to sparse matrices only.
+
+
+
val_id
+
TEXT. Name of 'val_id' column in sparse matrix representation defining the values of the nonzero entries. This parameter applies to sparse matrices only.
+
+
+
row_dim
+
INTEGER. The actual number of rows in the matrix. That is, if the matrix was transformed into dense format, this is the number of rows it would have. This parameter applies to sparse matrices only.
+
+
+
col_dim
+
INTEGER. The actual number of columns in the matrix. That is, if the matrix was transformed into dense format, this is the number of columns it would have. This parameter applies to sparse matrices only.
+
Note
The parameters 'row_dim' and 'col_dim' could actually be inferred from the sparse matrix representation, so they will be removed in the future. For now they are maintained for backward compatability so you must enter them. Making 'row_dim' or 'col_dim' larger than the actual matrix has the effect of padding it with zeros, which is probably not useful.
+
+
residual_table (optional)
+
TEXT, default: NULL. Name of the optional residual table.
+
The residual_table encodes a dense residual matrix. The table has the following columns:
+
+
+
row_id
Row id of the output matrix.
+
+
row_vec
A vector containing elements in the row of the residual matrix.
+
+
+
+
result_summary_table (optional)
+
TEXT, default: NULL. Name of the optional summary table.
+
The result_summary_table contains information about the performance time of the PCA projection. The table has the following columns:
+
+
exec_time
Elapsed time (ms) for execution of the function.
+
+
residual_norm
Absolute error of the residuals.
+
+
relative_residual_norm
Relative error of the residuals.
+
+
+
+
Examples
+
View online help for the PCA projection function:
+SELECT madlib.pca_project();
+
+
Create sample data in dense matrix form:
+DROP TABLE IF EXISTS mat;
+CREATE TABLE mat (id integer,
+ row_vec double precision[]
+ );
+INSERT INTO mat VALUES
+(1, '{1,2,3}'),
+(2, '{2,1,2}'),
+(3, '{3,2,1}');
+
+
Run the PCA function for a specified number of principal components and view the results:
+DROP TABLE IF EXISTS result_table, result_table_mean;
+SELECT madlib.pca_train('mat', -- Source table
+ 'result_table', -- Output table
+ 'id', -- Row id of source table
+ 2); -- Number of principal components
+SELECT * FROM result_table ORDER BY row_id;
+
Now we use grouping in dense form to learn different models for different groups. First, we create sample data in dense matrix form with a grouping column. Note we actually have different matrix sizes for the different groups, which is allowed for dense:
Run the PCA projection on subsets of an input table based on grouping columns. Note that the parameter 'pc_table' used for projection must be generated in training using the same grouping columns.
+DROP TABLE IF EXISTS mat_group_projected;
+SELECT madlib.pca_project('mat_group',
+ 'result_table_group',
+ 'mat_group_projected',
+ 'id');
+SELECT * FROM mat_group_projected ORDER BY matrix_id, row_id;
+
Run the PCA sparse function for a specified number of principal components and view the results:
DROP TABLE IF EXISTS result_table, result_table_mean;
+SELECT madlib.pca_sparse_train( 'mat_sparse', -- Source table
+ 'result_table', -- Output table
+ 'row_id', -- Row id of source table
+ 'col_id', -- Column id of source table
+ 'value', -- Value of matrix at row_id, col_id
+ 4, -- Actual number of rows in the matrix
+ 5, -- Actual number of columns in the matrix
+ 3); -- Number of principal components
+SELECT * FROM result_table ORDER BY row_id;
+
Result (with principal components truncated for readability):
Now we use grouping in sparse form to learn different models for different groups. First, we create sample data in sparse matrix form with a grouping column:
Run the PCA function with grouping for a specified proportion of variance and view the results:
+DROP TABLE IF EXISTS result_table_group, result_table_group_mean;
+SELECT madlib.pca_sparse_train( 'mat_sparse_group', -- Source table
+ 'result_table_group', -- Output table
+ 'row_id', -- Row id of source table
+ 'col_id', -- Column id of source table
+ 'value', -- Value of matrix at row_id, col_id
+ 4, -- Actual number of rows in the matrix
+ 5, -- Actual number of columns in the matrix
+ 0.8, -- Proportion of variance
+ 'matrix_id');
+SELECT * FROM result_table_group ORDER BY matrix_id, row_id;
+
Result (with principal components truncated for readability):
This function is intended to operate on the principal component tables generated by pca_train or pca_sparse_train. The MADlib PCA functions generate a table containing the column-means in addition to a table containing the principal components. If this table is not found by the MADlib projection function, it will trigger an error. As long the principal component tables are created with MADlib functions, then the column-means table will be automatically found by the MADlib projection functions.
+
Because of the centering step in PCA projection (see "Technical Background"), sparse matrices almost always become dense during the projection process. Thus, this implementation automatically densifies sparse matrix input, and there should be no expected performance improvement in using sparse matrix input over dense matrix input.
+
Table names can be optionally schema qualified (current_schemas() is searched if a schema name is not provided) and all table and column names should follow case-sensitivity and quoting rules per the database. (For instance, 'mytable' and 'MyTable' both resolve to the same entity, i.e. 'mytable'. If mixed-case or multi-byte characters are desired for entity names then the string should be double-quoted; in this case the input would be '"MyTable"').
+
If the input table for pca_project (pca_sparse_project) contains grouping columns, the same grouping columns must be used in the training function used to generate the principal components too.
+
+
+
Technical Background
+
Given a table containing some principal components \( \boldsymbol P \) and some input data \( \boldsymbol X \), the low-dimensional representation \( {\boldsymbol X}' \) is computed as
where \(\hat{x} \) is the column means of \( \boldsymbol X \) and \( \vec{e} \) is the vector of all ones. This step is equivalent to centering the data around the origin.
+
The residual table \( \boldsymbol R \) is a measure of how well the low-dimensional representation approximates the true input data, and is computed as
Principal component analysis (PCA) is a mathematical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called principal components. This transformation is defined in such a way that the first principal component has the largest possible variance (i.e., accounts for as much of the variability in the data as possible), and each succeeding component in turn has the highest variance possible under the constraint that it be orthogonal to (i.e., uncorrelated with) the preceding components.
+
See the Technical Background for an introduction to principal component analysis.
+
Training Function
The training functions are slightly different for dense and sparse matrices. For dense matrices:
TEXT. Name of the input table containing the data for PCA training. The input data matrix should have \( N \) rows and \( M \) columns, where \( N \) is the number of data points, and \( M \) is the number of features for each data point.
+
A dense input table is expected to be in the one of the two standard MADlib dense matrix formats, and a sparse input table should be in the standard MADlib sparse matrix format.
Note that the column name row_id is taken as an input parameter, and should contain a continguous list of row indices (starting at 1) for the input matrix.
+
The input table for sparse PCA is expected to be in the form:
The row_id and col_id columns specify which entries in the matrix are nonzero, and the val_id column defines the values of the nonzero entries.
+
Please refer to the Matrix Operations documentation for more details on defining matrices.
+
+
+
out_table
+
TEXT. The name of the table that will contain the output. There are three possible output tables as described below.
+
The primary output table (out_table) encodes the principal components with the k highest eigenvalues where k is either directly provided by the user or computed according to the proportion of variance. The table has the following columns:
+
+
row_id
Eigenvalue rank in descending order of the eigenvalue size.
+
+
principal_components
Vectors containing elements of the principal components.
+
+
std_dev
The standard deviation of each principal component.
+
+
proportion
The proportion of variance covered by the principal component.
+
+
The table out_table_mean contains the column means. This table has just one column:
+
+
column_mean
A vector containing the column means for the input matrix.
+
+
The optional table result_summary_table contains information about the performance of the PCA. The contents of this table are described under the result_summary_table argument.
+
+
+
row_id
+
TEXT. Column name containing the row IDs in the input source table. The column should be of type INT (or a type that can be cast to INT) and should only contain values between 1 and N. For dense matrix format, it should contain all continguous integers from 1 to N describing the full matrix.
+
+
+
col_id
+
TEXT. Column name containing the column IDs in sparse matrix representation. The column should be of type INT (or a type that can be cast to INT) and should only contain values between 1 and M. This parameter applies to sparse matrices only.
+
+
+
val_id
+
TEXT. Name of 'val_id' column in sparse matrix representation defining the values of the nonzero entries. This parameter applies to sparse matrices only.
+
+
+
row_dim
+
INTEGER. The actual number of rows in the matrix. That is, if the matrix was transformed into dense format, this is the number of rows it would have. This parameter applies to sparse matrices only.
+
+
+
col_dim
+
INTEGER. The actual number of columns in the matrix. That is, if the matrix was transformed into dense format, this is the number of columns it would have. This parameter applies to sparse matrices only.
+
Note
The parameters 'row_dim' and 'col_dim' could actually be inferred from the sparse matrix representation, so they will be removed in the future. For now they are maintained for backward compatability so you must enter them. Making 'row_dim' or 'col_dim' larger than the actual matrix has the effect of padding it with zeros, which is probably not useful.
+
+
components_param
+
INTEGER or FLOAT. The parameter to control the number of principal components to calculate from the input data. If 'components_param' is INTEGER, it is used to denote the number of principal components (k) to compute. If 'components_param' is FLOAT, the algorithm will return enough principal vectors so that the ratio of the sum of the eigenvalues collected thus far to the sum of all eigenvalues is greater than this parameter (proportion of variance). The value of 'components_param' must be either a positive INTEGER or a FLOAT in the range (0.0,1.0]
+
Note
The difference in interpretation between INTEGER and FLOAT was introduced to maintain backward campatibility after the proportion of variance feature was introduced. A special case to be aware of: 'components_param' = 1 (INTEGER) will return 1 principal component, but 'components_param' = 1.0 (FLOAT) will return all principal components, i.e., proportion of variance of 100%.
+
+Also, please note that the number of principal components (k) is global, even in the case where grouping is used (see 'grouping_cols' below). In the case of grouping, proportion of variance might be a better choice; this could result in different numbers of principal components for different groups.
+
+
grouping_cols (optional)
+
TEXT, default: NULL. A comma-separated list of column names, with the source data grouped using the combination of all the columns. An independent PCA model will be computed for each combination of the grouping columns.
+
Note
Dense matrices can be different sizes for different groups if desired. Sparse matrices cannot be different sizes for different groups, because the 'row_dim' and 'col_dim' parameters used for sparse matrices are global across all groups.
+
+
lanczos_iter (optional)
+
INTEGER, default: minimum of {k+40, smallest matrix dimension} where k is the number of principal components specified in the parameter 'components_param'. This parameter defines the number of Lanczos iterations for the SVD calculation. The Lanczos iteration number roughly corresponds to the accuracy of the SVD calculation, and a higher number of iterations corresponds to greater accuracy but longer computation time. The number of iterations must be at least as large as the value of k, but no larger than the smallest dimension of the matrix. If the number of iterations is set to zero, then the default number of iterations will be used.
+
Note
If both 'lanczos_iter' and proportion of variance (via the 'components_param' parameter) are defined, 'lanczos_iter' will take precedence in determining the number of principal components (i.e. the number of principal components will not be greater than 'lanczos_iter' even if the target proportion had not been reached).
+
+
use_correlation (optional)
+
BOOLEAN, default FALSE. Whether to use the correlation matrix for calculating the principal components instead of the covariance matrix. Currently use_correlation is a placeholder for forward compatibility, so this value must be set to false.
+
+
+
result_summary_table (optional)
+
TEXT, default NULL. Name of the optional summary table. When NULL, no summary table is generated.
+
This sumary table has the following columns:
+
+
rows_used
INTEGER. Number of data points in the input.
+
+
exec_time (ms)
FLOAT8. Number of milliseconds for the PCA calculation to run.
+
+
iter
INTEGER. Number of iterations used in the SVD calculation.
+
+
recon_error
FLOAT8. The absolute error in the SVD approximation.
+
+
relative_recon_error
FLOAT8. The relative error in the SVD approximation.
+
+
use_correlation
BOOLEAN. Indicates if the correlation matrix was used.
+
+
+
+
Examples
+
+
View online help for the PCA training functions:
+SELECT madlib.pca_train();
+
or
+SELECT madlib.pca_sparse_train();
+
+
Create sample data in dense matrix form:
+DROP TABLE IF EXISTS mat;
+CREATE TABLE mat (id integer,
+ row_vec double precision[]
+ );
+INSERT INTO mat VALUES
+(1, '{1,2,3}'),
+(2, '{2,1,2}'),
+(3, '{3,2,1}');
+
+
Run the PCA function for a specified number of principal components and view the results:
+DROP TABLE IF EXISTS result_table, result_table_mean;
+SELECT madlib.pca_train('mat', -- Source table
+ 'result_table', -- Output table
+ 'id', -- Row id of source table
+ 2); -- Number of principal components
+SELECT * FROM result_table ORDER BY row_id;
+
Now we use grouping in dense form to learn different models for different groups. First, we create sample data in dense matrix form with a grouping column. Note we actually have different matrix sizes for the different groups, which is allowed for dense:
Run the PCA sparse function for a specified number of principal components and view the results:
DROP TABLE IF EXISTS result_table, result_table_mean;
+SELECT madlib.pca_sparse_train( 'mat_sparse', -- Source table
+ 'result_table', -- Output table
+ 'row_id', -- Row id of source table
+ 'col_id', -- Column id of source table
+ 'value', -- Value of matrix at row_id, col_id
+ 4, -- Actual number of rows in the matrix
+ 5, -- Actual number of columns in the matrix
+ 3); -- Number of principal components
+SELECT * FROM result_table ORDER BY row_id;
+
Result (with principal components truncated for readability):
Now we use grouping in sparse form to learn different models for different groups. First, we create sample data in sparse matrix form with a grouping column:
Run the PCA function with grouping for a specified proportion of variance and view the results:
+DROP TABLE IF EXISTS result_table_group, result_table_group_mean;
+SELECT madlib.pca_sparse_train( 'mat_sparse_group', -- Source table
+ 'result_table_group', -- Output table
+ 'row_id', -- Row id of source table
+ 'col_id', -- Column id of source table
+ 'value', -- Value of matrix at row_id, col_id
+ 4, -- Actual number of rows in the matrix
+ 5, -- Actual number of columns in the matrix
+ 0.8, -- Proportion of variance
+ 'matrix_id');
+SELECT * FROM result_table_group ORDER BY matrix_id, row_id;
+
Result (with principal components truncated for readability):
Table names can be optionally schema qualified (current_schemas() would be searched if a schema name is not provided) and all table and column names should follow case-sensitivity and quoting rules per the database. (For instance, 'mytable' and 'MyTable' both resolve to the same entity, i.e. 'mytable'. If mixed-case or multi-byte characters are desired for entity names then the string should be double-quoted; in this case the input would be '"MyTable"').
+
Because of the centering step in PCA (see Technical Background), sparse matrices almost always become dense during the training process. Since this implementation automatically densifies sparse matrix input, there should be no expected performance improvement in using sparse matrix input over dense matrix input.
+
For the parameter 'components_param', INTEGER and FLOAT are interpreted differently. A special case to be aware of: 'components_param' = 1 (INTEGER) will return 1 principal component, but 'components_param' = 1.0 (FLOAT) will return all principal components, i.e., proportion of variance of 100%.
+
If both 'lanczos_iter' and proportion of variance (via the 'components_param' parameter) are defined, 'lanczos_iter' will take precedence in determining the number of principal components (i.e. the number of principal components will not be greater than 'lanczos_iter' even if the target proportion had not been reached).
+
+
Technical Background
+
The PCA implemented here uses a distributed SVD decomposition implementation to recover the principal components (as opposed to the directly computing the eigenvectors of the covariance matrix). Let \( \boldsymbol X \) be the data matrix, and let \( \hat{x} \) be a vector of the column averages of \( \boldsymbol{X}\). PCA computes the matrix \( \hat{\boldsymbol X} \) as
where \( {\boldsymbol \Sigma} \) is a diagonal matrix. The eigenvalues are recovered as the entries of \( {\boldsymbol \Sigma}/(\sqrt{(N-1)} \), and the principal components are the rows of \( {\boldsymbol V} \). The reasoning behind using N − 1 instead of N to calculate the covariance is Bessel's correction.
+
Note
It is important to note that this PCA implementation assumes that the user will use only the principal components that have non-zero eigenvalues. The SVD calculation is done with the Lanczos method, which does not guarantee correctness for singular vectors with zero-valued eigenvalues. Consequently, principal components with zero-valued eigenvalues are not guaranteed to be correct. Generally, this will not be problem unless the user wants to use the principal components for the entire eigenspectrum.
The goal of the MADlib pivot function is to provide a data summarization tool that can do basic OLAP type operations on data stored in one table and output the summarized data to a second table.
VARCHAR. Name of the source table (or view) containing data to pivot.
+
+
+
output_table
+
VARCHAR. Name of output table that contains the pivoted data. The output table contains all the columns present in the 'index' column list, plus additional columns for each distinct value in 'pivot_cols'.
+
Note
The names of the columns in the output table are auto-generated. Please see the examples section below to see how this works in practice. The convention used is to concatenate the following strings and separate each by an underscore '_' :
+
name of the value column 'pivot_values'
+
aggregate function
+
name of the pivot column 'pivot_cols'
+
values in the pivot column
+
+
+
+
index
+
VARCHAR. Comma-separated columns that will form the index of the output pivot table. By index we mean the values to group by; these are the rows in the output pivot table.
+
+
+
pivot_cols
+
VARCHAR. Comma-separated columns that will form the columns of the output pivot table.
+
+
+
pivot_values
+
VARCHAR. Comma-separated columns that contain the values to be summarized in the output pivot table.
+
+
+
aggregate_func (optional)
+
VARCHAR. default: 'AVG'. A comma-separated list of aggregates to be applied to values. These can be PostgreSQL built-in aggregates [1] or UDAs. It is possible to assign a set of aggregates per value column. Please refer to the examples 12-14 below for syntax details.
+
Note
Only aggregates with strict transition functions are permitted here. A strict transition function means rows with null values are ignored; the function is not called and the previous state value is retained. If you need some other behavior for null inputs, this should be done prior to calling the pivot function. Aggregates with strict transition functions are described in [2,3].
+
+
fill_value (optional)
+
VARCHAR. default: NULL. If specified, determines how to fill NULL values resulting from pivot operation. This is a global parameter (not applied per aggregate) and is applied post-aggregation to the output table.
+
+
+
keep_null (optional)
+
BOOLEAN. default: FALSE. If TRUE, then pivot columns are created corresponding to NULL categories. If FALSE, then no pivot columns will be created for NULL categories.
+
+
+
output_col_dictionary (optional)
+
BOOLEAN. default: FALSE. This parameter is used to handle auto-generated column names that exceed the PostgreSQL limit of 63 bytes (could be a common occurrence). If TRUE, column names will be set as numerical IDs and will create a dictionary table called output_table appended with _dictionary. If FALSE, will auto-generate column names in the usual way unless the limit of 63 bytes will be exceeded. In this case, a dictionary output file will be created and a message given to the user.
+
+
+
output_type (optional)
+
VARCHAR. default: 'column'. This parameter controls the output format of the pivoted variables. If 'column', a column is created for each pivot variable. PostgreSQL limits the number of columns in a table (250 - 1600 depending on column types). If the total number of output columns exceeds this limit, then make this parameter either 'array' (to combine the output columns into an array) or 'svec' (to cast the array output to 'madlib.svec' type). If you have an 'aggregate_func' that has an array return type, it cannot be combined with 'output_type'='array' or 'svec'.
+
A dictionary will be created (output_col_dictionary=TRUE) when 'output_type' is 'array' or 'svec' to define each index into the array.
+
+
+
+
Note
+
NULLs in the index column are treated like any other value.
+
NULLs in the pivot column are ignored unless keep_null is TRUE.
+
Only strict transition functions are allowed so NULLs are ignored.
+
It is not allowed to set the fill_value parameter without setting the aggregate_func parameter due to possible ambiguity. Set aggregate_func to NULL for the default behavior and use fill_value as desired. Please note that full_value must be of the same type as the output of the aggregate_func (or capable of being cast to the same type by PostgreSQL), or else an error will result.
+
It is not allowed to set the output_col_dictionary parameter without setting the keep_null parameter due to possible ambiguity. Set keep_null to NULL for the default behavior and use output_col_dictionary as desired.
+
Expressions (instead of column names) are not supported. Create a view with the desired expressions and pass it as the input table (see example 3 below).
+
It is allowed to pass a partial mapping for the aggregate_func parameter. The missing value columns will be aggregated using the default function (average).
+
+
+
Examples
+
+
Create a simple dataset to demonstrate a basic pivot:
+DROP TABLE IF EXISTS pivset CASCADE; -- View below may depend on table so use CASCADE
+CREATE TABLE pivset(
+ id INTEGER,
+ piv INTEGER,
+ val FLOAT8
+ );
+INSERT INTO pivset VALUES
+ (0, 10, 1),
+ (0, 10, 2),
+ (0, 20, 3),
+ (1, 20, 4),
+ (1, 30, 5),
+ (1, 30, 6),
+ (1, 10, 7),
+ (NULL, 10, 8),
+ (1, NULL, 9),
+ (1, 10, NULL);
+
+
Pivot the table:
+DROP TABLE IF EXISTS pivout;
+SELECT madlib.pivot('pivset', 'pivout', 'id', 'piv', 'val');
+SELECT * FROM pivout ORDER BY id;
+
Use multiple aggregate functions on the same value column (cross product):
+DROP TABLE IF EXISTS pivout;
+SELECT madlib.pivot('pivset_ext', 'pivout', 'id', 'piv', 'val', 'avg, sum');
+\x on
+SELECT * FROM pivout ORDER BY id;
+
Use different aggregate functions for different value columns:
+DROP TABLE IF EXISTS pivout;
+SELECT madlib.pivot('pivset_ext', 'pivout', 'id', 'piv', 'val, val2',
+ 'val=avg, val2=sum');
+\x on
+SELECT * FROM pivout ORDER BY id;
+
Use multiple aggregate functions for different value columns:
+DROP TABLE IF EXISTS pivout;
+SELECT madlib.pivot('pivset_ext', 'pivout', 'id', 'piv', 'val, val2',
+ 'val=avg, val2=[avg,sum]');
+\x on
+SELECT * FROM pivout ORDER BY id;
+
NOTE: The following links refer to documentation resources for the current PostgreSQL database version. Depending upon your database platform version, you may need to change "current" references in the links to your database version.
+
If your database platform uses the Greenplum Database (or related variants), please check with the project community and/or your database vendor to identify the PostgreSQL version it is based on.
The PMML export function in MADlib has the following syntax:
+pmml ( model_table,
+ name_spec
+ )
+
Arguments
+
model_table
+
VARCHAR. The name of the table containing the model.
+
+
+
name_spec (optional)
+
VARCHAR or VARCHAR[]. Names to be used in the Data Dictionary of the PMML. See pmml() for detailed explanation.
+
+
+
Output XML. The output of this function is a standard PMML document, some examples of which are covered in the next section.
+
Note
In PostgreSQL, users may be required to install their database with XML support in order to use this function.
+
Usually the user wants to export the resulting PMML contents into a PMML file so that external softwares can use it. The following method can be used (Note: the user needs to use unaligned table output mode for psql with '-A' flag. And inside psql client, both '\t' and '\o' should be used):
+
+> # under bash
+> psql -A my_database
+# -- in psql now
+# \t
+# \o test.pmml -- export to a file
+# select madlib.pmml('tree_out');
+# \o
+# \t
+
Note: If the second argument of 'pmml' function is not specified, a default suffix "_pmml_prediction" will be automatically append to the column name to be predicted. This can help avoid name conflicts.
+
The following example demonstrates grouping columns in the model table for the same dataset as the previous example.
+
+
Train a different regression model with 'treatment' as the grouping column.
Note: MADlib currently supports PMML export for Linear Regression, Logistic Regression, Generalized Linear Regression Model, Multinomial Logistic Regression, Ordinal Linear Regression, Decision Tree and Random Forests.
+
In Ordinal Regression, the signs of feature coefficients will be different in PMML export and in the default output model table from ordinal(). This is due to the difference of model settings.
