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Posted to commits@madlib.apache.org by ri...@apache.org on 2017/05/16 20:29:42 UTC
[27/51] [partial] incubator-madlib-site git commit: Add v1.11 docs
http://git-wip-us.apache.org/repos/asf/incubator-madlib-site/blob/b5b51c69/docs/v1.11/group__grp__dense__linear__solver.html
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+<title>MADlib: Dense Linear Systems</title>
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+ <td style="padding-left: 0.5em;">
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+ <span id="projectnumber">1.11</span>
+ </div>
+ <div id="projectbrief">User Documentation for MADlib</div>
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+
+<div class="header">
+ <div class="headertitle">
+<div class="title">Dense Linear Systems<div class="ingroups"><a class="el" href="group__grp__utility__functions.html">Utility Functions</a> » <a class="el" href="group__grp__linear__solver.html">Linear Solvers</a></div></div> </div>
+</div><!--header-->
+<div class="contents">
+<div class="toc"><b>Contents</b> <ul>
+<li class="level1">
+<a href="#dls_usage">Solution Function</a> </li>
+<li class="level1">
+<a href="#dls_opt_params">Optimizer Parameters</a> </li>
+<li class="level1">
+<a href="#dls_examples">Examples</a> </li>
+<li class="level1">
+<a href="#related">Related Topics</a> </li>
+</ul>
+</div><p>The linear systems module implements solution methods for systems of consistent linear equations. Systems of linear equations take the form: </p><p class="formulaDsp">
+<img class="formulaDsp" alt="\[ Ax = b \]" src="form_213.png"/>
+</p>
+<p>where <img class="formulaInl" alt="$x \in \mathbb{R}^{n}$" src="form_214.png"/>, <img class="formulaInl" alt="$A \in \mathbb{R}^{m \times n} $" src="form_215.png"/> and <img class="formulaInl" alt="$b \in \mathbb{R}^{m}$" src="form_216.png"/>. We assume that there are no rows of <img class="formulaInl" alt="$A$" src="form_42.png"/> where all elements are zero. The algorithms implemented in this module can handle large dense linear systems. Currently, the algorithms implemented in this module solve the linear system by a direct decomposition. Hence, these methods are known as <em>direct method</em>.</p>
+<p><a class="anchor" id="dls_usage"></a></p><dl class="section user"><dt>Solution Function</dt><dd><pre class="syntax">
+linear_solver_dense( tbl_source,
+ tbl_result,
+ row_id,
+ LHS,
+ RHS,
+ grouping_col,
+ optimizer,
+ optimizer_params
+ )
+</pre> <b>Arguments</b> <dl class="arglist">
+<dt>tbl_source </dt>
+<dd><p class="startdd">TEXT. The name of the table containing the training data. The input data is expected to be of the following form: </p><pre>{TABLE|VIEW} <em>sourceName</em> (
+ ...
+ <em>row_id</em> FLOAT8,
+ <em>left_hand_side</em> FLOAT8[],
+ <em>right_hand_side</em> FLOAT8,
+ ...
+)</pre><p>Each row represents a single equation. The <em>right_hand_side</em> column refers to the right hand side of the equations while the <em>left_hand_side</em> column refers to the multipliers on the variables on the left hand side of the same equations.</p>
+<p class="enddd"></p>
+</dd>
+<dt>tbl_result </dt>
+<dd><p class="startdd">TEXT. The name of the table where the output is saved. The output is stored in the table named by the <em>tbl_result</em> argument. It contains the following columns: </p><table class="output">
+<tr>
+<th>solution </th><td>FLOAT8[]. The solution variables in the same order as that provided as input in the 'left_hand_side' column name of the <em>source_table</em> </td></tr>
+<tr>
+<th>residual_norm </th><td>FLOAT8. The scaled residual norm, defined as <img class="formulaInl" alt="$ \frac{|Ax - b|}{|b|} $" src="form_217.png"/>. This value is an indication of the accuracy of the solution. </td></tr>
+<tr>
+<th>iters </th><td>INTEGER. Number of iterations required by the algorithm (only applicable for iterative algorithms). The output is NULL for 'direct' methods. </td></tr>
+</table>
+<p class="enddd"></p>
+</dd>
+<dt>row_id </dt>
+<dd><p class="startdd">TEXT. The name of the column storing the 'row id' of the equations.</p>
+<p>For a system with N equations, the row_id's must be a continuous range of integers from <img class="formulaInl" alt="$ 0 \ldots n-1 $" src="form_218.png"/>. </p>
+<p class="enddd"></p>
+</dd>
+<dt>LHS </dt>