+
Background
The Predictive Model Markup Language (PMML) is an XML-based file format that provides a way for applications to describe and exchange models produced by data mining and machine learning algorithms. A PMML file comprises the following components:
+
Header: Contains general information of the model, such as copyright information and model description.
+
Data Dictionary: Contains definitions of fields used in the model.
+
Data Transformations: Contains transformations for mapping user data into a form that can be used by the mining model.
+
Model: Contains definitions of the data mining model, which includes attributes such as the model name, function name, and algorithm name.
+
Mining Schema: Contains specific information for the fields used in the model, which includes the name and usage type.
+
Targets: Allows for post-processing of the predicted value.
+
Output: Allows for naming of output fields expected from the model.
This module provides a set of metrics to evaluate the quality of predictions of a model. A typical function will take a set of "prediction" and "observation" values and use them to calculate the desired metric, unless noted otherwise. Grouping is supported for all functions (except confusion matrix).
TEXT. Name of the output table. For consistency, a table is created for all metric outputs even when grouping is not used, which may mean there is only a single value in the output table in some cases.
+
prediction_col
+
TEXT. Name of the column of predicted values from input table.
+
observed_col
+
TEXT. Name of the column of observed values from input table.
INTEGER. The number of parameters in the predicting model, not counting the constant term.
+
training_size (for adjusted R-squared score only)
+
INTEGER. The number of rows used for training, excluding any NULL rows.
+
grouping_cols (optional)
+
TEXT, default: NULL. Name of the column of grouping values from input table.
+
+
Function Specific Details
+
R-squared Score
+
This function returns the coefficient of determination (R2) between the predicted and observed values. An R2 of 1 indicates that the regression line perfectly fits the data, while an R2 of 0 indicates that the line does not fit the data at all. Negative values of R2 may occur when fitting non-linear functions to data. Please refer to reference [1] for more details.
+
Adjusted R-squared Score
+
This function returns the adjusted R2 score in addition to the R-squared score described above. Adjusted R2 score is used to counter the problem of the R2 automatically increasing when extra explanatory variables are added to the model. It takes two additional parameters describing the degrees of freedom of the model (num_predictors) and the size of the training set over which it was developed (training_size):
+
+
num_predictors: Indicates the number of parameters the model has other than the constant term. For example, if it is set to '3' the model may take the following form as an example: 7 + 5x + 39y + 0.91z.
+
training_size: Indicates the number of rows in the training set (excluding any NULL rows).
+
+
Neither of these arguments can be deduced from the predicted values and the test data alone which is why they are explicit inputs. Please refer to reference [1] for more details.
+
Binary Classification
+
This function returns an output table with a number of metrics commonly used in binary classification.
+
The definitions of the various metrics are as follows:
+
+
\(\textit{tp}\) is the count of correctly-classified positives.
+
\(\textit{tn}\) is the count of correctly-classified negatives.
+
\(\textit{fp}\) is the count of misclassified negatives.
+
\(\textit{fn}\) is the count of misclassified positives.
This function returns the area under the Receiver Operating Characteristic curve for binary classification (the AUC). The ROC curve is the curve relating the classifier's TPR and FPR metrics. (See Binary Classification above for a definition of these metrics). Please refer to reference [2] for more details. Note that the binary classification function can be used to obtain the data (TPR and FPR values) required for drawing the ROC curve.
+
Note
For 'binary_classifier' and 'area_under_roc' functions:
+
The 'observed_col' column is assumed to be a numeric column with two values: 0 and 1, or a Boolean column. For the purposes of the metric calculation, 0 is considered to be negative and 1 to be positive.
+
The 'pred_col' column is expected to contain numeric values corresponding to likelihood/probability. A larger value corresponds to greater certainty that the observed value will be '1', and a lower value corresponds to a greater certainty that it will be '0'.
+
+
+
Confusion Matrix
+
This function returns the confusion matrix of a multi-class classification. Each column of the matrix represents the instances in a predicted class while each row represents the instances in an actual class. This allows more detailed analysis than mere proportion of correct guesses (accuracy). Please refer to the reference [3] for more details. Please note that grouping is not supported for the confusion matrix.
+
Examples
+
+
Create the sample data:
+DROP TABLE IF EXISTS test_set;
+CREATE TABLE test_set(
+ pred FLOAT8,
+ obs FLOAT8
+ );
+INSERT INTO test_set VALUES
+ (37.5,53.1), (12.3,34.2), (74.2,65.4), (91.1,82.1);
+
+
Run the Mean Absolute Error function:
+DROP TABLE IF EXISTS table_out;
+SELECT madlib.mean_abs_error( 'test_set', 'table_out', 'pred', 'obs');
+SELECT * FROM table_out;
+
Result
+ mean_abs_error
+ ----------------
+ 13.825
+
+
Run the Mean Absolute Percentage Error function:
+DROP TABLE IF EXISTS table_out;
+SELECT madlib.mean_abs_perc_error( 'test_set', 'table_out', 'pred', 'obs');
+SELECT * FROM table_out;
+
Create the sample data for binary classifier metrics:
+DROP TABLE IF EXISTS test_set;
+CREATE TABLE test_set AS
+ SELECT ((a*8)::integer)/8.0 pred,
+ ((a*0.5+random()*0.5)>0.5) obs
+ FROM (select random() as a from generate_series(1,100)) x;
+
The Probability Functions module provides cumulative distribution, density/mass, and quantile functions for a wide range of probability distributions.
+
Unless otherwise documented, all of these functions are wrappers around functionality provided by the boost C++ library [1, “Statistical Distributions and Functions”].
+
For convenience, all cumulative distribution and density/mass functions (CDFs and PDF/PMFs in short) are defined over the range of all floating-point numbers including infinity. Inputs that are NULL or NaN (not a number) will always produce a NULL or NaN result, respectively. Inputs that are plus or minus infinity will return the respective limits.
+
A quantile function for a probability distrution with CDF \( F \) takes a probability argument \( p \in [0,1] \) and returns the value \( x \) so that \( F(x) = p \), provided such an \( x \) exists and it is unique. If it does not, the result will be \( \sup \{ x \in D \mid F(x) \leq p \} \) (interpreted as 0 if the supremum is over an empty set) if \( p < 0.5 \), and \( \inf \{ x \in D \mid F(x) \geq p \} \) if \( p \geq 0.5 \). Here \( D \) denotes the domain of the distribution, which is the set of reals \( \mathbb R \) for continuous and the set of nonnegative integers \( \mathbb N_0 \) for discrete distributions.
+
Intuitively, the formulas in the previous paragraph deal with the following special cases. The 0-quantile will always be the “left end” of the support, and the 1-quantile will be the “right end” of the support of the distribution. For discrete distributions, most values of \( p \in [0,1] \) do not admit an \( x \) with \( F(x) = p \). Instead, there is an \( x \in \mathbb N_0 \) so that \( F(x) < p < F(x + 1) \). The above formulas mean that the value returned as \( p \)-quantile is \( x \) if \( p < 0.5 \), and it is \( x + 1 \) if \( p \geq 0.5 \). (As a special case, in order to ensure that quantiles are always within the support, the \( p \)-quantile will be 0 if \( p < F(0) \)).
+
The rationale for choosing this behavior is that \(p\)-quantiles for \( p < 0.5 \) are typically requested when interested in the value \( x \) such that with confidence level at least \( 1 - p \) a random variable will be \( > x \) (or equivalently, with probability at most \( p \), it will be \( \leq x \)). Likewise, \(p\)-quantiles for \( p \geq 0.5 \) are typically requested when interested in the value \( x \) such that with confidence level at least \( p \) a random variable will be \( \leq x \). See also [1, “Understanding Quantiles of Discrete Distributions”].
Random forest builds an ensemble of classifiers, each of which is a tree model constructed using bootstrapped samples from the input data. The results of these models are then combined to yield a single prediction, which, at the expense of some loss in interpretation, can be highly accurate. Refer to Breiman et al. [1][2][3] for details on the implementation used here.
text. Name of the table containing the training data.
+
+
+
output_table_name
+
TEXT. Name of the generated table containing the model. If a table with the same name already exists, an error will be returned. A summary table named <output_table_name>_summary and a grouping table named <output_table_name>_group are also created. These are described later on this page.
+
+
+
id_col_name
+
TEXT. Name of the column containing id information in the training data. This is a mandatory argument and is used for prediction and other purposes. The values are expected to be unique for each row.
+
+
+
dependent_variable
+
TEXT. Name of the column that contains the output (response) for training. Boolean, integer and text types are considered to be classification outputs, while double precision values are considered to be regression outputs. The response variable for a classification tree can be multinomial, but the time and space complexity of the training function increases linearly as the number of response classes increases.
+
+
+
list_of_features
+
TEXT. Comma-separated string of column names or expressions to use as predictors. Can also be a '*' implying all columns are to be used as predictors (except for the ones included in the next argument that lists exclusions). The types of the features can be mixed: boolean, integer, and text columns are considered categorical and double precision columns are considered continuous. Categorical variables are not encoded and used as is in the training.
+
Array columns can also be included in the list, where the array is expanded to treat each element of the array as a feature.
+
Note that not every combination of the levels of a categorical variable is checked when evaluating a split. The levels of the non-integer categorical variable are ordered by the entropy of the variable in predicting the response. The split at each node is evaluated between these ordered levels. Integer categorical variables, however, are simply ordered by their value.
+
+
+
list_of_features_to_exclude
+
TEXT. Comma-separated string of column names to exclude from the predictors list. If the dependent_variable is an expression (including cast of a column name), then this list should include the columns present in the dependent_variable expression, otherwise those columns will be included in the features (resulting in meaningless trees). The names in this parameter should be identical to the names used in the table and quoted appropriately.
+
+
+
grouping_cols (optional)
+
TEXT, default: NULL. Comma-separated list of column names to group the data by. This will produce multiple random forests, one for each group.
+
+
+
num_trees (optional)
+
INTEGER, default: 100. Maximum number of trees to grow in the random forest model. Actual number of trees grown may be different, depending on the data.
+
+
+
num_random_features (optional)
+
INTEGER, default: sqrt(n) for classification, n/3 for regression, where n is the number of features. This parameter is the number of features to randomly select at each split.
+
+
+
importance (optional)
+
BOOLEAN, default: true. Whether or not to calculate variable importance. If set to true, out-of-bag variable importance and impurity variable importance for categorical and continuous features will be output to the group table <model_table>_group. Note that total runtime will increase when variable importance is turned on. Refer to [1][2][3] for more information on variable importance.
+
+
+
num_permutations (optional)
+
INTEGER, default: 1. Number of times to permute each feature value while calculating the out-of-bag variable importance. Only applies when the 'importance' parameter is set to true.
+
Note
Variable importance for a feature is determined by permuting the variable and computing the drop in predictive accuracy using out-of-bag samples [1]. Setting this greater than 1 performs an average over multiple importance calculations, but increases total run time. In most cases, the default value of 1 is sufficient to compute the importance. Due to nature of permutation, the importance value can end up being negative if the number of levels for a categorical variable is small and is unbalanced. In such a scenario, the importance values are shifted to ensure that the lowest importance value is 0. To see importance values normalized to sum to 100 across all variables, use the importance display helper function described later on this page.
+
+
+
max_tree_depth (optional)
+
INTEGER, default: 7. Maximum depth of any node of a tree, with the root node counted as depth 0. A deeper tree can lead to better prediction but will also result in longer processing time and higher memory usage. Current allowed maximum is 15. Note that since random forest is an ensemble method, individual trees typically do not need to be deep.
+
+
+
min_split (optional)
+
INTEGER, default: 20. Minimum number of observations that must exist in a node for a split to be attempted.
+
+
+
min_bucket (optional)
+
INTEGER, default: min_split/3. Minimum number of observations in any terminal node. If only one of min_bucket or min_split is specified, min_split is set to min_bucket*3 or min_bucket to min_split/3, as appropriate.
+
+
+
num_splits (optional)
+
INTEGER, default: 20. Continuous-valued features are binned into discrete quantiles to compute split boundaries. This global parameter is used to compute the resolution of splits for continuous features. Higher number of bins will lead to better prediction, but will also result in longer processing time and higher memory usage.
+
+
+
null_handling_params (optional)
+
TEXT. Comma-separated string of key-value pairs controlling the behavior of various features handling missing values. One of the following can be used if desired (not both):
+
+
max_surrogates
Default: 0. Number of surrogates to store for each node. One approach to handling NULLs is to use surrogate splits for each node. A surrogate variable enables you to make better use of the data by using another predictor variable that is associated (correlated) with the primary split variable. The surrogate variable comes into use when the primary predictior value is NULL. Surrogate rules implemented here are based on reference [1].
+
+
null_as_category
Default: FALSE. Whether to treat NULL as a valid level for categorical features. FALSE means that NULL is not a valid level, which is probably the most common sitation.
+
If set to TRUE, NULL values are considered a categorical value and placed at the end of the ordering of categorical levels. Placing at the end ensures that NULL is never used as a value to split a node on. One reason to make NULL a category is that it allows you to predict on categorical levels that were not in the training data by lumping them into an "other bucket."
+
This parameter is ignored for continuous-valued features.
+
+
+
+
+
verbose (optional)
+
BOOLEAN, default: FALSE. Provides verbose output of the results of training.
+
+
+
sample_ratio (optional)
+
DOUBLE PRECISION, in the range of (0, 1], default: 1. If 'sample_ratio' is less than 1, a bootstrap sample size smaller than the data table is used for training each tree in the forest. A ratio that is close to 0 may result in trees with only the root node. This sample parameter allows users to quickly experiment with the random forest function since it reduces run time by using only some of the data.
+
+
Output
+
+
The model table produced by the training function contains the following columns:
+
+
+
gid
INTEGER. Group id that uniquely identifies a set of grouping column values.
+
+
sample_id
INTEGER. The id of the bootstrap sample that this tree is a part of.
+
+
tree
BYTEA8. Trained tree model stored in binary format (not human readable).
+
+
A summary table named <model_table>_summary is also created at the same time, which contains the following columns:
+
+
method
'forest_train'
+
+
+
+
is_classification
BOOLEAN. True if it is a classification model, false if for regression.
+
+
+
+
source_table
TEXT. Data source table name.
+
+
+
+
model_table
TEXT. Model table name.
+
+
+
+
id_col_name
TEXT. The ID column name.
+
+
+
+
dependent_varname
TEXT. Dependent variable.
+
+
+
+
independent_varnames
TEXT. Independent variables
+
+
+
+
cat_features
TEXT. List of categorical features as a comma-separated string.
+
+
+
+
con_features
TEXT. List of continuous feature as a comma-separated string.
+
+
+
+
grouping_cols
INTEGER. Names of grouping columns.
+
+
+
+
num_trees
INTEGER. Number of trees grown by the model.
+
+
+
+
num_random_features
INTEGER. Number of features randomly selected for each split.
+
+
+
+
max_tree_depth
INTEGER. Maximum depth of any tree in the random forest model_table.
+
+
+
+
min_split
INTEGER. Minimum number of observations in a node for it to be split.
+
+
+
+
min_bucket
INTEGER. Minimum number of observations in any terminal node.
+
+
+
+
num_splits
INTEGER. Number of buckets for continuous variables.
+
+
+
+
verbose
BOOLEAN. Whether or not to display debug info.
+
+
+
+
importance
BOOLEAN. Whether or not to calculate variable importance.
+
+
+
+
num_permutations
INTEGER. Number of times feature values are permuted while calculating out-of-bag variable importance.
+
+
+
+
num_all_groups
INTEGER. Number of groups during forest training.
+
+
+
+
num_failed_groups
INTEGER. Number of failed groups during forest training.
+
+
+
+
total_rows_processed
BIGINT. Total numbers of rows processed in all groups.
+
+
+
+
total_rows_skipped
BIGINT. Total numbers of rows skipped in all groups due to missing values or failures.
+
+
+
+
dependent_var_levels
TEXT. For classification, the distinct levels of the dependent variable.
+
+
+
+
dependent_var_type
TEXT. The type of dependent variable.
+
+
+
+
independent_var_types
TEXT. A comma separated string for the types of independent variables.
+
+
+
+
null_proxy
TEXT. Describes how NULLs are handled. If NULL is not treated as a separate categorical variable, this will be NULL. If NULL is treated as a separate categorical value, this will be set to "__NULL__"
+
+
A table named <model_table>_group is also created at the same time, even if no grouping is specified. It contains the following columns:
+
+
gid
integer. Group id that uniquely identifies a set of grouping column values. If grouping is not used, this will always be 1.
+
+
+
+
<...>
Same type as in the training data table 'grouping_cols'. This could be multiple columns depending on the 'grouping_cols' input.
+
+
+
+
success
BOOLEAN. Indicator of the success of the group.
+
+
+
+
cat_levels_in_text
TEXT[]. Ordered levels (values) of categorical variables corresponding to the categorical features in the 'list_of_features' argument above. Used to help interpret the trained tree. For example, if the categorical features specified are weather_outlook and windy in that order, then 'cat_levels_in_text' might be [overcast, rain, sunny, False, True].
+
+
+
+
cat_n_levels
INTEGER[]. Number of levels for each categorical variable. Used to help interpret the trained tree. In the example from above, 'cat_n_levels' would be [3, 2] since there are 3 levels for weather_outlook and 2 levels windy.
+
+
+
+
oob_error
DOUBLE PRECISION. Out-of-bag error for the random forest model.
+
+
+
+
oob_var_importance
DOUBLE PRECISION[]. Out-of-bag variable importance for both categorical and continuous features. The order corresponds to the order of the variables in 'independent_varnames' in <model_table>_summary.
+
+
+
+
impurity_var_importance
DOUBLE PRECISION[]. Impurity variable importance for both categorial and continuous features. The order corresponds to the order of the variables in 'independent_varnames' in <model_table>_summary.
+
+
+
+
Run-time and Memory Usage
+
The number of features and the number of class values per categorical feature have a direct impact on run-time and memory. In addition, here is a summary of the main parameters in the training function that affect run-time and memory:
+
+
+
Parameter
Run-time
Memory
Notes
+
+
'num_trees'
High
No or little effect.
Linear with number of trees. Notes that trees train sequentially one after another, though each tree is trained in parallel.
+
+
'importance'
Moderate
No or little effect.
Depends on number of features and 'num_permutations' parameter.
+
+
'num_permutations'
Moderate
No or little effect.
Depends on number of features.
+
+
'max_tree_depth'
High
High
Deeper trees can take longer to run and use more memory.
+
+
'min_split'
No or little effect, unless very small.
No or little effect, unless very small.
If too small, can impact run-time by building trees that are very thick.
+
+
'min_bucket'
No or little effect, unless very small.
No or little effect, unless very small.
If too small, can impact run-time by building trees that are very thick.
+
+
'num_splits'
High
High
Depends on number of continuous variables. Effectively adds more features as the binning becomes more granular.
+
+
'sample_ratio'
High
High
Reduces run time by using only some of the data.
+
+
If you experience long run-times or are hitting memory limits, consider reducing one or more of these parameters. One approach when building a random forest model is to start with a small number of trees and a low maximum depth value, and use suggested defaults for other parameters. This will give you a sense of run-time and test set accuracy. Then you can change number of trees and maximum depth in a systematic way as required to improve accuracy.
+
Prediction Function
The prediction function estimates the conditional mean given a new predictor. It has the following syntax:
text. Name of the table containing the random forest model from training.
+
+
+
new_data_table
+
TEXT. Name of the table containing prediction data. This table is expected to contain the same features that were used during training. The table should also contain id_col_name used for identifying each row.
+
+
+
output_table
+
TEXT. Name of the table to output prediction results. If this table already exists, an error is returned. The table contains the id_col_name column giving the 'id' for each prediction and the prediction columns for the dependent variable.
+
If type = 'response', then the table has a single additional column with the prediction value of the response. The type of this column depends on the type of the response variable used during training.
+
If type = 'prob', then the table has multiple additional columns, one for each possible value of the response variable. The columns are labeled as 'estimated_prob_dep_value', where dep_value represents each value of the response variable.
+
+
+
type (optional)
+
TEXT, optional, default: 'response'. For regression trees, the output is always the predicted value of the dependent variable. For classification trees, the type variable can be 'response', giving the classification prediction as output, or 'prob', giving the class probabilities as output. For each value of the dependent variable, a column with the probabilities is added to the output table.
+
+
Tree Display
The display function outputs a graph representation of a single tree of the random forest. The output can either be in the popular 'dot' format that can be visualized using various programs including those in the GraphViz package, or in a simple text format. The details of the text format are output with the tree.
This output contains the list of surrogate splits for each internal node. The nodes are sorted in ascending order by id. This is equivalent to viewing the tree in a breadth-first manner. For each surrogate, we output the surrogate split (variable and threshold) and also give the number of rows that were common between the primary split and the surrogate split. Finally, the number of rows present in the majority branch of the primary split is also shown. Only surrogates that perform better than this majority branch are included in the surrogate list. When the primary variable has a NULL value the surrogate variables are used in order to compute the split for that node. If all surrogates variables are NULL, then the majority branch is used to compute the split for a tuple.
+
Arguments
+
forest_model_table
+
TEXT. Name of the table containing the random forest model.
+
+
+
gid
+
INTEGER. Id of the group that this tree is part of.
+
+
+
sample_id
+
INTEGER. Id of the bootstrap sample that this tree is part of.
+
+
+
dot_format (optional)
+
BOOLEAN, default = TRUE. Output can either be in a dot format or a text format. If TRUE, the result is in the dot format, else output is in text format.
+
+
+
verbose (optional)
+
BOOLEAN, default = FALSE. If true, the dot format output will contain additional information (impurity, sample size, number of weighted rows for each response variable, classification or prediction if the tree was pruned at this level)
+
+
The output is always returned as a 'TEXT'. For the dot format, the output can be redirected to a file on the client side and then rendered using visualization programs.
+
To export the dot format result to an external file, use the method below. Please note that you should use unaligned table output mode for psql with '-A' flag, or else you may get an error when you try to convert the dot file to another format for viewing (e.g., PDF). And inside the psql client, both '\t' and '\o' should be used:
+
+> # under bash
+> psql -A my_database
+# -- in psql now
+# \t
+# \o test.dot -- export to a file
+# select madlib.tree_display('tree_out');
+# \o
+# \t
+
After the dot file has been generated, use third-party plotting software to plot the trees in a nice format:
+> # under bash, convert the dot file into a PDF file
+> dot -Tpdf test.dot > test.pdf
+> xpdf test.pdf&
+
This is a helper function that creates a table to more easily view out-of-bag and impurity variable importance values for a given model table. This function rescales the importance values to represent them as percentages i.e. importance values are scaled to sum to 100.
TEXT. Name of the table containing the random forest model.
+
output_table
+
TEXT. Name of the table to create for importance values.
+
+
The summary and group tables generated by the forest_train function are required for this function to work.
+
Examples
+
Note
+
Not all random forest parameters are demonstrated in the examples below. Some are shown in the decision tree user documentation since usage is similar.
+
Your results may look different than those below due the random nature of random forests.
+
+
+
Random Forest Classification Example
+
+
Load input data set related to whether to play golf or not:
The 'cat_levels_in_text' array shows the levels of the categorical variables "OUTLOOK" and windy, which have 3 and 2 levels respectively. Out-of-bag and impurity variable importance arrays are ordered according to the order of the variables in 'independent_varnames' in <model_table>_summary. A higher value means higher importance for the variable. We can use the helper function to get a normalized view of variable importance:
+\x off
+DROP TABLE IF EXISTS imp_output;
+SELECT madlib.get_var_importance('train_output','imp_output');
+SELECT * FROM imp_output ORDER BY oob_var_importance DESC;
+
Predict output categories. For the purpose of this example, we use the same data that was used for training:
+\x off
+DROP TABLE IF EXISTS prediction_results;
+SELECT madlib.forest_predict('train_output', -- tree model
+ 'rf_golf', -- new data table
+ 'prediction_results', -- output table
+ 'response'); -- show response
+SELECT g.id, class, estimated_class FROM prediction_results p,
+rf_golf g WHERE p.id = g.id ORDER BY g.id;
+
+ id | class | estimated_class
+----+------------+-----------------
+ 1 | Don't Play | Don't Play
+ 2 | Don't Play | Don't Play
+ 3 | Play | Play
+ 4 | Play | Play
+ 5 | Play | Play
+ 6 | Don't Play | Don't Play
+ 7 | Play | Play
+ 8 | Don't Play | Don't Play
+ 9 | Play | Play
+ 10 | Play | Play
+ 11 | Play | Play
+ 12 | Play | Play
+ 13 | Play | Play
+ 14 | Don't Play | Don't Play
+(14 rows)
+
To display the probabilities associated with each value of the dependent variable, set the 'type' parameter to 'prob':
+DROP TABLE IF EXISTS prediction_results;
+SELECT madlib.forest_predict('train_output', -- tree model
+ 'rf_golf', -- new data table
+ 'prediction_results', -- output table
+ 'prob'); -- show probability
+SELECT g.id, class, "estimated_prob_Don't Play", "estimated_prob_Play"
+FROM prediction_results p, rf_golf g WHERE p.id = g.id ORDER BY g.id;
+
+ id | class | estimated_prob_Don't Play | estimated_prob_Play
+----+------------+---------------------------+---------------------
+ 1 | Don't Play | 0.9 | 0.1
+ 2 | Don't Play | 0.85 | 0.15
+ 3 | Play | 0 | 1
+ 4 | Play | 0.35 | 0.65
+ 5 | Play | 0.05 | 0.95
+ 6 | Don't Play | 0.85 | 0.15
+ 7 | Play | 0.25 | 0.75
+ 8 | Don't Play | 0.85 | 0.15
+ 9 | Play | 0.15 | 0.85
+ 10 | Play | 0.15 | 0.85
+ 11 | Play | 0.35 | 0.65
+ 12 | Play | 0.1 | 0.9
+ 13 | Play | 0 | 1
+ 14 | Don't Play | 0.8 | 0.2
+(14 rows)
+
+
View a single tree in text format within the forest identified by 'gid' and 'sample_id', out of the several that were created:
+ -------------------------------------
+ - Each node represented by 'id' inside ().
+ - Leaf nodes have a * while internal nodes have the split condition at the end.
+ - For each internal node (i), it's children will be at (2i+1) and (2i+2).
+ - For each split the first indented child (2i+1) is the 'True' node and
+second indented child (2i+2) is the 'False' node.
+ - Number of (weighted) rows for each response variable inside [].
+ - Order of values = ['"Don\'t Play"', '"Play"']
+ -------------------------------------
+ (0)[ 5 10] windy in {False}
+ (1)[2 8] "OUTLOOK" in {overcast,sunny}
+ (3)[2 1] humidity <= 75
+ (7)[0 1] * --> "Play"
+ (8)[2 0] * --> "Don't Play"
+ (4)[0 7] * --> "Play"
+ (2)[3 2] temperature <= 75
+ (5)[1 2] humidity <= 70
+ (11)[1 1] * --> "Don't Play"
+ (12)[0 1] * --> "Play"
+ (6)[2 0] * --> "Don't Play"
+ -------------------------------------
+
Arrays of features. Categorical and continuous features can be array columns, in which case the array is expanded to treat each element of the array as a feature:
+DROP TABLE IF EXISTS train_output, train_output_group, train_output_summary;
+SELECT madlib.forest_train('rf_golf', -- source table
+ 'train_output', -- output model table
+ 'id', -- id column
+ 'class', -- response
+ '"Temp_Humidity", clouds_airquality', -- features
+ NULL, -- exclude columns
+ NULL, -- grouping columns
+ 20::integer, -- number of trees
+ 2::integer, -- number of random features
+ TRUE::boolean, -- variable importance
+ 1::integer, -- num_permutations
+ 8::integer, -- max depth
+ 3::integer, -- min split
+ 1::integer, -- min bucket
+ 10::integer -- number of splits per continuous variable
+ );
+\x on
+SELECT * FROM train_output_summary;
+
Load input data related to fuel consumption and 10 aspects of automobile design and performance for 32 automobiles (1973–74 models). Data was extracted from the 1974 Motor Trend US magazine.
Use the helper function to display normalized variable importance:
+\x off
+DROP TABLE IF EXISTS mt_imp_output;
+SELECT madlib.get_var_importance('mt_cars_output','mt_imp_output');
+SELECT * FROM mt_imp_output ORDER BY am, oob_var_importance DESC;
+
Predict for data not previously seen by assuming NULL value as the default:
+\x off
+DROP TABLE IF EXISTS table_test;
+CREATE TABLE table_test (
+ id integer,
+ country text,
+ city text,
+ weather text,
+ expected_response text
+);
+INSERT INTO table_test VALUES
+(1,'IN','MUM','cloudy','a'),
+(2,'US','HOU','humid','b'),
+(3,'US','NY','sunny','c'),
+(4,'US','NY','rainy','d');
+DROP TABLE IF EXISTS prediction_results;
+SELECT madlib.forest_predict('train_output',
+ 'table_test',
+ 'prediction_results',
+ 'response');
+SELECT s.id, expected_response, estimated_response
+FROM prediction_results p, table_test s
+WHERE s.id = p.id ORDER BY id;
+
+ id | expected_response | estimated_response
+----+-------------------+--------------------
+ 1 | a | a
+ 2 | b | b
+ 3 | c | c
+ 4 | d | d
+(4 rows)
+
There is only training data for country 'US' so the response for country 'IN' is 'a', corresponding to a NULL (not 'US') country level. Likewise, any city in the 'US' that is not 'NY' will predict response 'b', corresponding to a NULL (not 'NY') city level.
Generates a regularized regression model for variable selection in linear and logistic regression problems, combining the L1 and L2 penalties of the lasso and ridge methods.
Estimate generalized linear model (GLM). GLM is a flexible generalization of ordinary linear regression that allows for response variables that have error distribution models other than a normal distribution. The GLM generalizes linear regression by allowing the linear model to be related to the response variable via a link function and by allowing the magnitude of the variance of each measurement to be a function of its predicted value.
Models the relationship between one or more predictor variables and a binary categorical dependent variable by predicting the probability of the dependent variable using a logistic function.
The functions in this module calculate robust variance (Huber-White estimates) for linear regression, logistic regression, multinomial logistic regression, and Cox proportional hazards. They are useful in calculating variances in a dataset with potentially noisy outliers. The Huber-White implemented here is identical to the "HC0" sandwich operator in the R module "sandwich".
+
The interfaces for robust linear, logistic, and multinomial logistic regression are similar. Each regression type has its own training function. The regression results are saved in an output table with small differences, depending on the regression type.
+
Warning
Please note that the interface for Cox proportional hazards, unlike the interface of other regression methods, accepts an output model table produced by coxph_train() function.
VARCHAR. The name of the table containing the training data.
+
out_table
+
VARCHAR. Name of the generated table containing the output model. The output table contains the following columns.
+
+
coef
DOUBLE PRECISION[]. Vector of the coefficients of the regression.
+
+
std_err
DOUBLE PRECISION[]. Vector of the standard error of the coefficients.
+
+
t_stats
DOUBLE PRECISION[]. Vector of the t-stats of the coefficients.
+
+
p_values
DOUBLE PRECISION[]. Vector of the p-values of the coefficients.
+
+
A summary table named <out_table>_summary is also created, which is the same as the summary table created by linregr_train function. Please refer to the documentation for linear regression for details.
+
+
dependent_varname
+
VARCHAR. The name of the column containing the dependent variable.
+
independent_varname
+
VARCHAR. Expression list to evaluate for the independent variables. An intercept variable is not assumed. It is common to provide an explicit intercept term by including a single constant 1 term in the independent variable list.
+
grouping_cols (optional)
+
VARCHAR, default: NULL. An expression list used to group the input dataset into discrete groups, running one regression per group. Similar to the SQL "GROUP BY" clause. When this value is NULL, no grouping is used and a single result model is generated. Default value: NULL.
VARCHAR. The name of the table containing the training data.
+
out_table
+
VARCHAR. Name of the generated table containing the output model. The output table has the following columns:
+
+
coef
Vector of the coefficients of the regression.
+
+
std_err
Vector of the standard error of the coefficients.
+
+
z_stats
Vector of the z-stats of the coefficients.
+
+
p_values
Vector of the p-values of the coefficients.
+
+
A summary table named <out_table>_summary is also created, which is the same as the summary table created by logregr_train function. Please refer to the documentation for logistic regression for details.
+
+
dependent_varname
+
VARCHAR. The name of the column containing the independent variable.
+
independent_varname
+
VARCHAR. Expression list to evaluate for the independent variables. An intercept variable is not assumed. It is common to provide an explicit intercept term by including a single constant 1 term in the independent variable list.
+
grouping_cols (optional)
+
VARCHAR, default: NULL. An expression list used to group the input dataset into discrete groups, running one regression per group. Similar to the SQL "GROUP BY" clause. When this value is NULL, no grouping is used and a single result model is generated.
+
max_iter (optional)
+
INTEGER, default: 20. The maximum number of iterations that are allowed.
+
optimizer
+
VARCHAR, default: 'fista'. Name of optimizer, either 'fista' or 'igd'.
+
tolerance (optional)
+
DOUBLE PRECISION, default: 1e-6. The criteria to end iterations. Both the 'fista' and 'igd' optimizers compute the average difference between the coefficients of two consecutive iterations, and when the difference is smaller than tolerance or the iteration number is larger than max_iter, the computation stops.
+
verbose_mode (optional)
+
BOOLEAN, default: FALSE. Whether the regression fit should print any warning messages.
VARCHAR. The name of the table containing training data, properly qualified.
+
out_table
+
VARCHAR. The name of the table where the regression model will be stored. The output table has the following columns:
+
+
category
The category.
+
+
ref_category
The refererence category used for modeling.
+
+
coef
Vector of the coefficients of the regression.
+
+
std_err
Vector of the standard error of the coefficients.
+
+
z_stats
Vector of the z-stats of the coefficients.
+
+
p_values
Vector of the p-values of the coefficients.
+
+
A summary table named <out_table>_summary is also created, which is the same as the summary table created by mlogregr_train function. Please refer to the documentation for multinomial logistic regression for details.
+
+
dependent_varname
+
VARCHAR. The name of the column containing the dependent variable.
+
independent_varname
+
VARCHAR. Expression list to evaluate for the independent variables. An intercept variable is not assumed. It is common to provide an explicit intercept term by including a single constant 1 term in the independent variable list. The independent_varname can be the name of a column that contains an array of numeric values. It can also be a string with the format 'ARRAY[1, x1, x2, x3]', where x1, x2 and x3 are each column names.
+
ref_category (optional)
+
INTEGER, default: 0. The reference category.
+
grouping_cols (optional)
+
VARCHAR, default: NULL. Not currently implemented. Any non-NULL value is ignored. An expression list used to group the input dataset into discrete groups, running one regression per group. Similar to the SQL "GROUP BY" clause. When this value is NULL, no grouping is used and a single result model is generated.
+
optimizer_params (optional)
+
TEXT, default: NULL, which uses the default values of optimizer parameters: max_iter=20, optimizer='newton', tolerance=1e-4. It should be a string that contains pairs of 'key=value' separated by commas.
+
verbose_mode (optional)
+
BOOLEAN, default FALSE. Not currently implemented. TRUE if the regression fit should print warning messages.
+
+
Robust Variance Function For Cox Proportional Hazards
Alternatively, unnest the arrays in the results for easier reading of output.
+\x off
+SELECT unnest(array['intercept', 'treatment', 'trait_anxiety' ]) as attribute,
+ unnest(coef) as coefficient,
+ unnest(std_err) as standard_error,
+ unnest(z_stats) as z_stat,
+ unnest(p_values) as pvalue
+FROM patients_logregr;
+
+
+
Cox Proportional Hazards Example
+
View online help for the robust Cox Proportional hazards training method.
+SELECT madlib.robust_variance_coxph();
+
+
Create an input data set.
+DROP TABLE IF EXISTS sample_data;
+CREATE TABLE sample_data (
+ id INTEGER NOT NULL,
+ grp DOUBLE PRECISION,
+ wbc DOUBLE PRECISION,
+ timedeath INTEGER,
+ status BOOLEAN
+);
+COPY sample_data FROM STDIN DELIMITER '|';
+ 0 | 0 | 1.45 | 35 | t
+ 1 | 0 | 1.47 | 34 | t
+ 3 | 0 | 2.2 | 32 | t
+ 4 | 0 | 1.78 | 25 | t
+ 5 | 0 | 2.57 | 23 | t
+ 6 | 0 | 2.32 | 22 | t
+ 7 | 0 | 2.01 | 20 | t
+ 8 | 0 | 2.05 | 19 | t
+ 9 | 0 | 2.16 | 17 | t
+ 10 | 0 | 3.6 | 16 | t
+ 11 | 1 | 2.3 | 15 | t
+ 12 | 0 | 2.88 | 13 | t
+ 13 | 1 | 1.5 | 12 | t
+ 14 | 0 | 2.6 | 11 | t
+ 15 | 0 | 2.7 | 10 | t
+ 16 | 0 | 2.8 | 9 | t
+ 17 | 1 | 2.32 | 8 | t
+ 18 | 0 | 4.43 | 7 | t
+ 19 | 0 | 2.31 | 6 | t
+ 20 | 1 | 3.49 | 5 | t
+ 21 | 1 | 2.42 | 4 | t
+ 22 | 1 | 4.01 | 3 | t
+ 23 | 1 | 4.91 | 2 | t
+ 24 | 1 | 5 | 1 | t
+\.
+
When doing regression analysis, we are sometimes interested in the variance of the computed coefficients \( \boldsymbol c \). While the built-in regression functions provide variance estimates, we may prefer a robust variance estimate.
+
The robust variance calculation can be expressed in a sandwich formation, which is the form
+\[ S( \boldsymbol c) = B( \boldsymbol c) M( \boldsymbol c) B( \boldsymbol c) \]
+
+
where \( B( \boldsymbol c)\) and \( M( \boldsymbol c)\) are matrices. The \( B( \boldsymbol c) \) matrix, also known as the bread, is relatively straight forward, and can be computed as
+\[ B( \boldsymbol c) = n\left(\sum_i^n -H(y_i, x_i, \boldsymbol c) \right)^{-1} \]
+
+
where \( H \) is the hessian matrix.
+
The \( M( \boldsymbol c)\) matrix has several variations, each with different robustness properties. The form implemented here is the Huber-White sandwich operator, which takes the form
The above method for calculating robust variance (Huber-White estimates) is implemented for linear regression, logistic regression, and multinomial logistic regression. It is useful in calculating variances in a dataset with potentially noisy outliers. The Huber-White implemented here is identical to the "HC0" sandwich operator in the R module "sandwich".
+
When multinomial logistic regression is computed before the multinomial robust regression, it uses a default reference category of zero and the regression coefficients are included in the output table. The regression coefficients in the output are in the same order as the multinomial logistic regression function, which is described below. For a problem with \( K \) dependent variables \( (1, ..., K) \) and \( J \) categories \( (0, ..., J-1) \), let \( {m_{k,j}} \) denote the coefficient for dependent variable \( k \) and category \( j \) . The output is \( {m_{k_1, j_0}, m_{k_1, j_1} \ldots m_{k_1, j_{J-1}}, m_{k_2, j_0}, m_{k_2, j_1} \ldots m_{k_K, j_{J-1}}} \). The order is NOT CONSISTENT with the multinomial regression marginal effect calculation with function marginal_mlogregr. This is deliberate because the interfaces of all multinomial regressions (robust, clustered, ...) will be moved to match that used in marginal.
+
The robust variance of Cox proportional hazards is more complex because coeeficients are trained by maximizing a partial log-likelihood. Therefore, one cannot directly use the formula for \( M( \boldsymbol c) \) as in Huber-White robust estimator. Extra terms are needed. See [4] for details.
[3] Achim Zeileis: Object-oriented Computation of Sandwich Estimators. Research Report Series / Department of Statistics and Mathematics, 37. Department of Statistics and Mathematics, WU Vienna University of Economics and Business, Vienna. http://cran.r-project.org/web/packages/sandwich/vignettes/sandwich-OOP.pdf
+
[4] D. Y. Lin and L . J. Wei, The Robust Inference for the Cox Proportional Hazards Model, Journal of the American Statistical Association, Vol. 84, No. 408, p.1074 (1989).
This MADlib method is still in early stage development. Interface and implementation are subject to change.
+
The random sampling module consists of useful utility functions for sampling operations. These functions can be used while implementing new algorithms.
+
Functions
+
Sample a single row according to weights.
+weighted_sample( value,
+ weight
+ )
+
Arguments
+
value
+
BIGINT or FLOAT8[]. Value of row. Uniqueness is not enforced. If a value occurs multiple times, the probability of sampling this value is proportional to the sum of its weights.
+
weight
+
FLOAT8. Weight for row. A negative value here is treated has zero weight.
+
+
Refer to the file for documentation on each of the utility functions.
A method to independently sample classes to produce a balanced data set. This is commonly used when classes are imbalanced, to ensure that subclasses are adequately represented in the sample.
The MADlib sessionize function performs time-oriented session reconstruction on a data set comprising a sequence of events. A defined period of inactivity indicates the end of one session and beginning of the next session. Sessions can be useful in many domains including web analytics [1], network security, manufacturing, finance, and operational analytics.
VARCHAR. Name of the source table that contains the data to be sessionized.
+
+
+
output_table
+
VARCHAR. Name of the output view or table. (The parameter create_view described below defines whether the output is actually a view or a table.) In addition to the columns in the source_table, the output also contains a new column called session_id:
+
+session_id=1,2,...n where n is the number of the session in the partition.
+
+
+
+
partition_expr
+
VARCHAR. The 'partition_expr' is a single column or a list of comma-separated columns/expressions to divide all rows into groups, or partitions. Sessionization is applied across the rows that fall into the same partition. This parameter can be set to NULL or '' to indicate the sessionization operation is to be applied to the whole input table.
+
+
+
time_stamp
+
VARCHAR. The time stamp column name that is used for sessionization calculation. Note that the time_stamp column will be sorted in ascending order before the session reconstruction is done within a partition.
+
+
+
max_time
+
INTERVAL. Maximum delta time (i.e., time out) between subsequent events that define a session. If the elapsed time between subsequent events is longer than max_time, a new session is created.
+
Note
Note that max_time is of time type INTERVAL which is a PostgreSQL way of describing elapsed time. For more information on INTERVAL please refer to reference [2].
+
+
output_cols (optional)
+
VARCHAR. An optional comma separated list of columns to be written to the output_table. Must be a valid SELECT expression. This is set to '*' by default, which means all columns in the input table will be written to the output_table plus the session_id column. Note that this parameter could include a list containing the partition_expr or any other expressions of interest. E.g., '*, expr1, expr2, etc.' where this means output all columns from the input table plus the expressions listed plus the session_id column.
+
+
+
create_view (optional)
+
BOOLEAN default: TRUE. Determines whether to create a view or materialize the output as a table. If you only need session info once, creating a view could be significantly faster than materializing as a table. Please note that if you set create_view to NULL (allowed by PostgreSQL) it will get set to the default value of TRUE.
+
+
Examples
+
The data set describes shopper behavior on a notional web site that sells beer and wine. A beacon fires an event to a log file when the shopper visits different pages on the site: landing page, beer selection page, wine selection page, and checkout. Each user is identified by a a user id, and every time a page is visited, the page and time stamp are logged.
+ DROP VIEW IF EXISTS sessionize_output_view;
+ SELECT madlib.sessionize(
+ 'eventlog', -- Name of input table
+ 'sessionize_output_view', -- View to store sessionize results
+ 'user_id', -- Partition input table by user id
+ 'event_timestamp', -- Time column used to compute sessions
+ '0:30:0' -- Use 30 minute time out to define sessions
+ );
+SELECT * FROM sessionize_output_view ORDER BY user_id, event_timestamp;
+
Now let's say we want to see 3 minute sessions by a group of users with a certain range of user IDs. To do this, we need to sessionize the table based on a partition expression. Also, we want to persist a table output with a reduced set of columns in the table.
+ DROP TABLE IF EXISTS sessionize_output_table;
+ SELECT madlib.sessionize(
+ 'eventlog', -- Name of input table
+ 'sessionize_output_table', -- Table to store sessionize results
+ 'user_id < 200000', -- Partition input table by subset of users
+ 'event_timestamp', -- Order partitions in input table by time
+ '180', -- Use 180 second time out to define sessions (same as '0:03:0')
+ 'event_timestamp, user_id, user_id < 200000 AS "Department-A1"', -- Select only user_id and event_timestamp columns, along with the session id as output
+ 'f' -- create a table
+ );
+ SELECT * FROM sessionize_output_table WHERE "Department-A1"='TRUE' ORDER BY event_timestamp;
+
Result showing 2 users and 3 total sessions across the group:
NOTE: The following PostgreSQL link refers to documentation resources for the current PostgreSQL database version. Depending upon your database platform version, you may need to change "current" reference in the link to your database version.
+
If your database platform uses the Greenplum Database (or related variants), please check with the project community and/or your database vendor to identify the PostgreSQL version it is based on.
A collection of methods to estimate the number of unique values contained in the data.
+
Sketches (sometimes called "synopsis data structures") are small randomized in-memory data structures that capture statistical properties of a large set of values (e.g., a column of a table). Sketches can be formed in a single pass of the data, and used to approximate a variety of descriptive statistics.
+
We implement sketches as SQL User-Defined Aggregates (UDAs). Because they are single-pass, small-space and parallelized, a single query can use many sketches to gather summary statistics on many columns of a table efficiently.
+
This module currently implements user-defined aggregates based on three main sketch methods:
+
Count-Min (CM) sketches, which can be used to approximate a number of descriptive statistics including
+
COUNT of rows whose column value matches a given value in a set
+
COUNT of rows whose column value falls in a range (*)
+
order statistics including median and centiles (*)
+
histograms: both equi-width and equi-depth (*)
+
+
+
Flajolet-Martin (FM) sketches for approximating COUNT(DISTINCT).
+
Most Frequent Value (MFV) sketches, which output the most frequently-occuring values in a column, along with their associated counts.
+
+
Note: Features marked with a star (*) only work for discrete types that can be cast to int8.
+
The sketch methods consist of a number of SQL UDAs (user-defined aggregates) and UDFs (user-defined functions), to be used directly in SQL queries.
The sparse linear systems module implements solution methods for systems of consistent linear equations. Systems of linear equations take the form:
+\[ Ax = b \]
+
+
where \(x \in \mathbb{R}^{n}\), \(A \in \mathbb{R}^{m \times n} \) and \(b \in \mathbb{R}^{m}\). This module accepts sparse matrix input formats for \(A\) and \(b\). We assume that there are no rows of \(A\) where all elements are zero.
+
Note
Algorithms with fail if there is an row of the input matrix containing all zeros.
+
The algorithms implemented in this module can handle large sparse square linear systems. Currently, the algorithms implemented in this module solve the linear system using direct or iterative methods.
Each row represents a single equation. The rhs columns refer to the right hand side of the equations and the lhs columns refer to the multipliers on the variables on the left hand side of the same equations.
+
+
+
tbl_source_rhs
+
TEXT. The name of the table containing the right hand side vector. For the RHS matrix, the input data is expected to be of the following form:
Each row represents a single equation. The rhs columns refer to the right hand side of the equations while the lhs columns refers to the multipliers on the variables on the left hand side of the same equations.
+
+
+
tbl_result
+
TEXT. The name of the table where the output is saved. Output is stored in the tabled named by the tbl_result argument. The table contains the following columns. The output contains the following columns:
+
+
solution
FLOAT8[]. The solution is an array with the variables in the same order as that provided as input in the 'left_hand_side' column name of the 'source_table'
+
+
residual_norm
FLOAT8. Scaled residual norm, defined as \( \frac{|Ax - b|}{|b|} \). This value is an indication of the accuracy of the solution.
+
+
iters
INTEGER. Number of iterations required by the algorithm (only applicable for iterative algorithms) . The output is NULL for 'direct' methods.
+
+
+
+
lhs_row_id
+
TEXT. The name of the column storing the 'row id' of the equations.