+<dd><p class="startdd">TEXT. The name of the column storing the 'left hand side' of the equations, stored as an array.</p>
+<p class="enddd"></p>
+</dd>
+<dt>RHS </dt>
+<dd><p class="startdd">TEXT. The name of the column storing the 'right hand side' of the equations.</p>
+<p class="enddd"></p>
+</dd>
+<dt>grouping_cols (optional) </dt>
+<dd>TEXT, default: NULL. Group by column names. <em>Not currently implemented. Any non-NULL value is ignored.</em> </dd>
+<dt>optimizer (optional) </dt>
+<dd><p class="startdd">TEXT, default: 'direct'. The type of optimizer.</p>
+<p class="enddd"></p>
+</dd>
+<dt>optimizer_params (optional) </dt>
+<dd>TEXT, default: NULL. Optimizer specific parameters. </dd>
+</dl>
+</dd></dl>
+<p><a class="anchor" id="dls_opt_params"></a></p><dl class="section user"><dt>Optimizer Parameters</dt><dd></dd></dl>
+<p>For each optimizer, there are specific parameters that can be tuned for better performance.</p>
+<dl class="arglist">
+<dt>algorithm (default: householderqr) </dt>
+<dd><p class="startdd">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.</p>
+<p>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. </p><pre class="fragment"> Algorithm | Conditions on A | Speed | Accuracy
+ ----------------------------------------------------------
+ householderqr | None | ++ | +
+ partialpivlu | Invertable | ++ | +
+ fullpivlu | None | - | +++
+ colpivhouseholderqr | None | + | ++
+ fullpivhouseholderqr | None | - | +++
+ llt | Pos. Definite | +++ | +
+ ldlt | Pos. or Neg Def | +++ | ++
+</pre><p>For speed '++' is faster than '+', which is faster than '-'. For accuracy '+++' is better than '++'.</p>
+<p class="enddd">More details about the individual algorithms can be found in the <a href="http://eigen.tuxfamily.org/dox-devel/group__TutorialLinearAlgebra.html">Eigen documentation</a>. Eigen is an open source library for linear algebra. </p>
+</dd>
+</dl>
+<p><a class="anchor" id="dls_examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl>
+<ol type="1">
+<li>View online help for the linear systems solver function. <pre class="example">
+SELECT madlib.linear_solver_dense();
+</pre></li>
+<li>Create the sample data set. <pre class="example">
+CREATE TABLE linear_systems_test_data( id INTEGER NOT NULL,
+ lhs DOUBLE PRECISION[],
+ rhs DOUBLE PRECISION
+ );
+INSERT INTO linear_systems_test_data(id, lhs, rhs)
+ VALUES
+ (0, ARRAY[1,0,0], 20),
+ (1, ARRAY[0,1,0], 15),
+ (2, ARRAY[0,0,1], 20);
+</pre></li>
+<li>Solve the linear systems with default parameters. <pre class="example">
+SELECT madlib.linear_solver_dense( 'linear_systems_test_data',
+ 'output_table',
+ 'id',
+ 'lhs',
+ 'rhs'
+ );
+</pre></li>
+<li>Obtain the output from the output table. <pre class="example">
+\x on
+SELECT * FROM output_table;
+</pre> Result: <pre class="result">
+--------------------+-------------------------------------
+solution | {20,15,20}
+residual_norm | 0
+iters | NULL
+</pre></li>
+<li>Choose an algorithm different than the default. <pre class="example">
+DROP TABLE IF EXISTS result_table;
+SELECT madlib.linear_solver_dense( 'linear_systems_test_data',
+ 'result_table',
+ 'id',
+ 'lhs',
+ 'rhs',
+ NULL,
+ 'direct',
+ 'algorithm=llt'
+ );
+</pre></li>
+</ol>
+<p><a class="anchor" id="related"></a></p><dl class="section user"><dt>Related Topics</dt><dd>File <a class="el" href="dense__linear__systems_8sql__in.html" title="SQL functions for linear systems. ">dense_linear_systems.sql_in</a> documenting the SQL functions</dd></dl>
+</div><!-- contents -->
+</div><!-- doc-content -->
+<!-- start footer part -->
+<div id="nav-path" class="navpath"><!-- id is needed for treeview function! -->
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+ <li class="footer">Generated on Tue May 16 2017 13:24:39 for MADlib by
+ <a href="http://www.doxygen.org/index.html">
+ <img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.13 </li>
+ </ul>
+</div>
+</body>
+</html>
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+<!-- HTML header for doxygen 1.8.4-->
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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+ <td style="padding-left: 0.5em;">
+ <div id="projectname">
+ <span id="projectnumber">1.11</span>
+ </div>
+ <div id="projectbrief">User Documentation for MADlib</div>
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+ onmouseover="return searchBox.OnSearchSelectShow()"