Note
For a system with N equations, the row_id's must be a continuous range of integers from \( 0 \ldots n-1 \).
+
+
lhs_col_id
+
TEXT. The name of the column (in tbl_source_lhs) storing the 'col id' of the equations.
+
+
+
lhs_value
+
TEXT. The name of the column (in tbl_source_lhs) storing the 'value' of the equations.
+
+
+
rhs_row_id
+
TEXT. The name of the column (in tbl_source_rhs) storing the 'col id' of the equations.
+
+
+
rhs_value
+
TEXT. The name of the column (in tbl_source_rhs) storing the 'value' of the equations.
+
+
+
num_vars
+
INTEGER. The number of variables in the linear system equations.
+
+
+
grouping_col (optional)
+
TEXT, default: NULL. Group by column names.
Note
The grouping feature is currently not implemented and this parameter is only a placeholder.
+
+
optimizer (optional)
+
TEXT, default: 'direct'. Type of optimizer.
+
+
+
optimizer_params (optional)
+
TEXT, default: NULL. Optimizer specific parameters.
+
+
Optimizer Parameters
+
For each optimizer, there are specific parameters that can be tuned for better performance.
+
+
algorithm (default: ldlt)
+
+
There are several algorithms that can be classified as 'direct' methods of solving linear systems. Madlib functions provide various algorithmic options available for users.
+
The following table provides a guideline on the choice of algorithm based on conditions on the A matrix, speed of the algorithms and numerical stability.
+
Algorithm | Conditions on A | Speed | Memory
+ ----------------------------------------------------------
+ llt | Sym. Pos Def | ++ | ---
+ ldlt | Sym. Pos Def | ++ | ---
+
+ For speed '++' is faster than '+', which is faster than '-'.
+ For accuracy '+++' is better than '++'.
+ For memory, '-' uses less memory than '--'.
+
+ Note: ldlt is often preferred over llt
+
There are several algorithms that can be classified as 'iterative' methods of solving linear systems. Madlib functions provide various algorithmic options available for users.
+
The following table provides a guideline on the choice of algorithm based on conditions on the A matrix, speed of the algorithms and numerical stability.
+
Algorithm | Conditions on A | Speed | Memory | Convergence
+ ----------------------------------------------------------------------
+ cg-mem | Sym. Pos Def | +++ | - | ++
+ bicgstab-mem | Square | ++ | - | +
+ precond-cg-mem | Sym. Pos Def | ++ | - | +++
+ precond-bicgstab-mem | Square | + | - | ++
+
+ For memory, '-' uses less memory than '--'.
+ For speed, '++' is faster than '+'.
+
Algorithm Details:
+
+
cg-mem
In memory conjugate gradient with diagonal preconditioners.
+
+
bicgstab-mem
Bi-conjugate gradient (equivalent to performing CG on the least squares formulation of Ax=b) with incomplete LU preconditioners.
+
+
precond-cg-mem
In memory conjugate gradient with diagonal preconditioners.
+
+
bicgstab-mem
Bi-conjugate gradient (equivalent to performing CG on the least squares formulation of Ax=b) with incomplete LU preconditioners.
+
+
+
+
toler (default: 1e-5)
+
Termination tolerance (applicable only for iterative methods) which determines the stopping criterion (with respect to residual norm) for iterative methods.
+
+
+
+
Examples
+
+
View online help for the sparse linear systems solver function.
+SELECT madlib.linear_solver_sparse();
+
+
Create the sample data set.
+DROP TABLE IF EXISTS sparse_linear_systems_lhs;
+CREATE TABLE sparse_linear_systems_lhs (
+ rid INTEGER NOT NULL,
+ cid INTEGER,
+ val DOUBLE PRECISION
+);
+DROP TABLE IF EXISTS sparse_linear_systems_rhs;
+CREATE TABLE sparse_linear_systems_rhs (
+ rid INTEGER NOT NULL,
+ val DOUBLE PRECISION
+);
+INSERT INTO sparse_linear_systems_lhs(rid, cid, val) VALUES
+(0, 0, 1),
+(1, 1, 1),
+(2, 2, 1),
+(3, 3, 1);
+INSERT INTO sparse_linear_systems_rhs(rid, val) VALUES
+(0, 10),
+(1, 20),
+(2, 30);
+
Given a graph and a source vertex, the single source shortest path (SSSP) algorithm finds a path from the source vertex to every other vertex in the graph, such that the sum of the weights of the path edges is minimized.
TEXT. Name of the table containing the vertex data for the graph. Must contain the column specified in the 'vertex_id' parameter below.
+
+
+
vertex_id
+
TEXT, default = 'id'. Name of the column in 'vertex_table' containing vertex ids. The vertex ids can be of type INTEGER or BIGINT with no duplicates. They do not need to be contiguous.
+
+
+
edge_table
+
TEXT. Name of the table containing the edge data. The edge table must contain columns for source vertex, destination vertex and edge weight. Column naming convention is described below in the 'edge_args' parameter.
+
+
+
edge_args
+
TEXT. A comma-delimited string containing multiple named arguments of the form "name=value". The following parameters are supported for this string argument:
+
src (INTEGER or BIGINT): Name of the column containing the source vertex ids in the edge table. Default column name is 'src'.
+
dest (INTEGER or BIGINT): Name of the column containing the destination vertex ids in the edge table. Default column name is 'dest'.
+
weight (FLOAT8): Name of the column containing the edge weights in the edge table. Default column name is 'weight'.
+
+
+
+
source_vertex
+
INTEGER or BIGINT. The source vertex id for the algorithm to start. This vertex id must exist in the 'vertex_id' column of 'vertex_table'.
+
+
+
out_table
+
TEXT. Name of the table to store the result of SSSP. It contains a row for every vertex of every group and have the following columns (in addition to the grouping columns):
+
vertex_id : The id for the destination. Will use the input parameter 'vertex_id' for column naming.
+
weight : The total weight of the shortest path from the source vertex to this particular vertex. Will use the input parameter 'weight' for column naming.
+
parent : The parent of this vertex in the shortest path from source. Will use 'parent' for column naming.
+
+
A summary table named <out_table>_summary is also created. This is an internal table that keeps a record of the input parameters and is used by the path function described below.
+
+
+
grouping_cols (optional)
+
TEXT, default = NULL. List of columns used to group the input into discrete subgraphs. These columns must exist in the edge table. When this value is null, no grouping is used and a single SSSP result is generated.
+
+
Path Retrieval
+
The path retrieval function returns the shortest path from the source vertex to a specified desination vertex.
TEXT. Name of the table that contains the SSSP output.
+
+
+
dest_vertex
+
INTEGER or BIGINT. The vertex that will be the destination of the desired path.
+
+
+
path_table
+
TEXT. Name of the output table that contains the path. It contains a row for every group and has the following columns:
+
grouping_cols : The grouping columns given in the creation of the SSSP table. If there are no grouping columns, these columns will not exist and the table will have a single row.
+
path (ARRAY) : The shortest path from the source vertex (as specified in the SSSP execution) to the destination vertex.
+
+
+
+
+
Examples
+
+
Create vertex and edge tables to represent the graph:
+DROP TABLE IF EXISTS out_path;
+SELECT madlib.graph_sssp_get_path('out',5,'out_path');
+SELECT * FROM out_path;
+
+ path
+---------
+ {0,2,5}
+
+
Now let's do a similar example except using different column names in the tables (i.e., not the defaults). Create the vertex and edge tables:
+DROP TABLE IF EXISTS vertex_alt, edge_alt;
+CREATE TABLE vertex_alt AS SELECT id AS v_id FROM vertex;
+CREATE TABLE edge_alt AS SELECT src AS e_src, dest, weight AS e_weight FROM edge;
+
+
Get the shortest path from vertex 1:
+DROP TABLE IF EXISTS out_alt, out_alt_summary;
+SELECT madlib.graph_sssp(
+ 'vertex_alt', -- Vertex table
+ 'v_id', -- Vertex id column (NULL means use default naming)
+ 'edge_alt', -- Edge table
+ 'src=e_src, weight=e_weight', -- Edge arguments (NULL means use default naming)
+ 1, -- Source vertex for path calculation
+ 'out_alt'); -- Output table of shortest paths
+SELECT * FROM out_alt ORDER BY v_id;
+
+DROP TABLE IF EXISTS edge_gr;
+CREATE TABLE edge_gr AS
+(
+ SELECT *, 0 AS grp FROM edge
+ UNION
+ SELECT *, 1 AS grp FROM edge WHERE src < 6 AND dest < 6
+);
+INSERT INTO edge_gr VALUES
+(4,5,-20,1);
+
+
Find SSSP for all groups
+DROP TABLE IF EXISTS out_gr, out_gr_summary;
+SELECT madlib.graph_sssp(
+ 'vertex', -- Vertex table
+ NULL, -- Vertex id column (NULL means use default naming)
+ 'edge_gr', -- Edge table
+ NULL, -- Edge arguments (NULL means use default naming)
+ 0, -- Source vertex for path calculation
+ 'out_gr', -- Output table of shortest paths
+ 'grp' -- Grouping columns
+);
+SELECT * FROM out_gr ORDER BY grp,id;
+
On a Greenplum cluster, the edge table should be distributed by the source vertex id column for better performance.
+
The Bellman-Ford algorithm [1] is used to implement SSSP. This algorithm allows negative edges but not negative cycles. In the case of graphs with negative cycles, an error will be given and no output table will be generated.
+
Also see the Grail project [2] for more background on graph analytics processing in relational databases.
Hypothesis tests are used to confirm or reject a null hypothesis \( H_0 \) about the distribution of random variables, given realizations of these random variables. Since in general it is not possible to make statements with certainty, one is interested in the probability \( p \) of seeing random variates at least as extreme as the ones observed, assuming that \( H_0 \) is true. If this probability \( p \) is small, \( H_0 \) will be rejected by the test with significance level \( p \). Falsifying \( H_0 \) is the canonic goal when employing a hypothesis test. That is, hypothesis tests are typically used in order to substantiate that instead the alternative hypothesis \( H_1 \) is true.
+
Hypothesis tests may be divided into parametric and non-parametric tests. A parametric test assumes certain distributions and makes inferences about parameters of the distributions (e.g., the mean of a normal distribution). Formally, there is a given domain of possible parameters \( \Gamma \) and the null hypothesis \( H_0 \) is the event that the true parameter \( \gamma_0 \in \Gamma_0 \), where \( \Gamma_0 \subsetneq \Gamma \). Non-parametric tests, on the other hand, do not assume any particular distribution of the sample (e.g., a non-parametric test may simply test if two distributions are similar).
+
The first step of a hypothesis test is to compute a test statistic, which is a function of the random variates, i.e., a random variate itself. A hypothesis test relies on the distribution of the test statistic being (approximately) known. Now, the \( p \)-value is the probability of seeing a test statistic at least as extreme as the one observed, assuming that \( H_0 \) is true. In a case where the null hypothesis corresponds to a family of distributions (e.g., in a parametric test where \( \Gamma_0 \) is not a singleton set), the \( p \)-value is the supremum, over all possible distributions according to the null hypothesis, of these probabilities.
+
Note
Please refer to hypothesis_tests.sql_in for additional technical information on the MADlib implementation of hypothesis tests, and for detailed function signatures for all tests.
+
Input
+
Input data is assumed to be normalized with all values stored row-wise. In general, the following inputs are expected.
{TABLE|VIEW} source (
+ ...
+ first BOOLEAN,
+ value DOUBLE PRECISION
+ ...
+)
The first column indicates whether a value is from the first sample (if TRUE) or the second sample (if FALSE).
+
Many-sample tests expect the following form:
{TABLE|VIEW} source (
+ ...
+ group INTEGER,
+ value DOUBLE PRECISION
+ ...
+)
Usage
+
All tests are implemented as aggregate functions. The non-parametric (rank-based) tests are implemented as ordered aggregate functions and thus necessitate an ORDER BY clause. In the following, the most simple forms of usage are given. Specific function signatures, as described in hypothesis_tests.sql_in, may require more arguments or a different ORDER BY clause.
+
+
Run a parametric one-sample test:
SELECT test(value) FROM source
where 'test' can be one of
+
t_test_one (one-sample or dependent paired Student's t-test)
+
chi2_gof_test (Pearson's chi-squared goodness of fit test, also used for chi-squared independence test as shown in example section below)
+
+
+
Run a parametric two-sample/multi-sample test:
SELECT test(first/group, value) FROM source
where 'test' can be one of
+
f_test (Fisher F-test)
+
t_test_two_pooled (two-sample pooled Student’s t-test, i.e. equal variances)
+
t_test_two_unpooled (two-sample unpooled t-test, i.e., unequal variances, also known as Welch's t-test)
+
one_way_anova (one-way analysis of variance, multi-sample)
+
+
+
Run a non-parametric two-sample/multi-sample test:
SELECT test(first/group, value ORDER BY value) FROM source
Note on non-parametric tests: Kolomogov-Smirnov two-sample test is based on the asymptotic theory. The p-value is given by comparing the test statistics with the Kolomogov distribution. The p-value is also adjusted for data with heavy tail distribution, which may give different results than those given by R function's ks.test. See [3] for a detailed explanation. The literature is not unanimous about the definitions of the Wilcoxon rank sum and Mann-Whitney tests. There are two possible definitions for the statistic; MADlib outputs the minimum of the two and uses it for significance testing. This might give different results for both mw_test and wsr_test compared to statistical functions in other popular packages (like R's wilcox.test function). See [4] for a detailed explanation.
+
+
+
Examples
+
+
One-sample and two-sample t-test (data is subset of mpg data from NIST/SEMATECH)
+-- Create table for one sample tests
+DROP TABLE IF EXISTS auto83b_one_sample;
+CREATE TABLE auto83b_one_sample AS
+ SELECT mpg_us AS mpg
+ FROM auto83b
+ WHERE mpg_us is not NULL;
+-- Print table
+SELECT * FROM auto83b_one_sample;
+
+-- Create table for two sample tests
+DROP TABLE IF EXISTS auto83b_two_sample;
+CREATE TABLE auto83b_two_sample AS
+SELECT TRUE AS is_us, mpg_us AS mpg
+ FROM auto83b
+ WHERE mpg_us is not NULL
+ UNION ALL
+ SELECT FALSE, mpg_j
+ FROM auto83b
+ WHERE mpg_j is not NULL;
+-- Print table
+SELECT * FROM auto83b_two_sample;
+
+ is_us | mpg
+-------+-----
+ t | 18
+ t | 15
+ t | 18
+ t | 16
+ t | 17
+ t | 15
+ t | 14
+ t | 14
+ t | 21
+ t | 10
+ t | 10
+ t | 11
+ t | 9
+ t | 13
+ t | 12
+ t | 18
+ t | 21
+ t | 19
+ t | 21
+ t | 15
+ t | 16
+ t | 15
+ t | 11
+ t | 20
+ t | 21
+ t | 19
+ t | 15
+ f | 24
+ f | 27
+ f | 27
+ f | 25
+ f | 31
+ f | 35
+ f | 24
+ f | 19
+ f | 31
+ f | 32
+ f | 24
+ f | 26
+ f | 9
+ f | 32
+ f | 37
+ f | 38
+ f | 34
+ f | 34
+ f | 32
+ f | 33
+ f | 32
+ f | 25
+ f | 24
+ f | 37
+(51 rows)
+
+-- One sample tests
+SELECT (madlib.t_test_one(mpg - 20)).* FROM auto83b_one_sample; -- test rejected for mean = 20
+
The Chi-squared independence test uses the Chi-squared goodness-of-fit function, as shown in the example below. The expected value needs to be computed and passed to the goodness-of-fit function. The expected value for MADlib is computed as sum of rows * sum of columns, for each element of the input matrix. For e.g., expected value for element (2,1) would be sum of row 2 * sum of column 1.
-- Input table is expected to be unpivoted, so need to pivot it
+CREATE TABLE chi2_test_friendly_unpivoted AS
+SELECT id_x, id_y, values[id_y] AS observed
+FROM
+ chi2_test_friendly,
+ generate_series(1,4) AS id_y;
-- Compute Chi-squared independence statistic, by calculating expected value in the SQL and calling the goodness-of-fit function
+SELECT (madlib.chi2_gof_test(observed, expected, deg_freedom)).*
+FROM (
+ -- Compute expected values and degrees of freedom
+ SELECT
+ observed,
+ sum(observed) OVER (PARTITION BY id_x)::DOUBLE PRECISION *
+ sum(observed) OVER (PARTITION BY id_y) AS expected
+ FROM chi2_test_friendly_unpivoted
+) p, (
+ SELECT
+ (count(DISTINCT id_x) - 1) * (count(DISTINCT id_y) - 1) AS deg_freedom
+ FROM chi2_test_friendly_unpivoted
+) q;
+
SELECT (madlib.one_way_anova(level, value)).* FROM (
+ SELECT level, resistance[level] AS value
+ FROM
+ nist_anova_test, (SELECT * FROM generate_series(1,3) level) q1
+) q2;
+
SELECT (madlib.ks_test(first, value,
+ (SELECT count(value) FROM ks_sample_1 WHERE first),
+ (SELECT count(value) FROM ks_sample_1 WHERE NOT first)
+ ORDER BY value)).*
+FROM ks_sample_1;
+
[1] M. Hollander, D. Wolfe: Nonparametric Statistical Methods, 2nd edition, Wiley, 1999
+
[2] E. Lehmann, J. Romano: Testing Statistical Hypotheses, 3rd edition, Springer, 2005
+
[3] M. Stephens: Use of the Kolmogorov-Smirnov, Cramer-Von Mises and related statistics without extensive tables, Journal of the Royal Statistical Society. Series B (Methodological) (1970): 115-122.
This module provides a basic stemming operation for text input. It is a support module for several machine learning algorithms that require a stemmer. Currently, it only supports English words.
+
This function is a SQL interface to the implementation of the Porter Stemming Algorithm. The original stemming algorithm is written and maintained by Martin Porter
+
Implementation Notes
+
All functions described in this module work with text OR text array.
+
Several of the function require TEXT VALUES, and returns NULL for a NULL input. See details in description of individual functions.
Stratified sampling is a method for independently sampling subpopulations (strata). It is commonly used to reduce sampling error by ensuring that subgroups are adequately represented in the sample.
TEXT. Name of the table containing the input data.
+
+
+
output_table
+
TEXT. Name of output table that contains the sampled data. The output table contains all columns present in the source table unless otherwise specified in the 'target_cols' parameter below.
+
+
+
proportion
+
FLOAT8 in the range (0,1). Each stratum is sampled independently.
+
+
+
grouping_cols (optional)
+
TEXT, default: NULL. A single column or a list of comma-separated columns that defines the strata. When this parameter is NULL, no grouping is used so the sampling is non-stratified, that is, the whole table is treated as a single group.
+
+
+
target_cols (optional)
+
TEXT, default NULL. A comma-separated list of columns to appear in the 'output_table'. If NULL or '*', all columns from the 'source_table' will appear in the 'output_table'.
+
Note
Do not include 'grouping_cols' in the parameter 'target_cols', because they are always included in the 'output_table'.
+
+
with_replacement (optional)
+
BOOLEAN, default FALSE. Determines whether to sample with replacement or without replacement (default). With replacement means that it is possible that the same row may appear in the sample set more than once. Without replacement means a given row can be selected only once.
+
+
Examples
+
Please note that due to the random nature of sampling, your results may look different from those below.
The MADlib summary() function produces summary statistics for any data table. The function invokes various methods from the MADlib library to provide the data overview.
The summary() function returns a composite type containing three fields:
+
+
output_table
TEXT. The name of the output table.
+
+
num_col_summarized
INTEGER. The number of columns from the source table that have been summarized.
+
+
duration
FLOAT8. The time taken (in seconds) to compute the summary.
+
+
Arguments
+
source_table
+
TEXT. Name of the table containing the input data.
+
+
+
output_table
+
TEXT. Name of the table for the output summary statistics. This table contains the following columns:
+
+
group_by
Group-by column name. NULL if none provided.
+
+
group_by_value
Value of the group-by column. NULL if there is no grouping.
+
+
target_column
Targeted column values for which summary is requested.
+
+
column_number
Physical column number for the target column, as described in pg_attribute catalog.
+
+
data_type
Data type of the target column. Standard GPDB type descriptors are displayed.
+
+
row_count
Number of rows for the target column.
+
+
distinct_values
Number of distinct values in the target column. If the summary() function is called with the get_estimates argument set to TRUE (default), then this is an estimated statistic based on the Flajolet-Martin distinct count estimator. If the get_estimates argument set to FALSE, will use PostgreSQL COUNT DISTINCT.
+
+
missing_values
Number of missing values in the target column.
+
+
blank_values
Number of blank values. Blanks are defined by this regular expression:
'^\w*$'
+
+
fraction_missing
Percentage of total rows that are missing, as a decimal value, e.g. 0.3.
+
+
fraction_blank
Percentage of total rows that are blank, as a decimal value, e.g. 0.3.
+
+
positive_values
Number of positive values in the target column if target is numeric, otherwise NULL.
+
+
negative_values
Number of negative values in the target column if target is numeric, otherwise NULL.
+
+
zero_values
Number of zero values in the target column if target is numeric, otherwise NULL. Note that we are reporting exact equality to 0.0 here, so even if you have a float value that is extremely small (say due to rounding), it will not be reported as a zero value.
+
+
mean
Mean value of target column if target is numeric, otherwise NULL.
+
+
variance
Variance of target column if target is numeric, otherwise NULL.
+
+
confidence_interval
Confidence interval (95% using z-score) of the mean value for the target column if target is numeric, otherwise NULL. Presented as an array of two elements in the form {lower bound, upper bound}.
+
+
min
Minimum value of target column. For strings this is the length of the shortest string.
+
+
max
Maximum value of target column. For strings this is the length of the longest string.
+
+
first_quartile
First quartile (25th percentile), only for numeric columns. (Unavailable for PostgreSQL 9.3 or lower.)
+
+
median
Median value of target column, if target is numeric, otherwise NULL. (Unavailable for PostgreSQL 9.3 or lower.)
+
+
third_quartile
Third quartile (25th percentile), only for numeric columns. (Unavailable for PostgreSQL 9.3 or lower.)
+
+
quantile_array
Percentile values corresponding to ntile_array. (Unavailable for PostgreSQL 9.3 or lower.)
+
+
most_frequent_values
An array containing the most frequently occurring values. The how_many_mfv argument determines the length of the array, which is 10 by default. If the summary() function is called with the get_estimates argument set to TRUE (default), the frequent values computation is performed using a parallel aggregation method that is faster, but in some cases may fail to detect the exact most frequent values.
+
+
mfv_frequencies
Array containing the frequency count for each of the most frequent values.
+
+
+
+
target_columns (optional)
+
TEXT, default NULL. A comma-separated list of columns to summarize. If NULL, summaries are produced for all columns.
+
+
+
grouping_cols (optional)
+
TEXT, default: null. A comma-separated list of columns on which to group results. If NULL, summaries are produced for the complete table.
Note
Please note that summary statistics are calculated for each grouping column independently. That is, grouping columns are not combined together as in the regular PostgreSQL style GROUP BY directive. (This was done to reduce long run time and huge output table size which would otherwise result in the case of large input tables with a lot of grouping_cols and target_cols specified.)
+
+
get_distinct (optional)
+
BOOLEAN, default TRUE. If true, distinct values are counted. The method for computing distinct values depends on the setting of the 'get_estimates' parameter below.
+
+
+
get_quartiles (optional)
+
BOOLEAN, default TRUE. If TRUE, quartiles are computed.
+
+
+
ntile_array (optional)
+
FLOAT8[], default NULL. An array of quantile values to compute. If NULL, quantile values are not computed.
Note
Quartile and quantile functions are not available in PostgreSQL 9.3 or lower. If you are using PostgreSQL 9.3 or lower, the output table will not contain these values, even if you set 'get_quartiles' = TRUE or provide an array of quantile values for the parameter 'ntile_array'.
+
+
how_many_mfv (optional)
+
INTEGER, default: 10. The number of most-frequent-values to compute. The method for computing MFV depends on the setting of the 'get_estimates' parameter below.