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+
+<div class="header">
+ <div class="summary">
+<a href="#groups">Modules</a> </div>
+ <div class="headertitle">
+<div class="title">Deprecated Modules</div> </div>
+</div><!--header-->
+<div class="contents">
+<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
+<table class="memberdecls">
+<tr class="heading"><td colspan="2"><h2 class="groupheader"><a name="groups"></a>
+Modules</h2></td></tr>
+<tr class="memitem:group__grp__indicator"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__indicator.html">Create Indicator Variables</a></td></tr>
+<tr class="memdesc:group__grp__indicator"><td class="mdescLeft"> </td><td class="mdescRight">Provides utility functions helpful for data preparation before modeling. <br /></td></tr>
+<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
+<tr class="memitem:group__grp__mlogreg"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__mlogreg.html">Multinomial Logistic Regression</a></td></tr>
+<tr class="memdesc:group__grp__mlogreg"><td class="mdescLeft"> </td><td class="mdescRight">Also called as softmax regression, models the relationship between one or more independent variables and a categorical dependent variable. <br /></td></tr>
+<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
+</table>
+</div><!-- contents -->
+</div><!-- doc-content -->
+<!-- start footer part -->
+<div id="nav-path" class="navpath"><!-- id is needed for treeview function! -->
+ <ul>
+ <li class="footer">Generated on Tue May 16 2017 13:24:39 for MADlib by
+ <a href="http://www.doxygen.org/index.html">
+ <img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.13 </li>
+ </ul>
+</div>
+</body>
+</html>
http://git-wip-us.apache.org/repos/asf/incubator-madlib-site/blob/b5b51c69/docs/v1.11/group__grp__deprecated.js
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@@ -0,0 +1,5 @@
+var group__grp__deprecated =
+[
+ [ "Create Indicator Variables", "group__grp__indicator.html", null ],
+ [ "Multinomial Logistic Regression", "group__grp__mlogreg.html", null ]
+];
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+<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
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+<tr class="memdesc:group__grp__correlation"><td class="mdescLeft"> </td><td class="mdescRight">Generates a cross-correlation matrix for all pairs of numeric columns in a table. <br /></td></tr>
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+ <div class="headertitle">
+<div class="title">Early Stage Development</div> </div>
+</div><!--header-->
+<div class="contents">
+<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
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+<tr class="memdesc:group__grp__sketches"><td class="mdescLeft"> </td><td class="mdescRight">A collection of methods to estimate the number of unique values contained in the data. <br /></td></tr>
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+<tr class="memitem:group__grp__cg"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__cg.html">Conjugate Gradient</a></td></tr>
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+<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
+<tr class="memitem:group__grp__bayes"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__bayes.html">Naive Bayes Classification</a></td></tr>
+<tr class="memdesc:group__grp__bayes"><td class="mdescLeft"> </td><td class="mdescRight">Constructs a classification model from a dataset where each attribute independently contributes to the probability that a data point belongs to a category. <br /></td></tr>
+<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
+<tr class="memitem:group__grp__sample"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__sample.html">Random Sampling</a></td></tr>
+<tr class="memdesc:group__grp__sample"><td class="mdescLeft"> </td><td class="mdescRight">Provides utility functions for sampling operations. <br /></td></tr>
+<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
+<tr class="memitem:group__grp__nene"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__nene.html">Nearest Neighbors</a></td></tr>
+<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
+</table>
+</div><!-- contents -->
+</div><!-- doc-content -->
+<!-- start footer part -->
+<div id="nav-path" class="navpath"><!-- id is needed for treeview function! -->
+ <ul>
+ <li class="footer">Generated on Tue May 16 2017 13:24:39 for MADlib by
+ <a href="http://www.doxygen.org/index.html">
+ <img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.13 </li>
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----------------------------------------------------------------------
diff --git a/docs/v1.11/group__grp__elasticnet.html b/docs/v1.11/group__grp__elasticnet.html
new file mode 100644
index 0000000..d6569e3
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@@ -0,0 +1,749 @@
+<!-- HTML header for doxygen 1.8.4-->
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
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+<meta http-equiv="Content-Type" content="text/xhtml;charset=UTF-8"/>
+<meta http-equiv="X-UA-Compatible" content="IE=9"/>
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+<meta name="keywords" content="madlib,postgres,greenplum,machine learning,data mining,deep learning,ensemble methods,data science,market basket analysis,affinity analysis,pca,lda,regression,elastic net,huber white,proportional hazards,k-means,latent dirichlet allocation,bayes,support vector machines,svm"/>
+<title>MADlib: Elastic Net Regularization</title>
+<link href="tabs.css" rel="stylesheet" type="text/css"/>
+<script type="text/javascript" src="jquery.js"></script>
+<script type="text/javascript" src="dynsections.js"></script>
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+ $(document).ready(function() { init_search(); });
+</script>
+<!-- hack in the navigation tree -->
+<script type="text/javascript" src="eigen_navtree_hacks.js"></script>
+<link href="doxygen.css" rel="stylesheet" type="text/css" />
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+ <span id="projectnumber">1.11</span>
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+</table>
+</div>
+<!-- end header part -->
+<!-- Generated by Doxygen 1.8.13 -->
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+</script>
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+ <div id="nav-tree">
+ <div id="nav-tree-contents">
+ <div id="nav-sync" class="sync"></div>
+ </div>
+ </div>
+ <div id="splitbar" style="-moz-user-select:none;"
+ class="ui-resizable-handle">
+ </div>
+</div>
+<script type="text/javascript">
+$(document).ready(function(){initNavTree('group__grp__elasticnet.html','');});
+</script>
+<div id="doc-content">
+<!-- window showing the filter options -->
+<div id="MSearchSelectWindow"
+ onmouseover="return searchBox.OnSearchSelectShow()"
+ onmouseout="return searchBox.OnSearchSelectHide()"
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+
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+ name="MSearchResults" id="MSearchResults">
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+ <div class="headertitle">
+<div class="title">Elastic Net Regularization<div class="ingroups"><a class="el" href="group__grp__super.html">Supervised Learning</a> » <a class="el" href="group__grp__regml.html">Regression Models</a></div></div> </div>
+</div><!--header-->
+<div class="contents">
+<div class="toc"><b>Contents</b><ul>
+<li class="level1">
+<a href="#train">Training Function</a> </li>
+<li class="level1">
+<a href="#optimizer">Optimizer Parameters</a> </li>
+<li class="level1">
+<a href="#predict">Prediction Functions</a> </li>
+<li class="level1">
+<a href="#examples">Examples</a> </li>
+<li class="level1">
+<a href="#background">Technical Background</a> </li>
+<li class="level1">
+<a href="#literature">Literature</a> </li>
+<li class="level1">
+<a href="#related">Related Topics</a> </li>
+</ul>
+</div><p>This module implements elastic net regularization [1] for linear and logistic regression. Regularization is a technique often used to prevent overfitting.</p>
+<p><a class="anchor" id="train"></a></p><dl class="section user"><dt>Training Function</dt><dd>The training function has the following syntax: <pre class="syntax">
+elastic_net_train( tbl_source,
+ tbl_result,
+ col_dep_var,
+ col_ind_var,
+ regress_family,
+ alpha,
+ lambda_value,
+ standardize,
+ grouping_col,
+ optimizer,
+ optimizer_params,
+ excluded,
+ max_iter,
+ tolerance
+ )
+</pre></dd></dl>
+<p><b>Arguments</b> </p><dl class="arglist">
+<dt>tbl_source </dt>
+<dd><p class="startdd">TEXT. The name of the table containing the training data.</p>