+
+
+
get_estimates (optional)
+
BOOLEAN, default TRUE. If TRUE, estimated values are produced for distinct values and most frequent values. If FALSE, exact values are calculated which will take longer to run, with the impact depending on data size.
+
+
+
n_cols_per_run (optional)
+
INTEGER, default: 15. The number of columns to collect summary statistics in one pass of the data. This parameter determines the number of passes through the data. For e.g., with a total of 40 columns to summarize and 'n_cols_per_run = 15', there will be 3 passes through the data, with each pass summarizing a maximum of 15 columns.
Note
This parameter should be used with caution. Increasing this parameter could decrease the total run time (if number of passes decreases), but will increase the memory consumption during each run. Since PostgreSQL limits the memory available for a single aggregate run, this increased memory consumption could result in an out-of-memory termination error.
Note that for the text column in record 6, some statistics are n/a, and the min and max values represent the length of the shortest and longest strings respectively.
+
Now group by the class of iris:
+DROP TABLE IF EXISTS iris_summary;
+SELECT * FROM madlib.summary( 'iris', -- Source table
+ 'iris_summary', -- Output table
+ 'sepal_length, sepal_width', -- Columns to summarize
+ 'class_name' -- Grouping column
+ );
+SELECT * FROM iris_summary;
+
Table names can be optionally schema qualified (current_schemas() would be searched if a schema name is not provided) and table and column names should follow case-sensitivity and quoting rules per the database. (For instance, 'mytable' and 'MyTable' both resolve to the same entity, i.e. 'mytable'. If mixed-case or multi-byte characters are desired for entity names then the string should be double-quoted; in this case the input would be '"MyTable"').
+
The get_estimates parameter controls computation for both distinct count and most frequent values:
+
If get_estimates is TRUE then the distinct value computation is estimated using Flajolet-Martin. MFV is computed using a fast method that does parallel aggregation in Greenplum Database at the expense of missing or duplicating some of the most frequent values.
+
If get_estimates is FALSE then the distinct values are computed in a slower but exact method using PostgreSQL COUNT DISTINCT. MFV is computed using a faithful implementation that preserves the approximation guarantees of the Cormode/Muthukrishnan method (more information at MFV (Most Frequent Values)).
Finds \(k\) nearest data points to the given data point and outputs majority vote value of output classes for classification, or average value of target values for regression.
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics.
+
Let \(A\) be a \(mxn\) matrix, where \(m \ge n\). Then \(A\) can be decomposed as follows:
+\[ A = U \Sigma V^T, \]
+
+
where \(U\) is a \(m \times n\) orthonormal matrix, \(\Sigma\) is a \(n \times n\) diagonal matrix, and \(V\) is an \(n \times n\) orthonormal matrix. The diagonal elements of \(\Sigma\) are called the singular values.
+
SVD Functions
+
SVD factorizations are provided for dense and sparse matrices. In addition, a native implementation is provided for very sparse matrices for improved performance.
The table contains a row_id column that identifies each row, with numbering starting from 1. The other columns contain the data for the matrix. There are two possible dense formats as illustrated by the 2x2 matrix example below. You can use either of these dense formats:
+
+ row_id col1 col2
+row1 1 1 0
+row2 2 0 1
+
+
+ row_id row_vec
+row1 1 {1, 0}
+row2 2 {0, 1}
+
+
+
+
output_table_prefix
+
TEXT. Prefix for output tables. See Output Tables below for a description of the convention used.
+
row_id
+
TEXT. ID for each row.
+
k
+
INTEGER. Number of singular values to compute.
+
n_iterations (optional).
+
INTEGER. Number of iterations to run.
Note
The number of iterations must be in the range [k, column dimension], where k is number of singular values.
+
+
result_summary_table (optional)
+
TEXT. The name of the table to store the result summary.
+
+
+
SVD Function for Sparse Matrices
+
Use this function for matrices that are represented in the sparse-matrix format (example below). Note that the input matrix is converted to a dense matrix before the SVD operation, for efficient computation reasons.
A sparse matrix is represented using the row and column indices for each non-zero entry of the matrix. This representation is useful for matrices containing multiple zero elements. Below is an example of a sparse 4x7 matrix with just 6 out of 28 entries being non-zero. The dimensionality of the matrix is inferred using the max value in row and col columns. Note the last entry is included (even though it is 0) to provide the dimensionality of the matrix, indicating that the 4th row and 7th column contain all zeros.
TEXT. Prefix for output tables. See Output Tables below for a description of the convention used.
+
row_id
+
TEXT. Name of the column containing the row index for each entry in sparse matrix.
+
col_id
+
TEXT. Name of the column containing the column index for each entry in sparse matrix.
+
value
+
TEXT. Name of column containing the non-zero values of the sparse matrix.
+
row_dim
+
INTEGER. Number of rows in matrix.
+
col_dim
+
INTEGER. Number of columns in matrix.
+
k
+
INTEGER. Number of singular values to compute.
+
n_iterations (optional)
+
INTEGER. Number of iterations to run.
Note
The number of iterations must be in the range [k, column dimension], where k is number of singular values.
+
+
result_summary_table (optional)
+
TEXT. The name of the table to store the result summary.
+
+
+
Native Implementation for Sparse Matrices
+
Use this function for matrices that are represented in the sparse-matrix format (see sparse matrix example above). This function uses the native sparse representation while computing the SVD.
Note
Note that this function should be favored if the matrix is highly sparse, since it computes very sparse matrices efficiently.
TEXT. Source table name (sparse matrix - see example above).
+
output_table_prefix
+
TEXT. Prefix for output tables. See Output Tables below for a description of the convention used.
+
row_id
+
TEXT. ID for each row.
+
col_id
+
TEXT. ID for each column.
+
value
+
TEXT. Non-zero values of the sparse matrix.
+
row_dim
+
INTEGER. Row dimension of sparse matrix.
+
col_dim
+
INTEGER. Col dimension of sparse matrix.
+
k
+
INTEGER. Number of singular values to compute.
+
n_iterations (optional)
+
INTEGER. Number of iterations to run.
Note
The number of iterations must be in the range [k, column dimension], where k is number of singular values.
+
+
result_summary_table (optional)
+
TEXT. Table name to store result summary.
+
+
+
Output Tables
+
Output for eigenvectors/values is in the following three tables:
+
Left singular matrix: Table is named <output_table_prefix>_u (e.g. ‘netflix_u’)
+
Right singular matrix: Table is named <output_table_prefix>_v (e.g. ‘netflix_v’)
+
Singular values: Table is named <output_table_prefix>_s (e.g. ‘netflix_s’)
+
+
The left and right singular vector tables are of the format:
+
+
row_id
INTEGER. The ID corresponding to each eigenvalue (in decreasing order).
+
+
row_vec
FLOAT8[]. Singular vector elements for this row_id. Each array is of size k.
+
+
The singular values table is in sparse table format, since only the diagonal elements of the matrix are non-zero:
+
+
row_id
INTEGER. i for ith eigenvalue.
+
+
col_id
INTEGER. i for ith eigenvalue (same as row_id).
+
+
value
FLOAT8. Eigenvalue.
+
+
All row_id and col_id in the above tables start from 1.
+
The result summary table has the following columns:
+
+
rows_used
INTEGER. Number of rows used for SVD calculation.
+
+
exec_time
FLOAT8. Total time for executing SVD.
+
+
iter
INTEGER. Total number of iterations run.
+
+
recon_error
FLOAT8. Total quality score (i.e. approximation quality) for this set of orthonormal basis.
+
+
relative_recon_error
FLOAT8. Relative quality score.
+
+
In the result summary table, the reconstruction error is computed as \( \sqrt{mean((X - USV^T)_{ij}^2)} \), where the average is over all elements of the matrices. The relative reconstruction error is then computed as ratio of the reconstruction error and \( \sqrt{mean(X_{ij}^2)} \).
+SELECT madlib.svd( 'mat', -- Input table
+ 'svd', -- Output table prefix
+ 'row_id', -- Column name with row index
+ 10, -- Number of singular values to compute
+ NULL, -- Use default number of iterations
+ 'svd_summary_table' -- Result summary table
+ );
+
+
Print out the singular values and the summary table. For the singular values:
+SELECT madlib.svd_sparse( 'mat_sparse', -- Input table
+ 'svd', -- Output table prefix
+ 'row_id', -- Column name with row index
+ 'col_id', -- Column name with column index
+ 'value', -- Matrix cell value
+ 16, -- Number of rows in matrix
+ 10, -- Number of columns in matrix
+ 10 -- Number of singular values to compute
+ );
+
+
Run the SVD function for a very sparse matrix.
+SELECT madlib.svd_sparse_native ( 'mat_sparse', -- Input table
+ 'svd', -- Output table prefix
+ 'row_id', -- Column name with row index
+ 'col_id', -- Column name with column index
+ 'value', -- Matrix cell value
+ 16, -- Number of rows in matrix
+ 10, -- Number of columns in matrix
+ 10 -- Number of singular values to compute
+ );
+
Technical Background
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics. Let \(A\) be a \(m \times n\) matrix, where \(m \ge n\). Then \(A\) can be decomposed as follows:
+\[ A = U \Sigma V^T, \]
+
+ where \(U\) is a \(m \times n\) orthonormal matrix, \(\Sigma\) is a \(n \times n\) diagonal matrix, and \(V\) is an \(n \times n\) orthonormal matrix. The diagonal elements of \(\Sigma\) are called the singular values. It is possible to formulate the problem of computing the singular triplets ( \(\sigma_i, u_i, v_i\)) of \(A\) as an eigenvalue problem involving a Hermitian matrix related to \(A\). There are two possible ways of achieving this:
+
+
+
+
With the cross product matrix, \(A^TA\) and \(AA^T\)
+ The singular values are the nonnegative square roots of the eigenvalues of the cross product matrix. This approach may imply a severe loss of accuracy in the smallest singular values. The cyclic matrix approach is an alternative that avoids this problem, but at the expense of significantly increasing the cost of the computation. Computing the cross product matrix explicitly is not recommended, especially in the case of sparse A. Bidiagonalization was proposed by Golub and Kahan [citation?] as a way of tridiagonalizing the cross product matrix without forming it explicitly. Consider the following decomposition
+\[ A = P B Q^T, \]
+
+ where \(P\) and \(Q\) are unitary matrices and \(B\) is an \(m \times n\) upper bidiagonal matrix. Then the tridiagonal matrix \(B*B\) is unitarily similar to \(A*A\). Additionally, specific methods exist that compute the singular values of \(B\) without forming \(B*B\). Therefore, after computing the SVD of B,
+\[ B = X\Sigma Y^T, \]
+
+ it only remains to compute the SVD of the original matrix with \(U = PX\) and \(V = QY\).
+
+
+
+
+
+
+
Generated on Mon Aug 28 2023 08:03:55 for MADlib by
+
+ 1.8.13
This module implements a sparse vector data type, named "svec", which provides compressed storage of vectors that have many duplicate elements.
+
Arrays of floating point numbers for various calculations sometimes have long runs of zeros (or some other default value). This is common in applications like scientific computing, retail optimization, and text processing. Each floating point number takes 8 bytes of storage in memory and/or disk, so saving those zeros is often worthwhile. There are also many computations that can benefit from skipping over the zeros.
+
Consider, for example, the following array of doubles stored as a Postgres/Greenplum "float8[]" data type:
This array would occupy slightly more than 320KB of memory or disk, most of it zeros. Even if we were to exploit the null bitmap and store the zeros as nulls, we would still end up with a 5KB null bitmap, which is still not nearly as memory efficient as we'd like. Also, as we perform various operations on the array, we do work on 40,000 fields that turn out to be unimportant.
+
To solve the problems associated with the processing of vectors discussed above, the svec type employs a simple Run Length Encoding (RLE) scheme to represent sparse vectors as pairs of count-value arrays. For example, the array above would be represented as
+
+'{1,1,40000,1,1}:{0,33,0,12,22}'::madlib.svec
+
which says there is 1 occurrence of 0, followed by 1 occurrence of 33, followed by 40,000 occurrences of 0, etc. This uses just 5 integers and 5 floating point numbers to store the array. Further, it is easy to implement vector operations that can take advantage of the RLE representation to make computations faster. The SVEC module provides a library of such functions.
+
The current version only supports sparse vectors of float8 values. Future versions will support other base types.
+
Using Sparse Vectors
+
An SVEC can be constructed directly with a constant expression, as follows:
+SELECT madlib.svec_agg(v1) FROM generate_series(1,k);
+
An SVEC can be created using the madlib.svec_cast_positions_float8arr() function by supplying an array of positions and an array of values at those positions:
+SELECT madlib.svec_cast_positions_float8arr(
+ array[n1,n2,...nk], -- positions of values in vector
+ array[v1,v2,...vk], -- values at each position
+ length, -- length of vector
+ base) -- value at unspecified positions
+
Add madlib to the search_path to use the svec operators defined in the module.
+
Document Vectorization into Sparse Vectors
This module implements an efficient way for document vectorization, converting text documents into sparse vector representation (MADlib.svec), required by various machine learning algorithms in MADlib.
+
The function accepts two tables as input, dictionary table and documents table, and produces the specified output table containing sparse vectors for the represented documents (in documents table).
TEXT. Name of the dictionary table containing features.
+
+
dict_id_col
TEXT. Name of the id column in the dictionary_tbl.
+ Expected Type: INTEGER or BIGINT.
+ NOTE: Values must be continuous ranging from 0 to total number of elements in the dictionary - 1.
+
+
dict_term_col
TEXT. Name of the column containing term (features) in dictionary_tbl.
+
+
+
+
documents_tbl
+
TEXT. Name of the documents table representing documents.
+
+
doc_id_col
TEXT. Name of the id column in the documents_tbl.
+
+
doc_term_col
TEXT. Name of the term column in the documents_tbl.
+
+
doc_term_info_col
TEXT. Name of the term info column in documents_tbl. The expected type of this column should be:
+ - INTEGER, BIGINT or DOUBLE PRECISION: Values directly used to populate vector.
+ - ARRAY: Length of the array used to populate the vector.
+ ** For an example use case on using these types of column types, please refer to the example below.
+
+
+
+
Example:
+ Consider a corpus consisting of set of documents consisting of features (terms) along with doc ids:
We can use operations with svec type like <, >, *, **, /, =, +, SUM, etc, and they have meanings associated with typical vector operations. For example, the plus (+) operator adds each of the terms of two vectors having the same dimension together.
A dot product (%*%) between the two vectors will result in a scalar result of type float8. The dot product should be (0*4 + 1*3 + 5*2) = 13, like this:
Special vector aggregate functions are also available. SUM is self explanatory. SVEC_COUNT_NONZERO evaluates the count of non-zero terms in each column found in a set of n-dimensional svecs and returns an svec with the counts. For instance, if we have the vectors {0,1,5}, {10,0,3},{0,0,3},{0,1,0}, then executing the SVEC_COUNT_NONZERO() aggregate function would result in {1,2,3}:
+
+CREATE TABLE list (a madlib.svec);
+INSERT INTO list VALUES ('{0,1,5}'::float8[]), ('{10,0,3}'::float8[]), ('{0,0,3}'::float8[]),('{0,1,0}'::float8[]);
+SELECT madlib.svec_count_nonzero(a)::float8[] FROM list;
+
We do not use null bitmaps in the svec data type. A null value in an svec is represented explicitly as an NVP (No Value Present) value. For example, we have:
+SELECT '{1,2,3}:{4,null,5}'::madlib.svec;
+
Result:
+ svec
+ ------------------
+ {1,2,3}:{4,NVP,5}
+
Adding svecs with null values results in NVPs in the sum:
The elements/subvector of an svec can be changed using the function svec_change(). It takes three arguments: an m-dimensional svec sv1, a start index j, and an n-dimensional svec sv2 such that j + n - 1 <= m, and returns an svec like sv1 but with the subvector sv1[j:j+n-1] replaced by sv2. An example follows:
The full list of functions available for operating on svecs are available in svec.sql-in.
+
A More Extensive Example
+
For a text classification example, let's assume we have a dictionary composed of words in a sorted text array:
+CREATE TABLE features (a text[]);
+INSERT INTO features VALUES
+ ('{am,before,being,bothered,corpus,document,i,in,is,me,
+ never,now,one,really,second,the,third,this,until}');
+
We have a set of documents, each represented as an array of words:
Now we have a dictionary and some documents, we would like to do some document categorization using vector arithmetic on word counts and proportions of dictionary words in each document.
+
To start this process, we'll need to find the dictionary words in each document. We'll prepare what is called a Sparse Feature Vector or SFV for each document. An SFV is a vector of dimension N, where N is the number of dictionary words, and in each cell of an SFV is a count of each dictionary word in the document.
+
Inside the sparse vector library, we have a function that will create an SFV from a document, so we can just do this (For a more efficient way for converting documents into sparse vectors, especially for larger datasets, please refer to Document Vectorization into Sparse Vectors):
+
+SELECT madlib.svec_sfv((SELECT a FROM features LIMIT 1),b)::float8[]
+ FROM documents;
+
Note that the output of madlib.svec_sfv() is an svec for each document containing the count of each of the dictionary words in the ordinal positions of the dictionary. This can more easily be understood by lining up the feature vector and text like this:
+
+SELECT madlib.svec_sfv((SELECT a FROM features LIMIT 1),b)::float8[]
+ , b
+ FROM documents;
+
+ a
+ -------------------------------------------------------------------------------------------------------
+{am,before,being,bothered,corpus,document,i,in,is,me,never,now,one,really,second,the,third,this,until}
+
Now when we look at the document "i am the second document in the corpus", its SFV is {1,3*0,1,1,1,1,6*0,1,2}. The word "am" is the first ordinate in the dictionary and there is 1 instance of it in the SFV. The word "before" has no instances in the document, so its value is "0" and so on.
+
The function madlib.svec_sfv() can process large numbers of documents into their SFVs in parallel at high speed.
+
The rest of the categorization process is all vector math. The actual count is hardly ever used. Instead, it's turned into a weight. The most common weight is called tf/idf for Term Frequency / Inverse Document Frequency. The calculation for a given term in a given document is
+
+{#Times in document} * log {#Documents / #Documents the term appears in}.
+
For instance, the term "document" in document A would have weight 1 * log (4/3). In document D, it would have weight 2 * log (4/3). Terms that appear in every document would have tf/idf weight 0, since log (4/4) = log(1) = 0. (Our example has no term like that.) That usually sends a lot of values to 0.
+
For this part of the processing, we'll need to have a sparse vector of the dictionary dimension (19) with the values
+log(#documents/#Documents each term appears in).
+
There will be one such vector for the whole list of documents (aka the "corpus"). The #documents is just a count of all of the documents, in this case 4, but there is one divisor for each dictionary word and its value is the count of all the times that word appears in the document. This single vector for the whole corpus can then be scalar product multiplied by each document SFV to produce the Term Frequency/Inverse Document Frequency weights.
+
This can be done as follows:
+CREATE TABLE corpus AS
+ (SELECT a, madlib.svec_sfv((SELECT a FROM features LIMIT 1),b) sfv
+ FROM documents);
+CREATE TABLE weights AS
+ (SELECT a docnum, madlib.svec_mult(sfv, logidf) tf_idf
+ FROM (SELECT madlib.svec_log(madlib.svec_div(count(sfv)::madlib.svec,madlib.svec_count_nonzero(sfv))) logidf
+ FROM corpus) foo, corpus ORDER BYdocnum);
+SELECT * FROM weights;
+
We can now get the "angular distance" between one document and the rest of the documents using the ACOS of the dot product of the document vectors: The following calculates the angular distance between the first document and each of the other documents:
+SELECT docnum,
+ 180. * ( ACOS( madlib.svec_dmin( 1., madlib.svec_dot(tf_idf, testdoc)
+ / (madlib.svec_l2norm(tf_idf)*madlib.svec_l2norm(testdoc))))/3.141592654) angular_distance
+ FROM weights,(SELECT tf_idf testdoc FROM weights WHERE docnum = 1 LIMIT 1) foo
+ ORDER BY 1;
+
We can see that the angular distance between document 1 and itself is 0 degrees and between document 1 and 3 is 90 degrees because they share no features at all. The angular distance can now be plugged into machine learning algorithms that rely on a distance measure between data points.
+
SVEC also provides functionality for declaring array given an array of positions and array of values, intermediate values betweens those are declared to be base value that user provides in the same function call. In the example below the fist array of integers represents the positions for the array two (array of floats). Positions do not need to come in the sorted order. Third value represents desired maximum size of the array. This assures that array is of that size even if last position is not. If max size < 1 that value is ignored and array will end at the last position in the position vector. Final value is a float representing the base value to be used between the declared ones (0 would be a common candidate):
Support vector machines are models for regression and classification tasks. SVM models have two particularly desirable features: robustness in the presence of noisy data and applicability to a variety of data configurations. At its core, a linear SVM model is a hyperplane separating two distinct classes of data (in the case of classification problems), in such a way that the distance between the hyperplane and the nearest training data point (called the margin) is maximized. Vectors that lie on this margin are called support vectors. With the support vectors fixed, perturbations of vectors beyond the margin will not affect the model; this contributes to the model’s robustness. By substituting a kernel function for the usual inner product, one can approximate a large variety of decision boundaries in addition to linear hyperplanes.
Classification Training Function
The SVM binary classification training function has the following format:
TEXT. Name of the table containing the training data.
+
+
+
model_table
+
TEXT. Name of the output table containing the model. Details of the output tables are provided below.
+
+
+
dependent_varname
+
TEXT. Name of the dependent variable column. For classification, this column can contain values of any type, but must assume exactly two distinct values since only binary classification is currently supported.
+
+
+
independent_varname
+
TEXT. Expression list to evaluate for the independent variables. An intercept variable should not be included as part of this expression. See 'fit_intercept' in the kernel params for info on intercepts. Please note that expression should be able to be cast to DOUBLE PRECISION[].
+
+
+
kernel_func (optional)
+
TEXT, default: 'linear'. Type of kernel. Currently three kernel types are supported: 'linear', 'gaussian', and 'polynomial'. The text can be any subset of the three strings; for e.g., kernel_func='ga' will create a Gaussian kernel.
+
+
+
kernel_params (optional)
+
TEXT, defaults: NULL. Parameters for non-linear kernel in a comma-separated string of key-value pairs. The actual parameters differ depending on the value of kernel_func. See the description below for details.
+
+
+
grouping_col (optional)
+
TEXT, default: NULL. An expression list used to group the input dataset into discrete groups, which results in running one model per group. Similar to the SQL "GROUP BY" clause. When this value is NULL, no grouping is used and a single model is generated. Please note that cross validation is not supported if grouping is used.
+
+
+
params (optional)
+
TEXT, default: NULL. Parameters for optimization and regularization in a comma-separated string of key-value pairs. If a list of values is provided, then cross-validation will be performed to select the best value from the list. See the description below for details.
+
+
+
verbose (optional)
+
BOOLEAN default: FALSE. Verbose output of the results of training.
+
+
+
Output tables
+ The model table produced by SVM contains the following columns:
+
+
coef
FLOAT8. Vector of coefficients.
+
+
grouping_key
TEXT Identifies the group to which the datum belongs.
+
+
num_rows_processed
BIGINT. Numbers of rows processed.
+
+
num_rows_skipped
BIGINT. Numbers of rows skipped due to missing values or failures.
+
+
num_iterations
INTEGER. Number of iterations completed by stochastic gradient descent algorithm. The algorithm either converged in this number of iterations or hit the maximum number specified in the optimization parameters.
+
+
loss
FLOAT8. Value of the objective function of SVM, expressed as an average loss per row over the source_table. See Technical Background section below for more details.
+
+
norm_of_gradient
FLOAT8. Value of the L2-norm of the (sub)-gradient of the objective function.
+
+
__dep_var_mapping
TEXT[]. Vector of dependent variable labels. The first entry corresponds to -1 and the second to +1. For internal use only.
+
+
An auxiliary table named <model_table>_random is created if the kernel is not linear. It contains data needed to embed test data into a random feature space (see references [2,3]). This data is used internally by svm_predict and not meaningful on its own to the user, so you can ignore it.