+<p class="enddd"></p>
+</dd>
+<dt>tbl_result </dt>
+<dd><p class="startdd">TEXT. Name of the output table containing output model. The output table produced by the <a class="el" href="elastic__net_8sql__in.html#a735038a5090c112505c740a90a203e83" title="Interface for elastic net. ">elastic_net_train()</a> function has the following columns: </p><table class="output">
+<tr>
+<th>regress_family </th><td>The regression type: 'gaussian' or 'binomial'. </td></tr>
+<tr>
+<th>features </th><td>Array of features (independent variables) passed to the algorithm. </td></tr>
+<tr>
+<th>features_selected </th><td>Array of features selected by the algorithm. </td></tr>
+<tr>
+<th>coef_nonzero </th><td>Coefficients of the selected features. </td></tr>
+<tr>
+<th>coef_all </th><td>Coefficients of all features, both selected and unselected. </td></tr>
+<tr>
+<th>intercept </th><td>Intercept for the model. </td></tr>
+<tr>
+<th>log_likelihood </th><td>Log of the likelihood value produced by the algorithm. </td></tr>
+<tr>
+<th>standardize </th><td>BOOLEAN. If data has been normalized, will be set to TRUE. </td></tr>
+<tr>
+<th>iteration_run </th><td>The number of iterations executed. </td></tr>
+</table>
+<p class="enddd"></p>
+</dd>
+<dt>col_dep_var </dt>
+<dd><p class="startdd">TEXT. An expression for the dependent variable.</p>
+<dl class="section note"><dt>Note</dt><dd>Both <em>col_dep_var</em> and <em>col_ind_var</em> can be valid PostgreSQL expressions. For example, <code>col_dep_var = 'log(y+1)'</code>, and <code>col_ind_var = 'array[exp(x[1]), x[2], 1/(1+x[3])]'</code>. In the binomial case, you can use a Boolean expression, for example, <code>col_dep_var = 'y < 0'</code>.</dd></dl>
+</dd>
+<dt>col_ind_var </dt>
+<dd><p class="startdd">TEXT. An expression for the independent variables. Use <code>'*'</code> to specify all columns of <em>tbl_source</em> except those listed in the <em>excluded</em> string described below. If <em>col_dep_var</em> is a column name, it is automatically excluded from the independent variables. However, if <em>col_dep_var</em> is a valid PostgreSQL expression, any column names used within the expression are only excluded if they are explicitly listed in the <em>excluded</em> argument. Therefore, it is a good idea to add all column names involved in the dependent variable expression to the <em>excluded</em> string.</p>
+<p class="enddd"></p>
+</dd>
+<dt>regress_family </dt>
+<dd><p class="startdd">TEXT. For regression type, specify either 'gaussian' ('linear') or 'binomial' ('logistic').</p>
+<p class="enddd"></p>
+</dd>
+<dt>alpha </dt>
+<dd><p class="startdd">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.</p>
+<p class="enddd"></p>
+</dd>
+<dt>lambda_value </dt>
+<dd><p class="startdd">FLOAT8. Regularization parameter (must be positive).</p>
+<p class="enddd"></p>
+</dd>
+<dt>standardize (optional) </dt>
+<dd><p class="startdd">BOOLEAN, default: TRUE. Whether to normalize the data or not. Setting to TRUE usually yields better results and faster convergence.</p>
+<p class="enddd"></p>
+</dd>
+<dt>grouping_col (optional) </dt>
+<dd><p class="startdd">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.</p>
+<dl class="section note"><dt>Note</dt><dd>Expressions are not currently supported for 'grouping_col'.</dd></dl>
+</dd>
+<dt>optimizer (optional) </dt>
+<dd><p class="startdd">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].</p>
+<p class="enddd"></p>
+</dd>
+<dt>optimizer_params (optional) </dt>
+<dd><p class="startdd">TEXT, default: NULL. Optimizer parameters, delimited with commas. These parameters differ depending on the value of <em>optimizer</em> parameter. See the descriptions below for details.</p>
+<p class="enddd"></p>
+</dd>
+<dt>excluded (optional) </dt>
+<dd><p class="startdd">TEXT, default: NULL. If the <em>col_ind_var</em> input is '*' then <em>excluded</em> can be provided as a comma-delimited list of column names that are to be excluded from the features. For example, <code>'col1, col2'</code>. If the <em>col_ind_var</em> is an array, <em>excluded</em> must be a list of the integer array positions to exclude, for example <code>'1,2'</code>. If this argument is NULL or an empty string, no columns are excluded.</p>
+<p class="enddd"></p>
+</dd>
+<dt>max_iter (optional) </dt>