+
A summary table named <model_table>_summary is also created, which has the following columns:
+
+
method
'svm'
+
+
version_number
Version of MADlib which was used to generate the model.
+
+
source_table
The data source table name.
+
+
model_table
The model table name.
+
+
dependent_varname
The dependent variable.
+
+
independent_varname
The independent variables.
+
+
kernel_func
The kernel function.
+
+
kernel_parameters
The kernel parameters, as well as random feature map data.
+
+
grouping_col
Columns on which to group.
+
+
optim_params
A string containing the optimization parameters.
+
+
reg_params
A string containing the regularization parameters.
+
+
num_all_groups
Number of groups in SVM training.
+
+
num_failed_groups
Number of failed groups in SVM training.
+
+
total_rows_processed
Total numbers of rows processed in all groups.
+
+
total_rows_skipped
Total numbers of rows skipped in all groups due to missing values or failures.
+
+
If cross validation is used, a table is created with a user specified name having the following columns:
+
+
...
Names of cross validation parameters
+
+
mean_score
Mean value of accuracy when predicted on the validation fold, averaged over all folds and all rows.
+
+
std_dev_score
Standard deviation of accuracy when predicted on the validation fold, averaged over all folds and all rows.
+
+
Regression Training Function
The SVM regression training function has the following format:
Specifications for regression are largely the same as for classification. In the model table, there is no dependent variable mapping. The following arguments have specifications which differ from svm_classification:
+
dependent_varname
+
TEXT. Name of the dependent variable column. For regression, this column can contain only values or expressions that can be cast to DOUBLE PRECISION. Otherwise, an error will be thrown.
+
params (optional)
+
TEXT, default: NULL. The parameters epsilon and eps_table are only meaningful for regression. See description below for more details.
+
+
Novelty Detection Training Function
The novelty detection function is a one-class SVM classifier, and has the following format:
Specifications for novelty detection are largely the same as for classification, except the dependent variable name is not specified. The model table is the same as that for classification.
+
Kernel Parameters
Kernel parameters are supplied in a string containing a comma-delimited list of name-value pairs. All of these named parameters are optional, and their order does not matter. You must use the format "<param_name> = <value>" to specify the value of a parameter, otherwise the parameter is ignored.
+
+
Parameters common to all kernels
+
+
fit_intercept
+
Default: True. The parameter fit_intercept is an indicator to add an intercept to the independent_varname array expression. The intercept is added to the end of the feature list - thus the last element of the coefficient list is the intercept.
+
n_components
+
Default: max(100, 2*num_features). The dimensionality of the transformed feature space. A larger value lowers the variance of the estimate of the kernel but requires more memory and takes longer to train.
Note
Setting the n_components kernel parameter properly is important to generate an accurate decision boundary and can make the difference between a good model and a useless model. Try increasing the value of n_components if you are not getting an accurate decision boundary. This parameter arises from using the primal formulation, in which we map data into a relatively low-dimensional randomized feature space [2, 3]. The parameter n_components is the dimension of that feature space. We use the primal in MADlib to support scaling to large data sets, compared to R or other single node implementations that use the dual formulation and hence do not have this type of mapping, since the the dimensionality of the transformed feature space in the dual is effectively infinite.
+
+
random_state
+
Default: 1. Seed used by a random number generator.
+
+
+
Parameters for 'gaussian' kernel
+
+
gamma
+
Default: 1/num_features. The parameter \(\gamma\) in the Radius Basis Function kernel, i.e., \(\exp(-\gamma||x-y||^2)\). Choosing a proper value for gamma is critical to the performance of kernel machine; e.g., while a large gamma tends to cause overfitting, a small gamma will make the model too constrained to capture the complexity of the data.
+
+
+
Parameters for 'polynomial' kernel
+
+
coef0
+
Default: 1.0. The independent term \(q\) in \( (\langle x,y\rangle + q)^r \). Must be larger than or equal to 0. When it is 0, the polynomial kernel is in homogeneous form.
+
degree
+
Default: 3. The parameter \(r\) in \( (\langle x,y\rangle + q)^r \).
+
+
Other Parameters
Parameters in this section are supplied in the params argument as a string containing a comma-delimited list of name-value pairs. All of these named parameters are optional, and their order does not matter. You must use the format "<param_name> = <value>" to specify the value of a parameter, otherwise the parameter is ignored.
+
Hyperparameter optimization can be carried out using the built-in cross validation mechanism, which is activated by assigning a value greater than 1 to the parameter n_folds in params. Please note that cross validation is not supported if grouping is used.
+
The values of a parameter to cross validate should be provided in a list. For example, if one wanted to regularize with the L1 norm and use a lambda value from the set {0.3, 0.4, 0.5}, one might input 'lambda={0.3, 0.4, 0.5}, norm=L1, n_folds=10' in params. Note that the use of '{}' and '[]' are both valid here.
Note
Note that not all of the parameters below can be cross-validated. For parameters where cross validation is allowed, their default values are presented in list format; e.g., [0.01].
Default: [0.01]. Also known as the initial learning rate. A small value is usually desirable to ensure convergence, while a large value provides more room for progress during training. Since the best value depends on the condition number of the data, in practice one often searches in an exponential grid using built-in cross validation; e.g., "init_stepsize = [1, 0.1, 0.001]". To reduce training time, it is common to run cross validation on a subsampled dataset, since this usually provides a good estimate of the condition number of the whole dataset. Then the resulting init_stepsize can be run on the whole dataset.
+
+
+
+
decay_factor
+
Default: [0.9]. Control the learning rate schedule: 0 means constant rate; <-1 means inverse scaling, i.e., stepsize = init_stepsize / iteration; > 0 means <exponential decay, i.e., stepsize = init_stepsize * decay_factor^iteration.
+
+
+
max_iter
+
Default: [100]. The maximum number of iterations allowed.
+
+
+
tolerance
+
Default: 1e-10. The criterion to end iterations. The training stops whenever the difference between the training models of two consecutive iterations is smaller than tolerance or the iteration number is larger than max_iter.
+
+
+
lambda
+
Default: [0.01]. Regularization parameter. Must be non-negative.
+
+
+
norm
+
Default: 'L2'. Name of the regularization, either 'L2' or 'L1'.
+
+
+
epsilon
+
Default: [0.01]. Determines the \(\epsilon\) for \(\epsilon\)-regression. Ignored during classification. When training the model, differences of less than \(\epsilon\) between estimated labels and actual labels are ignored. A larger \(\epsilon\) will yield a model with fewer support vectors, but will not generalize as well to future data. Generally, it has been suggested that epsilon should increase with noisier data, and decrease with the number of samples. See [5].
+
+
+
eps_table
+
Default: NULL. Name of the input table that contains values of epsilon for different groups. Ignored when grouping_col is NULL. Define this input table if you want different epsilon values for different groups. The table consists of a column named epsilon which specifies the epsilon values, and one or more columns for grouping_col. Extra groups are ignored, and groups not present in this table will use the epsilon value specified in parameter epsilon.
+
+
+
validation_result
+
Default: NULL. Name of the table to store the cross validation scores. This table is only created if the name is not NULL. The cross validation scores are the mean and standard deviation of the accuracy when predicted on the validation fold, averaged over all folds and all rows. For classification, the accuracy metric used is the ratio of correct classifications. For regression, the accuracy metric used is the negative of mean squared error (negative to make it a concave problem, thus selecting max means the highest accuracy).
+
+
+
n_folds
+
Default: 0. Number of folds (k). Must be at least 2 to activate cross validation. If a value of k > 2 is specified, each fold is then used as a validation set once, while the other k - 1 folds form the training set.
+
+
+
class_weight
+
Default: NULL for classification, 'balanced' for one-class novelty detection, this param is not applicable for regression.
+
Set the weight for the classes. If not given (empty/NULL), all classes are set to have equal weight. If 'class_weight = balanced', values of y are automatically adjusted as inversely proportional to class frequencies in the input data i.e. the weights are set as n_samples / (2 * bincount(y)).
+
Alternatively, 'class_weight' can be a mapping, giving the weight for each class. E.g., for dependent variable values 'a' and 'b', the 'class_weight' might be {a: 1, b: 3}. This gives three times the weight to observations with class value 'b' compared to 'a'. (In the SVM algorithm, this translates into observations with class value 'b' contributing 3x to learning in the stochastic gradient step compared to 'a'.)
+
For regression, the class weights are always one.
+
+
+
Prediction Function
The prediction function is used to estimate the conditional mean given a new predictor. The same syntax is used for classification, regression and novelty detection:
TEXT. Model table produced by the training function.
+
+
+
new_data_table
+
TEXT. Name of the table containing the prediction data. This table is expected to contain the same features that were used during training. The table should also contain id_col_name used for identifying each row.
+
+
+
id_col_name
+
TEXT. The name of the id column in the input table.
+
+
+
output_table
+
TEXT. Name of the table where output predictions are written. If this table name is already in use, then an error is returned. Table contains:
+
+
id
Gives the 'id' for each prediction, corresponding to each row from the new_data_table.
+
+
prediction
Provides the prediction for each row in new_data_table. For regression this would be the same as decision_function. For classification, this will be one of the dependent variable values.
+
+
decision_function
Provides the distance between each point and the separating hyperplane.
Predict using linear model. We want to predict if house price is less than $100,000. We use the training data set for prediction as well, which is not usual but serves to show the syntax. The predicted results are in the prediction column and the actual data is in the actual column.
+DROP TABLE IF EXISTS houses_pred;
+SELECT madlib.svm_predict('houses_svm',
+ 'houses',
+ 'id',
+ 'houses_pred');
+\x off
+SELECT *, price < 100000 AS actual FROM houses JOIN houses_pred USING (id) ORDER BY id;
+
+ id | tax | bedroom | bath | price | size | lot | prediction | decision_function | actual
+----+------+---------+------+--------+------+-------+------------+--------------------+--------
+ 1 | 590 | 2 | 1 | 50000 | 770 | 22100 | t | 0.211310440574799 | t
+ 2 | 1050 | 3 | 2 | 85000 | 1410 | 12000 | t | 0.37546191651855 | t
+ 3 | 20 | 3 | 1 | 22500 | 1060 | 3500 | t | 2.4021783278516 | t
+ 4 | 870 | 2 | 2 | 90000 | 1300 | 17500 | t | 0.63967342411632 | t
+ 5 | 1320 | 3 | 2 | 133000 | 1500 | 30000 | f | -0.179964783767855 | f
+ 6 | 1350 | 2 | 1 | 90500 | 820 | 25700 | f | -1.78347623159173 | t
+ 7 | 2790 | 3 | 2.5 | 260000 | 2130 | 25000 | f | -2.86795504439645 | f
+ 8 | 680 | 2 | 1 | 142500 | 1170 | 22000 | t | 0.811108105668757 | f
+ 9 | 1840 | 3 | 2 | 160000 | 1500 | 19000 | f | -1.61739505790168 | f
+ 10 | 3680 | 4 | 2 | 240000 | 2790 | 20000 | f | -3.96700444824078 | f
+ 11 | 1660 | 3 | 1 | 87000 | 1030 | 17500 | f | -2.19489938920329 | t
+ 12 | 1620 | 3 | 2 | 118600 | 1250 | 20000 | f | -1.53961627668269 | f
+ 13 | 3100 | 3 | 2 | 140000 | 1760 | 38000 | f | -4.54881979553637 | f
+ 14 | 2070 | 2 | 3 | 148000 | 1550 | 14000 | f | -2.06911803381861 | f
+ 15 | 650 | 3 | 1.5 | 65000 | 1450 | 12000 | t | 1.52704061329968 | t
+(15 rows)
+
Count the miss-classifications:
+SELECT COUNT(*) FROM houses_pred JOIN houses USING (id)
+WHERE houses_pred.prediction != (houses.price < 100000);
+
+ count
+-------+
+ 3
+
+
Train using Gaussian kernel. This time we specify the initial step size and maximum number of iterations to run. As part of the kernel parameter, we choose 10 as the dimension of the space where we train SVM. As a result, the model will be a 10 dimensional vector, instead of 4 as in the case of linear model.
Prediction using the Gaussian model. The predicted results are in the prediction column and the actual data is in the actual column.
+DROP TABLE IF EXISTS houses_pred_gaussian;
+SELECT madlib.svm_predict('houses_svm_gaussian',
+ 'houses',
+ 'id',
+ 'houses_pred_gaussian');
+\x off
+SELECT *, price < 100000 AS actual FROM houses JOIN houses_pred_gaussian USING (id) ORDER BY id;
+
+ id | tax | bedroom | bath | price | size | lot | prediction | decision_function | actual
+----+------+---------+------+--------+------+-------+------------+--------------------+--------
+ 1 | 590 | 2 | 1 | 50000 | 770 | 22100 | t | 1.89855833083557 | t
+ 2 | 1050 | 3 | 2 | 85000 | 1410 | 12000 | t | 1.47736856649617 | t
+ 3 | 20 | 3 | 1 | 22500 | 1060 | 3500 | t | 0.999999992995691 | t
+ 4 | 870 | 2 | 2 | 90000 | 1300 | 17500 | t | 0.999999989634351 | t
+ 5 | 1320 | 3 | 2 | 133000 | 1500 | 30000 | f | -1.03645694166465 | f
+ 6 | 1350 | 2 | 1 | 90500 | 820 | 25700 | t | 1.16430515664766 | t
+ 7 | 2790 | 3 | 2.5 | 260000 | 2130 | 25000 | f | -0.545622670134529 | f
+ 8 | 680 | 2 | 1 | 142500 | 1170 | 22000 | f | -1.00000000207512 | f
+ 9 | 1840 | 3 | 2 | 160000 | 1500 | 19000 | f | -1.4748622470053 | f
+ 10 | 3680 | 4 | 2 | 240000 | 2790 | 20000 | f | -1.00085274698056 | f
+ 11 | 1660 | 3 | 1 | 87000 | 1030 | 17500 | t | 1.8614251155696 | t
+ 12 | 1620 | 3 | 2 | 118600 | 1250 | 20000 | f | -1.77616417509695 | f
+ 13 | 3100 | 3 | 2 | 140000 | 1760 | 38000 | f | -1.07759348149549 | f
+ 14 | 2070 | 2 | 3 | 148000 | 1550 | 14000 | f | -3.42475835116536 | f
+ 15 | 650 | 3 | 1.5 | 65000 | 1450 | 12000 | t | 1.00000008401961 | t
+(15 rows)
+
Count the miss-classifications. Note this produces a more accurate result than the linear case for this data set:
+SELECT COUNT(*) FROM houses_pred_gaussian JOIN houses USING (id)
+WHERE houses_pred_gaussian.prediction != (houses.price < 100000);
+
+ count
+-------+
+ 0
+(1 row)
+
+
In the case of an unbalanced class-size dataset, use the 'balanced' parameter to classify when building the model:
+DROP TABLE IF EXISTS abalone_regr;
+SELECT madlib.svm_predict('abalone_svm_regression',
+ 'abalone',
+ 'id',
+ 'abalone_regr');
+\x off
+SELECT * FROM abalone JOIN abalone_regr USING (id) ORDER BY id;
+
+DROP TABLE IF EXISTS abalone_gaussian_regr;
+SELECT madlib.svm_predict('abalone_svm_gaussian_regression',
+ 'abalone',
+ 'id',
+ 'abalone_gaussian_regr');
+\x off
+SELECT * FROM abalone JOIN abalone_gaussian_regr USING (id) ORDER BY id;
+
Now train a non-linear one-class SVM for novelty detection, using a Gaussian kernel. Note that the dependent variable is not a parameter for one-class:
For the novelty detection using one-class, let's create a test data set using the last 3 values from the training set plus an outlier at the end (10x price):
Now run prediction on the Gaussian one-class novelty detection model. Result shows the last row predicted to be novel:
+DROP TABLE IF EXISTS houses_pred;
+SELECT madlib.svm_predict('houses_one_class_gaussian',
+ 'houses_one_class_test',
+ 'id',
+ 'houses_pred');
+\x off
+SELECT * FROM houses_one_class_test JOIN houses_pred USING (id) ORDER BY id;
+
Result showing the last row predicted to be novel:
where \((x_1,y_1),\ldots,(x_n,y_n)\) are labeled training data and \(\ell(y,f(x))\) is a loss function. When performing classification, \(\ell(y,f(x)) = \max(0,1-yf(x))\) is the hinge loss. For regression, the loss function \(\ell(y,f(x)) = \max(0,|y-f(x)|-\epsilon)\) is used.
+
If \( f_{w,b}(x) = \langle w, x\rangle + b\) is linear, then the objective function is convex and incremental gradient descent (IGD, or SGD) can be applied to find a global minimum. See Feng, et al. [1] for more details.
+
To learn with Gaussian or polynomial kernels, the training data is first mapped via a random feature map in such a way that the usual inner product in the feature space approximates the kernel function in the input space. The linear SVM training function is then run on the resulting data. See the papers [2,3] for more information on random feature maps.
+
Also, see the book [4] by Scholkopf and Smola for more details on SVMs in general.
Term frequency computes the number of times that a word or term occurs in a document. Term frequency is often used as part of a larger text processing pipeline, which may include operations such as stemming, stop word removal and topic modelling.
TEXT. The name of the table containing the documents, with one document per row. Each row is in the form <doc_id, word_vector> where doc_id is an id unique to each document, and word_vector is a text array containing the words in the document. The word_vector should contain multiple entries of a word if the document contains multiple occurrence of that word.
+
+
+
doc_id_col
+
TEXT. The name of the column containing the document id.
+
+
+
word_col
+
TEXT. The name of the column containing the vector of words/terms in the document. This column should be of type that can be cast to TEXT[].
+
+
+
output_table
+
TEXT. The name of the table to store the term frequency output. The output table contains the following columns:
+
doc_id_col: This the document id column (name will be same as the one provided as input).
+
word: Word/term present in a document. Depending on the value of compute_vocab below, this is either the original word as it appears in word_col, or an id representing the word. Note that word id's start from 0 not 1.
+
count: The number of times this word is found in the document.
+
+
+
+
compute_vocab
+
BOOLEAN. (Optional, Default=FALSE) Flag to indicate if a vocabulary table is to be created. If TRUE, an additional output table is created containing the vocabulary of all words, with an id assigned to each word in alphabetical order. The table is called output_table_vocabulary (i.e., suffix added to the output_table name) and contains the following columns:
+
wordid: An id for each word in alphabetical order.
+
word: The word/term corresponding to the id.
+
+
+
+
Examples
+
+
First we create a document table with one document per row:
+DROP TABLE IF EXISTS documents;
+CREATE TABLE documents(docid INT4, contents TEXT);
+INSERT INTO documents VALUES
+(0, 'I like to eat broccoli and bananas. I ate a banana and spinach smoothie for breakfast.'),
+(1, 'Chinchillas and kittens are cute.'),
+(2, 'My sister adopted two kittens yesterday.'),
+(3, 'Look at this cute hamster munching on a piece of broccoli.');
+
You can apply stemming, stop word removal and tokenization at this point in order to prepare the documents for text processing. Depending upon your database version, various tools are available. Databases based on more recent versions of PostgreSQL may do something like:
+SELECT tsvector_to_array(to_tsvector('english',contents)) from documents;
+
In this example, we assume a database based on an older version of PostgreSQL and just perform basic punctuation removal and tokenization. The array of words is added as a new column to the documents table:
+ALTER TABLE documents ADD COLUMN words TEXT[];
+UPDATE documents SET words =
+ regexp_split_to_array(lower(
+ regexp_replace(contents, E'[,.;\']','', 'g')
+ ), E'[\\s+]');
+\x on
+SELECT * FROM documents ORDER BY docid;
+
+-[ RECORD 1 ]------------------------------------------------------------------------------------
+docid | 0
+contents | I like to eat broccoli and bananas. I ate a banana and spinach smoothie for breakfast.
+words | {i,like,to,eat,broccoli,and,bananas,i,ate,a,banana,and,spinach,smoothie,for,breakfast}
+-[ RECORD 2 ]------------------------------------------------------------------------------------
+docid | 1
+contents | Chinchillas and kittens are cute.
+words | {chinchillas,and,kittens,are,cute}
+-[ RECORD 3 ]------------------------------------------------------------------------------------
+docid | 2
+contents | My sister adopted two kittens yesterday.
+words | {my,sister,adopted,two,kittens,yesterday}
+-[ RECORD 4 ]------------------------------------------------------------------------------------
+docid | 3
+contents | Look at this cute hamster munching on a piece of broccoli.
+words | {look,at,this,cute,hamster,munching,on,a,piece,of,broccoli}
+
+
Compute the frequency of each word in each document:
+DROP TABLE IF EXISTS documents_tf, documents_tf_vocabulary;
+SELECT madlib.term_frequency('documents', -- input table
+ 'docid', -- document id column
+ 'words', -- vector of words in document
+ 'documents_tf' -- output table
+ );
+\x off
+SELECT * FROM documents_tf ORDER BY docid;
+
TEXT. Name of the table containing the input data.
+
+
+
output_table
+
Name of output table. A new INTEGER column on the right called 'split' will identify 1 for train set and 0 for test set, unless the 'separate_output_tables' parameter below is TRUE, in which case two output tables will be created using the 'output_table' name with the suffixes '_train' and '_test'. The output table contains all the columns present in the source table unless otherwise specified in the 'target_cols' parameter below.
+
+
+
train_proportion
+
FLOAT8 in the range (0,1). Proportion of the dataset to include in the train split. If the 'grouping_col' parameter is specified below, each group will be sampled independently using the train proportion, i.e., in a stratified fashion.
+
+
+
test_proportion (optional)
+
FLOAT8 in the range (0,1). Proportion of the dataset to include in the test split. Default is the complement to the train proportion (1-'train_proportion'). If the 'grouping_col' parameter is specified below, each group will be sampled independently using the train proportion, i.e., in a stratified fashion.
+
+
+
grouping_cols (optional)
+
TEXT, default: NULL. A single column or a list of comma-separated columns that defines how to stratify. When this parameter is NULL, the train-test split is not stratified.
+
+
+
target_cols (optional)
+
TEXT, default NULL. A comma-separated list of columns to appear in the 'output_table'. If NULL or '*', all columns from the 'source_table' will appear in the 'output_table'.
+
Note
Do not include 'grouping_cols' in the parameter 'target_cols', because they are always included in the 'output_table'.
+
+
with_replacement (optional)
+
BOOLEAN, default FALSE. Determines whether to sample with replacement or without replacement (default). With replacement means that it is possible that the same row may appear in the sample set more than once. Without replacement means a given row can be selected only once.
+
+
+
separate_output_tables (optional)
+
BOOLEAN, default FALSE. If TRUE, two output tables will be created using the 'output_table' name with the suffixes '_train' and '_test'.
+
+
Examples
+
Please note that due to the random nature of sampling, your results may look different from those below.
Random forest is an ensemble learning method for classification and regression that construct a multitude of decision trees at training time, then produces the class that is the mean (regression) or mode (classification) of the prediction produced by the individual trees.
+
+
+
+
+
+
+
+
Generated on Mon Aug 28 2023 08:03:57 for MADlib by
+
+ 1.8.13
+
+
+
+
diff --git a/docs/docs/v2.1.0/group__grp__tree.js b/docs/docs/v2.1.0/group__grp__tree.js
new file mode 100644
index 00000000..7e73e8c4
--- /dev/null
+++ b/docs/docs/v2.1.0/group__grp__tree.js
@@ -0,0 +1,5 @@
+var group__grp__tree =
+[
+ [ "Decision Tree", "group__grp__decision__tree.html", null ],
+ [ "Random Forest", "group__grp__random__forest.html", null ]
+];
\ No newline at end of file
diff --git a/docs/docs/v2.1.0/group__grp__tsa.html b/docs/docs/v2.1.0/group__grp__tsa.html
new file mode 100644
index 00000000..9fea4f70
--- /dev/null
+++ b/docs/docs/v2.1.0/group__grp__tsa.html
@@ -0,0 +1,130 @@
+
+
+
+
+
+
+
+
+MADlib: Time Series Analysis
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
Create a new table with a subset of the columns dropped from a source table.
+
+
+
+
Note: If the function cleanup_madlib_temp_tables() gives an Out-of-memory error, then the number of tables to be dropped is too high to execute in one transaction. In such a case, please follow the instructions provided with the error to execute the command in multiple transactions.
Estimates the fit of a predictive model given a data set and specifications for the training, prediction, and error estimation functions.
+
Cross validation, sometimes called rotation estimation, is a technique for assessing how the results of a statistical analysis will generalize to an independent data set. It is mainly used in settings where the goal is prediction, and you want to estimate how accurately a predictive model will perform in practice.
+
The cross-validation function provided by this module is very flexible and can work with algorithms you want to cross validate, including algorithms you write yourself. Among the inputs to the cross-validation function are specifications of the modelling, prediction, and error metric functions. These three-part specifications include the name of the function, an array of arguments to pass to the function, and an array of the data types of the arguments. This makes it possible to use functions from other MADlib modules or user-defined functions that you supply.
+
+
The modelling (training) function takes in a given data set with independent and dependent variables and produces a model, which is stored in an output table.
+
The prediction function takes in the model generated by the modelling function and a different data set with independent variables, and produces a prediction of the dependent variables based on the model, which is stored in an output table. The prediction function should take a unique ID column name in the data table as one of the inputs, so that the prediction result can be compared with the validation values. Note: Prediction function in some MADlib modules do not save results into an output table. These prediction functions are not suitable for this cross-validation module.
+
The error metric function compares the prediction results with the known values of the dependent variables in the data set that was fed into the prediction function. It computes the error metric using the specified error metric function, and stores the results in a table.
+
+
Other inputs include the output table name, k value for the k-fold cross validation, and how many folds to try. For example, you can choose to run a simple validation instead of a full cross validation.
VARCHAR. The name of the function that trains the model.
+
+
+
modelling_params
+
VARCHAR[]. An array of parameters to supply to the modelling function.
+
+
+
modelling_params_type
+
VARCHAR[]. An array of data type names for each of the parameters supplied to the modelling function.
+
+
+
param_explored
+
VARCHAR. The name of the parameter that will be checked to find the optimum value. The name must appear in the modelling_params array.
+
+
+
explore_values
+
VARCHAR. The name of the parameter whose values are to be studied.
+
+
+
predict_func
+
VARCHAR. The name of the prediction function.
+
+
+
predict_params
+
VARCHAR[]. An array of parameters to supply to the prediction function.
+
+
+
predict_params_type
+
VARCHAR[]. An array of data type names for each of the parameters supplied to the prediction function.
+
+
+
metric_func
+
VARCHAR. The name of the function for measuring errors.
+
+
+
metric_params
+
VARCHAR[]. An array of parameters to supply to the error metric function.
+
+
+
metric_params_type
+
VARCHAR[]. An array of data type names for each of the parameters supplied to the metric function.
+
+
+
data_tbl
+
VARCHAR. The name of the data table that will be split into training and validation parts.
+
+
+
data_id
+
VARCHAR. The name of the column containing a unique ID associated with each row, or NULL if the table has no such column.
+
Ideally, the data set has a unique ID for each row so that it is easier to partition the data set into the training part and the validation part. Set the id_is_random argument to inform the cross-validation function whether the ID value is randomly assigned to each row. If it is not randomly assigned, the cross-validation function generates a random ID for each row.
+
+
+
id_is_random
+
BOOLEAN. TRUE if the provided ID is randomly assigned to each row.
+
+
+
validation_result
+
VARCHAR. The name of the table to store the output of the cross-validation function. The output table has the following columns:
+
+
param_explored
The name of the parameter checked to find the optimum value. This is the same name specified in the param_explored argument of the cross_validation_general() function.
+
+
average error
The average of the errors computed by the error metric function.
+
+
standard deviation of error
The standard deviation of the errors.
+
+
+
+
data_cols
+
A comma-separated list of names of data columns to use in the calculation. When its value is NULL, the function will automatically figure out all the column names of the data table. This is only used if the data_id argument is NULL, otherwise it is ignored.
+
If the data set has no unique ID for each row, the cross-validation function copies the data set to a temporary table with a randomly assigned ID column. Setting this argument to the list of independent and dependent variables that are to be used in the calculation minimizes the copying workload by only copying the required data.
+
The new temporary table is dropped after the computation has finished.
+
+
fold_num
+
INTEGER, default: 10. Value of k. How many folds validation? Each validation uses 1/fold_num fraction of the data for validation.
+
+
The parameter arrays for the modelling, prediction and metric functions can include the following special keywords:
+
+
%data% – The argument position for training/validation data
+
%model% – The argument position for the output/input of modelling/prediction function
+
%id% – The argument position of the unique ID column (user-provided or generated by the cross-validation function, as described above)
+
%prediction% – The argument position for the output/input of prediction/metric function
+
%error% – The argument position for the output of the metric function
+
+
Note: If the argument explore_values is NULL or has zero length, then the cross-validation function will only run a data folding.
Use the general function to explore lambda values for elastic net. (Note that elastic net also has a built in cross validation function for selecting elastic net control parameter alpha and regularization value lambda.)
The lock management parameter max_locks_per_transaction, which usually is set to the default value of 64, limits the number of tables that can be dropped inside a single transaction (the cross-validation function). Thus, the number of different values of param_explored (or the length of the explored_values array) cannot be too large. For 10-fold cross validation, the limit of length(explored_values) is around 40. If the limit is exceeded, you may get an "out of shared memory" error because max_locks_per_transaction is exceeded.
+
One way to overcome this limitation is to run the cross-validation function multiple times, with each run covering a different region of values of the parameter.
+
Note that MADlib implements cross-validation functions within certain individual modules, where it is possible to optimize the calculation to avoid dropping tables and prevent exceeding the max_locks_per_transaction limitation. Since module-specific cross-validation functions depend upon the implementation details of the modules to perform the optimization, they will not be as flexible as the generalized cross-validation function provided here.
+
Technical Background
+
One round of cross validation involves partitioning a sample of data into complementary subsets, performing the analysis on one subset (called the training set), and validating the analysis on the other subset (called the validation set or test set). To reduce variability, multiple rounds of cross validation are performed using different partitions, and the validation results are averaged over the rounds.
+
In k-fold cross validation, the original sample is randomly partitioned into k equal sized subsamples. Of the k subsamples, a single subsample is retained as the validation data for testing the model, and the remaining k−1 subsamples are used as training data. The cross-validation process is repeated k times (the folds), with each of the k subsamples used exactly once as the validation data. The k results from the folds can be averaged (or otherwise combined) to produce a single estimation. The advantage of this method over repeated random sub-sampling is that all observations are used for both training and validation, and each observation is used for validation exactly once. 10-fold cross validation is commonly used, but in general k remains an unfixed parameter.
Converts a feature array in a single column into multiple columns. This process can be used to reverse the function cols2vec.
+
Given a table with a column of type array, this function will create an output table that splits this array into multiple columns, one per array element. It includes the option to name the new feature columns, and to include columns from the original table in the output.
TEXT. Name of the table containing the source data..
+
+
+
output_table
+
TEXT. Name of the generated table containing the output. If a table with the same name already exists, an error will be returned.
+
+
+
vector_col
+
TEXT. Name of the column containing the feature array. Must be a one-dimensional array.
+
+
+
feature_names (optional)
+
TEXT[]. Array of names associated with the feature array. Note that this array exists in the summary table created by the function 'cols2vec'. If the 'feature_names' array is not specified, column names will be automatically generated of the form 'f1, f2, ...fn'.
Note
If you specify the 'feature_names' parameter, you will get exactly that number of feature columns in the 'output_table'. It means feature arrays from the 'vector_col' may be padded or truncated, if a particular feature array size does not match the target number of feature columns.
+
+If you do not specify the 'feature names' parameter, the number of feature columns generated in the 'output_table' will be the maximum array size from 'vector_col'. Feature arrays that are less than this maximum will be padded.
+
+
cols_to_output (optional)
+
TEXT, default NULL. Comma-separated string of column names from the source table to keep in the output table, in addition to the feature columns. To keep all columns from the source table, use '*'. The total number of columns in a table cannot exceed the PostgreSQL limits.
+
+
Output table
+ The output table produced by the vec2cols function contains the following columns:
+
+
<...>
Columns from source table, depending on which ones are kept (if any).
+
+
feature columns
Columns for each of the features in 'vector_col'. Column type will depend on the feature array type in the source table. Column naming will depend on whether the parameter 'feature_names' is used.
Split the column "clouds_airquality" into new columns called "clouds" and "air_quality". Also keep columns id and "OUTLOOK" from the source table
+DROP TABLE IF EXISTS vec2cols_result;
+SELECT madlib.vec2cols(
+ 'golf', -- source table
+ 'vec2cols_result', -- output table
+ 'clouds_airquality', -- column with array entries to split
+ ARRAY['clouds', 'air_quality'], -- feature names
+ 'id, "OUTLOOK"' -- columns to keep from source table
+);
+SELECT * FROM vec2cols_result ORDER BY id;
+
+ id | OUTLOOK | clouds | air_quality
+----+----------+--------+-------------
+ 1 | sunny | none | unhealthy
+ 2 | sunny | none | moderate
+ 3 | overcast | low | moderate
+ 4 | rain | low | moderate
+ 5 | rain | medium | good
+ 6 | rain | low | unhealthy
+ 7 | overcast | medium | moderate
+ 8 | sunny | high | unhealthy
+ 9 | sunny | high | good
+ 10 | rain | medium | good
+ 11 | sunny | none | good
+ 12 | overcast | medium | moderate
+ 13 | overcast | medium | moderate
+ 14 | rain | low | unhealthy
+(14 rows)
+
+
Similar to the previous example, except now we keep all columns from source table and do not specify the feature names, so that default names are created.
+DROP TABLE IF EXISTS vec2cols_result;
+SELECT madlib.vec2cols(
+ 'golf', -- source table
+ 'vec2cols_result', -- output table
+ 'clouds_airquality', -- column with array entries to split
+ NULL, -- feature names
+ '*' -- columns to keep from source table
+);
+SELECT * FROM vec2cols_result ORDER BY id;
+
+ id | OUTLOOK | temperature | humidity | Temp_Humidity | clouds_airquality | windy | class | observation_weight | f1 | f2
+----+----------+-------------+----------+---------------+-------------------+-------+------------+--------------------+--------+-----------
+ 1 | sunny | 85 | 85 | {85,85} | {none,unhealthy} | f | Don't Play | 5 | none | unhealthy
+ 2 | sunny | 80 | 90 | {80,90} | {none,moderate} | t | Don't Play | 5 | none | moderate
+ 3 | overcast | 83 | 78 | {83,78} | {low,moderate} | f | Play | 1.5 | low | moderate
+ 4 | rain | 70 | 96 | {70,96} | {low,moderate} | f | Play | 1 | low | moderate
+ 5 | rain | 68 | 80 | {68,80} | {medium,good} | f | Play | 1 | medium | good
+ 6 | rain | 65 | 70 | {65,70} | {low,unhealthy} | t | Don't Play | 1 | low | unhealthy
+ 7 | overcast | 64 | 65 | {64,65} | {medium,moderate} | t | Play | 1.5 | medium | moderate
+ 8 | sunny | 72 | 95 | {72,95} | {high,unhealthy} | f | Don't Play | 5 | high | unhealthy
+ 9 | sunny | 69 | 70 | {69,70} | {high,good} | f | Play | 5 | high | good
+ 10 | rain | 75 | 80 | {75,80} | {medium,good} | f | Play | 1 | medium | good
+ 11 | sunny | 75 | 70 | {75,70} | {none,good} | t | Play | 5 | none | good
+ 12 | overcast | 72 | 90 | {72,90} | {medium,moderate} | t | Play | 1.5 | medium | moderate
+ 13 | overcast | 81 | 75 | {81,75} | {medium,moderate} | f | Play | 1.5 | medium | moderate
+ 14 | rain | 71 | 80 | {71,80} | {low,unhealthy} | t | Don't Play | 1 | low | unhealthy
+(14 rows)
+
+
Now let's run cols2vec then reverse it using vec2cols. In this case we will get feature names from the cols2vec summary table. First run cols2vec:
+DROP TABLE IF EXISTS cols2vec_result, cols2vec_result_summary;
+SELECT madlib.cols2vec(
+ 'golf',
+ 'cols2vec_result',
+ 'temperature, humidity',
+ NULL,
+ 'id, temperature, humidity'
+);
+SELECT * FROM cols2vec_result ORDER BY id;
+
Now use feature_names from the summary table above to name the columns of the split array:
+DROP TABLE IF EXISTS vec2cols_result;
+SELECT madlib.vec2cols(
+ 'cols2vec_result', -- source table containing the feature vector
+ 'vec2cols_result', -- output table
+ 'feature_vector', -- column with array entries to split
+ (SELECT feature_names from cols2vec_result_summary), -- feature_names from summary table of cols2vec
+ 'id' -- columns to keep from source table
+);
+SELECT * FROM vec2cols_result ORDER BY id;
+
Given a directed graph, a weakly connected component (WCC) is a subgraph of the original graph where all vertices are connected to each other by some path, ignoring the direction of edges. In case of an undirected graph, a weakly connected component is also a strongly connected component. This module also includes a number of helper functions that operate on the WCC output.
TEXT. Name of the table containing the vertex data for the graph. Must contain the column specified in the 'vertex_id' parameter below.
+
+
+
vertex_id
+
TEXT, default = 'id'. Name of the column(s) in 'vertex_table' containing vertex ids. The vertex ids can be of type INTEGER or BIGINT with no duplicates. They do not need to be contiguous. If multiple columns are used as vertex ids, they are passed in the following format: [<vertex_id1>,<vertex_id2>,...]
+
+
+
edge_table
+
TEXT. Name of the table containing the edge data. The edge table must contain columns for source vertex and destination vertex.
+
+
+
edge_args
+
TEXT. A comma-delimited string containing multiple named arguments of the form "name=value". The following parameters are supported for this string argument:
+
src (INTEGER or BIGINT): Name of the column(s) containing the source vertex ids in the edge table. Default column name is 'src'.
+
dest (INTEGER or BIGINT): Name of the column(s) containing the destination vertex ids in the edge table. Default column name is 'dest'.
+
+
+
+
out_table
+
TEXT. Name of the table to store the component ID associated with each vertex. It will contain a row for every vertex from 'vertex_table' with the following columns:
+
vertex_id : The id of a vertex. Will use the input parameter 'vertex_id' for column naming. If multiple columns are used for identifying vertices, this column will be an array named "id".
+
component_id : Component that the vertex belongs to. We use the convention where 'component_id' is the id of the first vertex in a particular group. It means that component ids are generally not contiguous.
+
grouping_cols : Grouping column (if any) values associated with the vertex_id.
+
+
A summary table named <out_table>_summary is also created. This is an internal table that keeps a record of some of the input parameters and is used by the weakly connected component helper functions.
+
+
+
grouping_cols (optional)
+
TEXT, default: NULL. A single column or a list of comma-separated columns that divides the input data into discrete groups, which are treated independently as separate graphs. When this value is NULL, no grouping is used and weakly connected components are generated for all data (single graph).
Note
Expressions are not currently supported for 'grouping_cols'.
+
+
iteration_limit (optional)
+
INTEGER, default: NULL. Maximum number of iterations to run wcc. This parameter is used to stop wcc early to limit the number of subtransactions created by wcc. For such subtx issues, it is advised to set this parameter to
+
A wcc run that stopped early by this parameter can resume its progress by using the warm_start parameter. An additional table named <out_table>_message is also created. This table is necessary in case the iteration_limit is reached and there are still vertices to update. It gets used when the wcc continues the process via warm_start and gets dropped when the wcc determines there are no more updates necessary. The user might determine if the wcc is completed or not by checking the nodes_to_update column of <out_table>_summary table where 0 means wcc is complete.
+
+
+
+
warm_start (optional)
+
BOOLEAN, default: NULL. If True, wcc will look for the <out_table>_message table and continue using it and the partial output from <out_table> to continue the wcc process.
+
+
+
+
Note
On a Greenplum cluster, the edge table should be distributed by the source vertex id column for better performance. In addition, the user should note that this function creates a duplicate of the edge table (on Greenplum cluster) for better performance.
+
Retrieve Largest Connected Component
+
The largest connected component retrieval function finds the largest weakly connected component(s) in a graph. If weakly connected components was run with grouping, the largest connected components are computed for each group.
TEXT. Name of the table that contains the output of weakly connected components.
+
+
+
largest_cpt_table
+
TEXT. Name of the output table that contains the largest component's information. It contains one or more rows for every group and has the following columns:
+
grouping_cols: The grouping columns given in the creation of wcc_table. If there are no grouping columns, this column is not created.
+
component_id: The ID of the largest component. Recall that we use the convention where 'component_id' is the id of the first vertex in a particular group. It means that component ids are generally not contiguous. If there are multiple components of the same size, a row is created for each component. If grouping_cols is specified, the largest component is computed for each group.
+
num_vertices: Number of vertices in the largest component.
+
+
+
+
Retrieve Histogram of Vertices Per Connected Component
+
This function creates a histogram of the number of vertices per connected component.
TEXT. Name of the table that contains the output of weakly connected components.
+
+
+
histogram_table
+
TEXT. Name of the output table that contains the number of vertices per component. A row is created for every comoponent in every group if grouping_cols was specified when running weakly connected components. The output table has the following columns:
+
grouping_cols: The grouping columns given during the creation of the wcc_table. If there are no grouping columns, this column is not created.
+
component_id: The ID of the component.
+
num_vertices: Number of vertices in the component specified by the component_id column.
+
+
+
+
+
Check if Two Vertices Belong to the Same Component
+
This function determines if two vertices belong to the same component.
TEXT. Name of the table that contains the output of weakly connected components.
+
+
+
vertex_pair
+
BIGINT[]. A pair of vertex IDs separated by a comma. If multiple columns are used for identifying vertices, a 2D array will be required for this parameter.
+
+
+
pair_table
+
TEXT. Name of the output table that specifies if the two vertices in vertex_pair belong to the same component. If wcc_table was generated using grouping_cols, all the components in all groups are considered. The output table has the following columns:
+
component_id: Component ID that contains both the vertices in vertex_pair.
+
grouping_cols: The grouping columns given in the creation of wcc_table. If there are no grouping columns, this column is not created.
+
+
+
+
+
Retrieve All Vertices Reachable from a Vertex
+
This function finds all the vertices that can be reached from a given vertex via weakly connected paths.
TEXT. Name of the table that contains the output of weakly connected components.
+
+
+
count_table
+
TEXT. Name of the output table that contains the total number of components per group in the graph, if there are any grouping_cols in wcc_table. The output table has the following columns:
+
grouping_cols : The grouping columns given in the creation of wcc_table. If there are no grouping columns, this column is not created, and count is with regard to the entire graph.
+
num_components : Count of weakly connected components in a graph, or the number of components within a group if grouping_cols is defined.
Note that vertex 4 is not identified as a separate component above. This is because there is no entry in the edge table for vertex 4 indicating which group it belongs to (though you could do that if you wanted to).
+
Retrieve the largest connected component:
+DROP TABLE IF EXISTS largest_cpt_table;
+SELECT madlib.graph_wcc_largest_cpt(
+ 'wcc_out', -- WCC output table
+ 'largest_cpt_table'); -- output table containing largest component ID
+SELECT * FROM largest_cpt_table ORDER BY component_id;
+
Check if two vertices belong to the same component:
+DROP TABLE IF EXISTS vc_table;
+SELECT madlib.graph_wcc_vertex_check(
+ 'wcc_out', -- WCC output table
+ '14,15', -- Pair of vertex IDs
+ 'vc_table'); -- output table containing components that contain the two vertices
+SELECT * FROM vc_table ORDER BY component_id;
+
+DROP TABLE IF EXISTS count_table;
+SELECT madlib.graph_wcc_num_cpts(
+ 'wcc_out', -- WCC output table
+ 'count_table'); -- output table containing number of components per group
+SELECT * FROM count_table;
+
XGBoost is an optimized distributed gradient boosting library. The MADlib implementation is designed to train multiple models with different parameters to achieve parallelism on Greenplum systems (PostgreSQL is also supported.)
+
The main use case supported is classification. Regression is not currently supported.
+
Please note that the XGBoost implementation has the following limitations:
+
All of the dataset is collected in a single segment, compressed by zlib and stored in a single cells (one column of one row). PostgreSQL has a 1 GB limit for each cell which means the supported dataset size is limited.
+
The collected data is passed to the XGBoost in a single segment. This means, without grid search, the XGBoost is using only a single segment which might increase the run time.
+
+
XGBoost requires the following python libraries available in every host machine:
+
pandas v0.42
+
scikit-learn v0.19
+
xgboost v0.82
+
+
Versions are provided to ensure a python2 supported libraries can be obtained. Other versions of these libraries might work just as well.
TEXT. Name of the table containing the training data.
+
+
+
output_table
+
TEXT. Name of the generated table containing the model. If a table with the same name already exists, an error will be returned. A summary table named <output_table>_summary is also created.
+
+
+
id_column
+
TEXT. Name of the column containing id information in the training data. This is a mandatory argument and the values are expected to be unique for each row. Suggested column types are INTEGER, BIGINT, and VARCHAR.
+
+
+
dependent_variable
+
TEXT. Name of the column that contains the output (response) for training. Boolean, integer and text types are accepted for classification.
+
+
+
list_of_features
+
TEXT. Comma-separated string of column names or expressions to use as predictors. Can also be a '*' implying all columns are to be used as predictors (except for the ones included in the next argument that lists exclusions).
+
+
+
list_of_features_to_exclude (optional)
+
TEXT[]. Text array of column names to exclude from the predictors list. The names in this parameter should be identical to the names used in the table and quoted appropriately. The id_column and dependent_variable are excluded by default.
+
+
+
params_str (optional)
+
TEXT. Comma-separated string of key-value pairs used for initializing XGBClassifier. Each parameter can be assigned an array of values. These values are expanded into multiple parameter sets for grid search.
+
Note that these key-value pairs are passed to the xgboost as is and xgboost accepts any parameter thanks to kwargs. If there is a typo in the list, it might get ignored by xgboost.
+
eval_metric is a unique key for this dictionary. In XGBoost v0.82, this parameter is passed to the fit function and not the initializer like the rest. After v1.6.0, it was moved to the initializer. MADlib interface keeps these parameters together to ensure that when we upgrade the supported version of the XGBoost, we will be able to maintain same the interface.
+
Default values set by MADlib: learning_rate: 0.3 max_depth: 6 n_estimators: 100 eval_metric: 'auc'
+
+
+
+
sample_weights (optional)
+
TEXT. Column name containing numerical weights for sample_weights parameter of XGBClassifier fit function.
+
+
+
train_set_size (optional)
+
DOUBLE PRECISION. The proportion of the dataset to be used in training.
+
+
+
train_set_split_var (optional)
+
TEXT. Column name containing information on whether the associated row will be used for training or testing. If this parameter is set, train_set_size will be ignored.
+
+
Output
+
+
The model table produced by the training function contains the following columns:
+
+
+
model
BYTEA8. Trained XGBoost model stored in binary format (not human readable).
+
+
features
TEXT[]. Ordered levels (values) of categorical variables corresponding to the categorical features. Used to help interpret the trained model.
+
+
params_index
INTEGER. The index of the model. During grid search, each parameter set is given an index to distinguish different models.
+
+
+
+
A summary table named <output_table>_summary is also created at the same time, which has the following columns:
+
+
+
mdl_train_ts
TEXT. The timestamp for the initiation of the xgboost command.
+
+
mdl_name
TEXT. The name of the model set to <source_table>_xgboost.
+
+
features
TEXT[]. The list of features inputed to the XGBoost.
+
+
params
TEXT. The XGBClassifier parameters used for this particular model.
+
+
fnames
TEXT. The list of features in sorted order. This column is used for importance
+
+
importance
DOUBLE PRECISION[]. Importance values for each feature in the sorted order.
+
+
recall
DOUBLE PRECISION[]. Recall values for each class.
+
+
fscore
DOUBLE PRECISION[]. fscore values for each class.
+
+
support
DOUBLE PRECISION[]. support values for each class.