+<dd><p class="startdd">INTEGER, default: 1000. The maximum number of iterations allowed.</p>
+<p class="enddd"></p>
+</dd>
+<dt>tolerance </dt>
+<dd>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 <em>tolerance</em> or the iteration number is larger than <em>max_iter</em>, the computation stops. </dd>
+</dl>
+<p><a class="anchor" id="optimizer"></a></p><dl class="section user"><dt>Other Parameters</dt><dd></dd></dl>
+<p>For <em>optimizer_params</em>, 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>".</p>
+<p>The parameters described below are organized by category: warmup, cross validation and optimization.</p>
+<p><em><b>Warmup parameters</b></em> </p><pre class="syntax">
+ $$
+ warmup = <value>,
+ warmup_lambdas = <value>,
+ warmup_lambda_no = <value>,
+ warmup_tolerance = <value>
+ $$
+</pre><dl class="arglist">
+<dt>warmup </dt>
+<dd><p class="startdd">Default: FALSE. If <em>warmup</em> 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.</p>
+<p class="enddd"></p>
+</dd>
+<dt>warmup_lambdas </dt>
+<dd><p class="startdd">Default: NULL. Set of lambda values to use when <em>warmup</em> is TRUE. The default is NULL, which means that lambda values will be automatically generated.</p>
+<p class="enddd"></p>
+</dd>
+<dt>warmup_lambda_no </dt>
+<dd><p class="startdd">Default: 15. Number of lambda values used in <em>warm-up</em>. If <em>warmup_lambdas</em> is not NULL, this value is overridden by the number of provided lambda values.</p>
+<p class="enddd"></p>
+</dd>
+<dt>warmup_tolerance </dt>
+<dd>The value of tolerance used during warmup. The default value is the same as the <em>tolerance</em> argument described above. </dd>
+</dl>
+<p><em><b>Cross validation parameters</b></em> </p><dl class="section note"><dt>Note</dt><dd>Please note that for performance reasons, warmup is disabled whenever cross validation is used. Also, cross validation is not supported if grouping is used.</dd></dl>
+<pre class="syntax">
+ $$
+ n_folds = <value>,
+ validation_result = <value>,
+ lambda_value = <value>,
+ n_lambdas = <value>,
+ alpha = <value>
+ $$
+</pre><p>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 <em>n_folds</em>. Misclassification error is used for classification and mean squared error is used for regression.</p>
+<p>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.</p>
+<dl class="arglist">
+<dt>n_folds </dt>
+<dd><p class="startdd">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. </p>
+<p class="enddd"></p>
+</dd>
+<dt>validation_result </dt>
+<dd><p class="startdd">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. </p>
+<p class="enddd"></p>
+</dd>
+<dt>lambda_value </dt>
+<dd><p class="startdd">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.</p>
+<p class="enddd"></p>
+</dd>
+<dt>n_lambdas </dt>
+<dd><p class="startdd">Default: 15. Number of lambdas to cross validate over. If a list of lambda values is not provided in the <em>lambda_value</em> set above, this parameter can be used to autogenerate the set of lambdas. If the <em>lambda_value</em> set is not NULL, this value is overridden by the number of provided lambda values. </p>
+<dl class="section note"><dt>Note</dt><dd>If you want to cross validate over alpha only and not lambda, then set <em>lambda_value</em> to NULL and <em>n_lambdas</em> to 0. In this case, cross validation will be done on the set of <em>alpha</em> 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.</dd></dl>
+</dd>
+<dt>alpha </dt>
+<dd>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. </dd>
+</dl>
+<p><em><b>Optimizer parameters</b></em></p>
+<p><b>FISTA</b> Parameters </p><pre class="syntax">
+ $$
+ max_stepsize = <value>,
+ eta = <value>,
+ use_active_set = <value>,
+ activeset_tolerance = <value>,
+ random_stepsize = <value>
+ $$
+</pre><dl class="arglist">
+<dt>max_stepsize </dt>
+<dd><p class="startdd">Default: 4.0. Initial backtracking step size. At each iteration, the algorithm first tries <em>stepsize = max_stepsize</em>, and if it does not work out, it then tries a smaller step size, <em>stepsize = stepsize/eta</em>, where <em>eta</em> 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 <em>max_stepsize</em> can decrease the computation time by more than 10 times.</p>
+<p class="enddd"></p>
+</dd>
+<dt>eta </dt>
+<dd><p class="startdd">Default: 2.0 If stepsize does not work, <em>stepsize/<em>eta</em> is</em> tried. Must be greater than 1. </p>
+<p class="enddd"></p>
+</dd>
+<dt>use_active_set </dt>
+<dd><p class="startdd">Default: FALSE. If <em>use_active_set</em> 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.</p>
+<p class="enddd"></p>
+</dd>
+<dt>activeset_tolerance </dt>
+<dd><p class="startdd">The value of tolerance used during active set calculation. The default value is the same as the <em>tolerance</em> argument described above. </p>
+<p class="enddd"></p>
+</dd>
+<dt>random_stepsize </dt>
+<dd>Default: FALSE. Whether to add some randomness to the step size. Sometimes, this can speed up the calculation. </dd>
+</dl>
+<p><b>IGD</b> parameters </p><pre class="syntax">
+ $$
+ stepsize = <value>,
+ step_decay = <value>,
+ threshold = <value>,
+ parallel = <value>
+ $$
+</pre> <dl class="arglist">
+<dt>stepsize </dt>
+<dd><p class="startdd">The default is 0.01.</p>
+<p class="enddd"></p>
+</dd>
+<dt>step_decay </dt>
+<dd><p class="startdd">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.</p>
+<p class="enddd"></p>
+</dd>
+<dt>threshold </dt>
+<dd><p class="startdd">Default: 1e-10. When a coefficient is really small, set this coefficient to be 0.</p>
+<p>Due to the stochastic nature of SGD, we can only obtain very small values for the fitting coefficients. Therefore, <em>threshold</em> 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 <em>threshold</em>, set the original coefficient to zero.</p>
+<p class="enddd"></p>
+</dd>
+<dt>parallel </dt>
+<dd><p class="startdd">Whether to run the computation on multiple segments. The default is TRUE.</p>
+<p class="enddd">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 <em>parallel</em> option to allow you to choose whether to do parallel computation. </p>
+</dd>
+</dl>
+<p><a class="anchor" id="predict"></a></p><dl class="section user"><dt>Prediction Function</dt><dd></dd></dl>
+<h4>Per-Tuple Prediction</h4>
+<p>The prediction function returns a double value for the Gaussian family and a Boolean value for the Binomial family.</p>
+<p>The predict function has the following syntax (<a class="el" href="elastic__net_8sql__in.html#a96db4ff4ba3ea363fafbf6c036c19fae" title="Prediction for linear models use learned coefficients for a given example. ">elastic_net_gaussian_predict()</a> and <a class="el" href="elastic__net_8sql__in.html#aa78cde79f1f2caa7c5b38f933001d793" title="Prediction for logistic models use learned coefficients for a given example. ">elastic_net_binomial_predict()</a>): </p><pre class="syntax">
+elastic_net_<family>_predict(
+ coefficients,
+ intercept,
+ ind_var
+ )
+</pre><p><b>Arguments</b> </p><dl class="arglist">
+<dt>coefficients </dt>
+<dd>DOUBLE PRECISION[]. Fitting coefficients, usually <em>coef_all</em> or <em>coef_nonzero</em>. </dd>
+<dt>intercept </dt>
+<dd>DOUBLE PRECISION. Intercept for the model. </dd>
+<dt>ind_var </dt>
+<dd>DOUBLE PRECISION[]. Independent variables that correspond to coefficients. Use <em>features</em> column in <em>tbl_result</em> for <em>coef_all</em>, and <em>features_selected</em> for <em>coef_nonzero</em>. See the <a href="#additional_example">examples for this case below</a>. <dl class="section note"><dt>Note</dt><dd>Unexpected results or errors may be returned in the case that this argument <em>ind_var</em> is not specified properly. </dd></dl>
+</dd>
+</dl>
+<p>For the binomial family, there is a function (<a class="el" href="elastic__net_8sql__in.html#a308718fd5234bc1007b971a639aadf71" title="Compute the probability of belonging to the True class for a given observation. ">elastic_net_binomial_prob()</a>) that outputs the probability of the instance being TRUE: </p><pre class="syntax">
+elastic_net_binomial_prob(
+ coefficients,
+ intercept,
+ ind_var
+ )
+</pre><h4>Per-Table Prediction</h4>
+<p>Alternatively, you can use another prediction function that stores the prediction result in a table (<a class="el" href="elastic__net_8sql__in.html#a3578608204ac9b2d3442ff42977f632b" title="Prediction and put the result in a table can be used together with General-CV. ">elastic_net_predict()</a>). This is useful if you want to use elastic net together with the general cross validation function. </p><pre class="syntax">
+elastic_net_predict( tbl_model,
+ tbl_new_sourcedata,
+ col_id,
+ tbl_predict
+ )
+</pre><p> <b>Arguments</b> </p><dl class="arglist">
+<dt>tbl_model </dt>