+
+
test_ids
INTEGER[]. The ids of the rows used for this particular model.
+
+
params_index
INTEGER. The index of the parameter set used for this model.
TEXT. Name of the table containing the xgboost train output.
+
+
+
predict_output_table
+
TEXT. Name of the generated table containing the prediction. If a table with the same name already exists, an error will be returned. A metrics table named <output_table_name>_metrics and an roc curve table named <output_table_name>_roc_curve are also created if class_label is not NULL.
+
+
+
id_column
+
TEXT. Name of the column containing id information in the test data. This is a mandatory argument and the values are expected to be unique for each row.
+
+
+
class_label (optional)
+
TEXT. Name of the column containing class_label for the metrics and roc_curve calculations.
+
+
+
model_filters (optional)
+
TEXT. The filter for the model_table in case the user wants to use a subset of the models from the grid search.
+
+
Output
+
XGBoost prediction function creates three tables: <predict_output_table>, <predict_output_table>_metrics and <predict_output_table>_roc_curve. Some fields of the metrics table as well as the roc_curve table are available only if a class_label column is provided.
+
<predict_output_table>
+
+
+
+
id_column
INTEGER. Name of the column containing id information in the test data.
+
+
class_label_predicted
TEXT. The predicted value for the given data id.
+
+
class_label_proba_predicted
DOUBLE PRECISION[]. The prediction probabilities for the given data id.
+
+
<predict_output_table>_metrics
+
+
precision
DOUBLE PRECISION[]. Precision values for each class in the sorted order.
+
+
recall
DOUBLE PRECISION[]. Recall values for each class.
+
+
fscore
DOUBLE PRECISION[]. fscore values for each class.
+
+
support
DOUBLE PRECISION[]. support values for each class.
+
+
roc_auc_scores
DOUBLE PRECISION[]. roc_auc scores for each class.
+
+
feature_names
TEXT[]. The list of features in sorted order. This column is used for importance
+
+
feature_importance_scores
DOUBLE PRECISION[]. Importance values for each feature in the sorted order.
+
+
<predict_output_table>_roc_curve
+
+
fpr
DOUBLE PRECISION[]. False positive rate for the roc curve.
+
+
tpr
DOUBLE PRECISION[]. True positive rate for the roc curve.
+
+
thresholds
DOUBLE PRECISION[]. Threshold values for the roc curve.
Run XGBoost for a single parameter set and show the summary table
+DROP TABLE IF EXISTS xgb_out, xgb_out_summary;
+SELECT xgboost(
+ 'abalone', -- Training table
+ 'id', -- Id column
+ 'sex', -- Class label column
+ '*', -- Independent variables
+ NULL, -- Columns to exclude from features
+ $$
+ {
+ 'learning_rate': [0.01], #Regularization on weights (eta). For smaller values, increase n_estimators
+ 'max_depth': [9],#Larger values could lead to overfitting
+ 'subsample': [0.85],#introduce randomness in samples picked to prevent overfitting
+ 'colsample_bytree': [0.85],#introduce randomness in features picked to prevent overfitting
+ 'min_child_weight': [10],#larger values will prevent over-fitting
+ 'n_estimators':[100] #More estimators, lesser variance (better fit on test set)
+ }
+ $$, -- XGBoost grid search parameters
+ 'xgb_out', -- Grid search results table.
+ '', -- Class weights
+ 0.8, -- Training set size ratio
+ NULL -- Variable used to do the test/train split.
+);
+\x on
+SELECT * FROM xgb_out_summary;
+
+DROP TABLE IF EXISTS xgb_out, xgb_out_summary;
+SELECT xgboost(
+ 'abalone', -- Training table
+ 'id', -- Id column
+ 'sex', -- Class label column
+ '*', -- Independent variables
+ NULL, -- Columns to exclude from features
+ $$
+ {
+ 'learning_rate': [0.01,0.1], #Regularization on weights (eta). For smaller values, increase n_estimators
+ 'max_depth': [9,12],#Larger values could lead to overfitting
+ 'subsample': [0.85],#introduce randomness in samples picked to prevent overfitting
+ 'colsample_bytree': [0.85],#introduce randomness in features picked to prevent overfitting
+ 'min_child_weight': [10],#larger values will prevent over-fitting
+ 'n_estimators':[100] #More estimators, lesser variance (better fit on test set)
+ }
+ $$, -- XGBoost grid search parameters
+ 'xgb_out', -- Grid search results table.
+ '', -- Class weights
+ 0.8, -- Training set size ratio
+ NULL -- Variable used to do the test/train split.
+);
+\x on
+SELECT * FROM xgb_out_summary;
+
+ id | sex_predicted | sex_proba_predicted
+----+---------------+------------------------------------------------
+ 5 | I | {0.178420484066,0.599636971951,0.221942588687}
+ 6 | I | {0.185853347182,0.602131128311,0.212015494704}
+ 9 | I | {0.253592431545,0.440728843212,0.30567869544}
+ 10 | M | {0.374555230141,0.249226689339,0.376218110323}
+ 11 | F | {0.402999937534,0.336779236794,0.260220855474}
+ 13 | I | {0.338803291321,0.36541134119,0.295785397291}
+ 14 | F | {0.377499818802,0.301990658045,0.320509523153}
+ 17 | I | {0.179169252515,0.602536559105,0.218294218183}
+ 21 | M | {0.27216938138,0.33275142312,0.395079195499}
+ 25 | F | {0.449239164591,0.187060594559,0.36370024085}
+
+
+
Literature
+
[1] Chen, T., & Guestrin, C. (2016). XGBoost: A Scalable Tree Boosting System. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (pp. 785–794). New York, NY, USA: ACM. https://doi.org/10.1145/2939672.2939785
+
+
+
+
+
+
Generated on Mon Aug 28 2023 08:03:58 for MADlib by
+
+ 1.8.13
Let \( n_1, \dots, n_k \) be a realization of a (vector) random variable \( N = (N_1, \dots, N_k) \) that follows the multinomial distribution with parameters \( k \) and \( p = (p_1, \dots, p_k) \). Test the null hypothesis \( H_0 : p = p^0 \).
+
Parameters
+
+
observed
Number \( n_i \) of observations of the current event/row
+
expected
Expected number of observations of current event/row. This number is not required to be normalized. That is, \( p^0_i \) will be taken as expected divided by sum(expected). Hence, if this parameter is not specified, chi2_test() will by default use \( p^0 = (\frac 1k, \dots, \frac 1k) \), i.e., test that \( p \) is a discrete uniform distribution.
+
df
Degrees of freedom. This is the number of events reduced by the degree of freedom lost by using the observed numbers for defining the expected number of observations. If this parameter is 0, the degree of freedom is taken as \( (k - 1) \).
+
+
+
+
Returns
A composite value as follows. Let \( n = \sum_{i=1}^n n_i \).
Test null hypothesis that all possible outcomes of a categorical variable are equally likely:
SELECT (chi2_gof_test(observed, 1, NULL)).* FROM source
+
Test null hypothesis that two categorical variables are independent. Such data is often shown in a contingency table (also known as crosstab). A crosstab is a matrix where possible values for the first variable correspond to rows and values for the second variable to columns. The matrix elements are the observation frequencies of the joint occurrence of the respective values. chi2_gof_test() assumes that the crosstab is stored in normalized form, i.e., there are three columns var1, var2, observed.
SELECT (chi2_gof_test(observed, expected, deg_freedom)).*
+FROM (
+ SELECT
+ observed,
+ sum(observed) OVER (PARTITION BY var1)::DOUBLE PRECISION
+ * sum(observed) OVER (PARTITION BY var2) AS expected
+ FROM source
+) p, (
+ SELECT
+ (count(DISTINCT var1) - 1) * (count(DISTINCT var2) - 1) AS deg_freedom
+ FROM source
+) q;
Given realizations \( x_1, \dots, x_m \) and \( y_1, \dots, y_n \) of i.i.d. random variables \( X_1, \dots, X_m \sim N(\mu_X, \sigma^2) \) and \( Y_1, \dots, Y_n \sim N(\mu_Y, \sigma^2) \) with unknown parameters \( \mu_X, \mu_Y, \) and \( \sigma^2 \), test the null hypotheses \( H_0 : \sigma_X < \sigma_Y \) and \( H_0 : \sigma_X = \sigma_Y \).
+
Parameters
+
+
first
Indicator whether value is from first sample \( x_1, \dots, x_m \) (if TRUE) or from second sample \( y_1, \dots, y_n \) (if FALSE)
+
value
Value of random variate \( x_i \) or \( y_i \)
+
+
+
+
Returns
A composite value as follows. We denote by \( \bar x, \bar y \) the sample means and by \( s_X^2, s_Y^2 \) the sample variances.
+
statistic FLOAT8 - Statistic
+\[ f = \frac{s_Y^2}{s_X^2} \]
+
+ The corresponding random variable is F-distributed with \( (n - 1) \) degrees of freedom in the numerator and \( (m - 1) \) degrees of freedom in the denominator.
+
df1 BIGINT - Degrees of freedom in the numerator \( (n - 1) \)
+
df2 BIGINT - Degrees of freedom in the denominator \( (m - 1) \)
+
p_value_one_sided FLOAT8 - Lower bound on one-sided p-value. In detail, the result is \( \Pr[F \geq f \mid \sigma_X = \sigma_Y] \), which is a lower bound on \( \Pr[F \geq f \mid \sigma_X \leq \sigma_Y] \). Computed as (1.0 - fisher_f_cdf(statistic)).
+
p_value_two_sided FLOAT8 - Two-sided p-value, i.e., \( 2 \cdot \min \{ p, 1 - p \} \) where \( p = \Pr[ F \geq f \mid \sigma_X = \sigma_Y] \). Computed as (min(p_value_one_sided, 1. - p_value_one_sided)).
+
+
+
Usage
+
Test null hypothesis that the variance of the first sample is at most (or equal to, respectively) the variance of the second sample:
Given realizations \( x_1, \dots, x_m \) and \( y_1, \dots, y_m \) of i.i.d. random variables \( X_1, \dots, X_m \) and i.i.d. \( Y_1, \dots, Y_n \), respectively, test the null hypothesis that the underlying distributions function \( F_X, F_Y \) are identical, i.e., \( H_0 : F_X = F_Y \).
+
Parameters
+
+
first
Determines whether the value belongs to the first (if TRUE) or the second sample (if FALSE)
+
value
Value of random variate \( x_i \) or \( y_i \)
+
m
Size \( m \) of the first sample. See usage instructions below.
+
n
Size of the second sample. See usage instructions below.
+ where \( F_x(t) := \frac 1m |\{ i \mid x_i \leq t \}| \) and \( F_y \) (defined likewise) are the empirical distribution functions.
+
k_statistic FLOAT8 - Kolmogorov statistic \( k = (r + 0.12 + \frac{0.11}{r}) \cdot d \) where \( r = \sqrt{\frac{m n}{m+n}}. \) and \( d \) is the statistic. Then \( k \) is approximately Kolmogorov distributed.
+
p_value FLOAT8 - Approximate p-value, i.e., an approximate value for \( \Pr[D \geq d \mid F_X = F_Y] \). Computed as (1.0 - kolmogorov_cdf(k_statistic)).
+
+
+
Usage
+
Test null hypothesis that two samples stem from the same distribution:
SELECT (ks_test(first, value,
+ (SELECT count(value) FROM source WHERE first),
+ (SELECT count(value) FROM source WHERE NOT first)
+ ORDER BY value
+)).* FROM source
+
+
+
Note
This aggregate must be used as an ordered aggregate (ORDER BY value) and will raise an exception if values are not ordered.
Given realizations \( x_{1,1}, \dots, x_{1, n_1}, x_{2,1}, \dots, x_{2,n_2}, \dots, x_{k,n_k} \) of i.i.d. random variables \( X_{i,j} \sim N(\mu_i, \sigma^2) \) with unknown parameters \( \mu_1, \dots, \mu_k \) and \( \sigma^2 \), test the null hypotheses \( H_0 : \mu_1 = \dots = \mu_k \).
+
Parameters
+
+
group
Group which value is from. Note that group can assume arbitary value not limited to a continguous range of integers.
+
value
Value of random variate \( x_{i,j} \)
+
+
+
+
Returns
A composite value as follows. Let \( n := \sum_{i=1}^k n_i \) be the total size of all samples. Denote by \( \bar x \) the grand mean, by \( \overline{x_i} \) the group sample means, and by \( s_i^2 \) the group sample variances.
+
sum_squares_between DOUBLE PRECISION - sum of squares between the group means, i.e., \( \mathit{SS}_b = \sum_{i=1}^k n_i (\overline{x_i} - \bar x)^2. \)
+
sum_squares_within DOUBLE PRECISION - sum of squares within the groups, i.e., \( \mathit{SS}_w = \sum_{i=1}^k (n_i - 1) s_i^2. \)
+
df_between BIGINT - degree of freedom for between-group variation \( (k-1) \)
+
df_within BIGINT - degree of freedom for within-group variation \( (n-k) \)
+
mean_squares_between DOUBLE PRECISION - mean square between groups, i.e., \( s_b^2 := \frac{\mathit{SS}_b}{k-1} \)
+
mean_squares_within DOUBLE PRECISION - mean square within groups, i.e., \( s_w^2 := \frac{\mathit{SS}_w}{n-k} \)
+
statistic DOUBLE PRECISION - Statistic computed as
+\[ f = \frac{s_b^2}{s_w^2}. \]
+
+ This statistic is Fisher F-distributed with \( (k-1) \) degrees of freedom in the numerator and \( (n-k) \) degrees of freedom in the denominator.
+
p_value DOUBLE PRECISION - p-value, i.e., \( \Pr[ F \geq f \mid H_0] \).
+
+
+
Usage
+
Test null hypothesis that the mean of the all samples is equal:
SELECT (one_way_anova(group, value)).* FROM source
Given realizations \( x_1, \dots, x_n \) of i.i.d. random variables \( X_1, \dots, X_n \) with unknown mean \( \mu \), test the null hypotheses \( H_0 : \mu \leq 0 \) and \( H_0 : \mu = 0 \).
+
Parameters
+
+
value
Value of random variate \( x_i \) or \( y_i \). Values of 0 are ignored (i.e., they do not count towards \( n \)).
+
precision
The precision \( \epsilon_i \) with which value is known. The precision determines the handling of ties. The current value \( v_i \) is regarded a tie with the previous value \( v_{i-1} \) if \( v_i - \epsilon_i \leq \max_{j=1, \dots, i-1} v_j + \epsilon_j \). If precision is negative, then it will be treated as value * 2^(-52). (Note that \( 2^{-52} \) is the machine epsilon for type DOUBLE PRECISION.)
+
+
+
+
Returns
A composite value:
+
statistic FLOAT8 - statistic computed as follows. Let \( w^+ = \sum_{i \mid x_i > 0} r_i \) and \( w^- = \sum_{i \mid x_i < 0} r_i \) be the signed rank sums where
+ where \( t_i \) is the number of values with absolute value equal to \( |x_i| \). The corresponding random variable is approximately standard normally distributed.
+
p_value_one_sided FLOAT8 - One-sided p-value i.e., \( \Pr[Z \geq z \mid \mu \leq 0] \). Computed as (1.0 - normal_cdf(z_statistic)).
One-sample test: Test null hypothesis that the mean of a sample is at most (or equal to, respectively) \( \mu_0 \):
SELECT (wsr_test(value - mu_0 ORDER BY abs(value))).* FROM source
+
Dependent paired test: Test null hypothesis that the mean difference between the first and second value in a pair is at most (or equal to, respectively) \( \mu_0 \):
SELECT (wsr_test(first - second - mu_0 ORDER BY abs(first - second))).* FROM source
If correctly determining ties is important (e.g., you may want to do so when comparing to software products that take first, second, and mu_0 as individual parameters), supply the precision parameter. This can be done as follows:
SELECT (wsr_test(
+ first - second - mu_0,
+ 3 * 2^(-52) * greatest(first, second, mu_0)
+ ORDER BY abs(first - second)
+)).* FROM source
Here \( 2^{-52} \) is the machine epsilon, which we scale to the magnitude of the input data and multiply with 3 because we have a sum with three terms.
+
+
+
Note
This aggregate must be used as an ordered aggregate (ORDER BY abs(value)) and will raise an exception if the absolute values are not ordered.
Given realizations \( x_1, \dots, x_n \) of i.i.d. random variables \( X_1, \dots, X_n \sim N(\mu, \sigma^2) \) with unknown parameters \( \mu \) and \( \sigma^2 \), test the null hypotheses \( H_0 : \mu \leq 0 \) and \( H_0 : \mu = 0 \).
+
Parameters
+
+
value
Value of random variate \( x_i \)
+
+
+
+
Returns
A composite value as follows. We denote by \( \bar x \) the sample mean and by \( s^2 \) the sample variance.
+
statistic FLOAT8 - Statistic
+\[ t = \frac{\sqrt n \cdot \bar x}{s} \]
+
+ The corresponding random variable is Student-t distributed with \( (n - 1) \) degrees of freedom.
+
df FLOAT8 - Degrees of freedom \( (n - 1) \)
+
p_value_one_sided FLOAT8 - Lower bound on one-sided p-value. In detail, the result is \( \Pr[\bar X \geq \bar x \mid \mu = 0] \), which is a lower bound on \( \Pr[\bar X \geq \bar x \mid \mu \leq 0] \). Computed as (1.0 - students_t_cdf(statistic)).
One-sample t-test: Test null hypothesis that the mean of a sample is at most (or equal to, respectively) \( \mu_0 \):
SELECT (t_test_one(value - mu_0)).* FROM source
+
Dependent paired t-test: Test null hypothesis that the mean difference between the first and second value in each pair is at most (or equal to, respectively) \( \mu_0 \):
SELECT (t_test_one(first - second - mu_0)).*
+ FROM source
+ is the pooled variance. The corresponding random variable is Student-t distributed with \( (n + m - 2) \) degrees of freedom.
+
df FLOAT8 - Degrees of freedom \( (n + m - 2) \)
+
p_value_one_sided FLOAT8 - Lower bound on one-sided p-value. In detail, the result is \( \Pr[\bar X - \bar Y \geq \bar x - \bar y \mid \mu_X = \mu_Y] \), which is a lower bound on \( \Pr[\bar X - \bar Y \geq \bar x - \bar y \mid \mu_X \leq \mu_Y] \). Computed as (1.0 - students_t_cdf(statistic)).
+
p_value_two_sided FLOAT8 - Two-sided p-value, i.e., \( \Pr[ |\bar X - \bar Y| \geq |\bar x - \bar y| \mid \mu_X = \mu_Y] \). Computed as (2 * students_t_cdf(-abs(statistic))).
+
+
+
Usage
+
Two-sample pooled t-test: Test null hypothesis that the mean of the first sample is at most (or equal to, respectively) the mean of the second sample:
SELECT (t_test_pooled(first, value)).* FROM source
+ degrees of freedom (Welch–Satterthwaite formula).
+
df FLOAT8 - Degrees of freedom (as above)
+
p_value_one_sided FLOAT8 - Lower bound on one-sided p-value. In detail, the result is \( \Pr[\bar X - \bar Y \geq \bar x - \bar y \mid \mu_X = \mu_Y] \), which is a lower bound on \( \Pr[\bar X - \bar Y \geq \bar x - \bar y \mid \mu_X \leq \mu_Y] \). Computed as (1.0 - students_t_cdf(statistic)).
+
p_value_two_sided FLOAT8 - Two-sided p-value, i.e., \( \Pr[ |\bar X - \bar Y| \geq |\bar x - \bar y| \mid \mu_X = \mu_Y] \). Computed as (2 * students_t_cdf(-abs(statistic))).
+
+
+
Usage
+
Two-sample unpooled t-test: Test null hypothesis that the mean of the first sample is at most (or equal to, respectively) the mean of the second sample:
SELECT (t_test_unpooled(first, value)).* FROM source
+ is defined as the rank of \( x_i \) in the combined list of all \( m+n \) observations. For ties, the average rank of all equal values is used.
+
p_value_one_sided FLOAT8 - Approximate one-sided p-value, i.e., an approximate value for \( \Pr[Z \geq z \mid H_0] \). Computed as (1.0 - normal_cdf(z_statistic)).
+
p_value_two_sided FLOAT8 - Approximate two-sided p-value, i.e., an approximate value for \( \Pr[|Z| \geq |z| \mid H_0] \). Computed as (2 * normal_cdf(-abs(z_statistic))).
+
+
+
Usage
+
Test null hypothesis that two samples stem from the same distribution:
SELECT (mw_test(first, value ORDER BY value)).* FROM source
+
+
+
Note
This aggregate must be used as an ordered aggregate (ORDER BY value) and will raise an exception if values are not ordered.
Apache MADlib is an open-source library for scalable in-database analytics. It provides data-parallel implementations of mathematical, statistical, graph and machine learning methods for structured and unstructured data.
Please refer to the ReadMe file for information about incorporated third-party material. License information regarding MADlib and included third-party libraries can be found in the License directory.
+
+
+
+
+
+
Generated on Mon Aug 28 2023 08:03:59 for MADlib by
+
+ 1.8.13
Generated on Mon Aug 28 2023 08:03:54 for MADlib by
+
+ 1.8.13
+
+
+
+
diff --git a/docs/docs/v2.1.0/jquery.js b/docs/docs/v2.1.0/jquery.js
new file mode 100644
index 00000000..f5343eda
--- /dev/null
+++ b/docs/docs/v2.1.0/jquery.js
@@ -0,0 +1,87 @@
+/*!
+ * jQuery JavaScript Library v1.7.1
+ * http://jquery.com/
+ *
+ * Copyright 2011, John Resig
+ * Dual licensed under the MIT or GPL Version 2 licenses.
+ * http://jquery.org/license
+ *
+ * Includes Sizzle.js
+ * http://sizzlejs.com/
+ * Copyright 2011, The Dojo Foundation
+ * Released under the MIT, BSD, and GPL Licenses.
+ *
+ * Date: Mon Nov 21 21:11:03 2011 -0500
+ */
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- 
- 
- 
- 
- 
- 
+
+ 
');
+}
+
+function showSyncOn(n,relpath)
+{
+ n.html('
');
+}
+
+function toggleSyncButton(relpath)
+{
+ var navSync = $('#nav-sync');
+ if (navSync.hasClass('sync')) {
+ navSync.removeClass('sync');
+ showSyncOff(navSync,relpath);
+ storeLink(stripPath2(pathName())+hashUrl());
+ } else {
+ navSync.addClass('sync');
+ showSyncOn(navSync,relpath);
+ deleteLink();
+ }
+}
+
+function initNavTree(toroot,relpath)
+{
+ var o = new Object();
+ o.toroot = toroot;
+ o.node = new Object();
+ o.node.li = document.getElementById("nav-tree-contents");
+ o.node.childrenData = NAVTREE;
+ o.node.children = new Array();
+ o.node.childrenUL = document.createElement("ul");
+ o.node.getChildrenUL = function() { return o.node.childrenUL; };
+ o.node.li.appendChild(o.node.childrenUL);
+ o.node.depth = 0;
+ o.node.relpath = relpath;
+ o.node.expanded = false;
+ o.node.isLast = true;
+ o.node.plus_img = document.createElement("span");
+ o.node.plus_img.className = 'arrow';
+ o.node.plus_img.innerHTML = arrowRight;
+
+ if (localStorageSupported()) {
+ var navSync = $('#nav-sync');
+ if (cachedLink()) {
+ showSyncOff(navSync,relpath);
+ navSync.removeClass('sync');
+ } else {
+ showSyncOn(navSync,relpath);
+ }
+ navSync.click(function(){ toggleSyncButton(relpath); });
+ }
+
+ $(window).load(function(){
+ navTo(o,toroot,hashUrl(),relpath);
+ showRoot();
+ });
+
+ $(window).bind('hashchange', function(){
+ if (window.location.hash && window.location.hash.length>1){
+ var a;
+ if ($(location).attr('hash')){
+ var clslink=stripPath(pathName())+':'+hashValue();
+ a=$('.item a[class$="'+clslink.replace(/
+
+
+
+
+
+
+
+
+
+