+<dd>TEXT. Name of the table containing the output from the training function. </dd>
+<dt>tbl_new_sourcedata </dt>
+<dd>TEXT. Name of the table containing the new source data. </dd>
+<dt>col_id </dt>
+<dd>TEXT. Unique ID associated with each row. </dd>
+<dt>tbl_predict </dt>
+<dd>TEXT. Name of table to store the prediction result. </dd>
+</dl>
+<p>You do not need to specify whether the model is "linear" or "logistic" because this information is already included in the <em>tbl_model</em> table.</p>
+<p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl>
+<ol type="1">
+<li>Display online help for the <a class="el" href="elastic__net_8sql__in.html#a735038a5090c112505c740a90a203e83" title="Interface for elastic net. ">elastic_net_train()</a> function: <pre class="example">
+SELECT madlib.elastic_net_train();
+</pre></li>
+<li>Create an input data set of house prices and features: <pre class="example">
+DROP TABLE IF EXISTS houses;
+CREATE TABLE houses ( id INT,
+ tax INT,
+ bedroom INT,
+ bath FLOAT,
+ price INT,
+ size INT,
+ lot INT,
+ zipcode INT);
+INSERT INTO houses (id, tax, bedroom, bath, price, size, lot, zipcode) VALUES
+(1 , 590 , 2 , 1 , 50000 , 770 , 22100 , 94301),
+(2 , 1050 , 3 , 2 , 85000 , 1410 , 12000 , 94301),
+(3 , 20 , 3 , 1 , 22500 , 1060 , 3500 , 94301),
+(4 , 870 , 2 , 2 , 90000 , 1300 , 17500 , 94301),
+(5 , 1320 , 3 , 2 , 133000 , 1500 , 30000 , 94301),
+(6 , 1350 , 2 , 1 , 90500 , 820 , 25700 , 94301),
+(7 , 2790 , 3 , 2.5 , 260000 , 2130 , 25000 , 94301),
+(8 , 680 , 2 , 1 , 142500 , 1170 , 22000 , 94301),
+(9 , 1840 , 3 , 2 , 160000 , 1500 , 19000 , 94301),
+(10 , 3680 , 4 , 2 , 240000 , 2790 , 20000 , 94301),
+(11 , 1660 , 3 , 1 , 87000 , 1030 , 17500 , 94301),
+(12 , 1620 , 3 , 2 , 118600 , 1250 , 20000 , 94301),
+(13 , 3100 , 3 , 2 , 140000 , 1760 , 38000 , 94301),
+(14 , 2070 , 2 , 3 , 148000 , 1550 , 14000 , 94301),
+(15 , 650 , 3 , 1.5 , 65000 , 1450 , 12000 , 94301),
+(16 , 770 , 2 , 2 , 91000 , 1300 , 17500 , 76010),
+(17 , 1220 , 3 , 2 , 132300 , 1500 , 30000 , 76010),
+(18 , 1150 , 2 , 1 , 91100 , 820 , 25700 , 76010),
+(19 , 2690 , 3 , 2.5 , 260011 , 2130 , 25000 , 76010),
+(20 , 780 , 2 , 1 , 141800 , 1170 , 22000 , 76010),
+(21 , 1910 , 3 , 2 , 160900 , 1500 , 19000 , 76010),
+(22 , 3600 , 4 , 2 , 239000 , 2790 , 20000 , 76010),
+(23 , 1600 , 3 , 1 , 81010 , 1030 , 17500 , 76010),
+(24 , 1590 , 3 , 2 , 117910 , 1250 , 20000 , 76010),
+(25 , 3200 , 3 , 2 , 141100 , 1760 , 38000 , 76010),
+(26 , 2270 , 2 , 3 , 148011 , 1550 , 14000 , 76010),
+(27 , 750 , 3 , 1.5 , 66000 , 1450 , 12000 , 76010);
+</pre></li>
+<li>Train the model: <pre class="example">
+DROP TABLE IF EXISTS houses_en, houses_en_summary;
+SELECT madlib.elastic_net_train( 'houses', -- Source table
+ 'houses_en', -- Result table
+ 'price', -- Dependent variable
+ 'array[tax, bath, size]', -- Independent variable
+ 'gaussian', -- Regression family
+ 0.5, -- Alpha value
+ 0.1, -- Lambda value
+ TRUE, -- Standardize
+ NULL, -- Grouping column(s)
+ 'fista', -- Optimizer
+ '', -- Optimizer parameters
+ NULL, -- Excluded columns
+ 10000, -- Maximum iterations
+ 1e-6 -- Tolerance value
+ );
+</pre></li>
+<li>View the resulting model: <pre class="example">
+-- Turn on expanded display to make it easier to read results.
+\x on
+SELECT * FROM houses_en;
+</pre> Result: <pre class="result">
+-[ RECORD 1 ]-----+-------------------------------------------
+family | gaussian
+features | {tax,bath,size}
+features_selected | {tax,bath,size}
+coef_nonzero | {22.785201806,10707.9664343,54.7959774173}
+coef_all | {22.785201806,10707.9664343,54.7959774173}
+intercept | -7798.71393905
+log_likelihood | -512248641.971
+standardize | t
+iteration_run | 10000
+</pre></li>
+<li>Use the prediction function to evaluate residuals: <pre class="example">
+\x off
+SELECT id, price, predict, price - predict AS residual
+FROM (
+ SELECT
+ houses.*,
+ madlib.elastic_net_gaussian_predict(
+ m.coef_all, -- Coefficients
+ m.intercept, -- Intercept
+ ARRAY[tax,bath,size] -- Features (corresponding to coefficients)
+ ) AS predict
+ FROM houses, houses_en m) s
+ORDER BY id;
+</pre> Result: <pre class="result">
+ id | price | predict | residual
+----+--------+------------------+-------------------
+ 1 | 50000 | 58545.391894031 | -8545.391894031
+ 2 | 85000 | 114804.077663003 | -29804.077663003
+ 3 | 22500 | 61448.835664388 | -38948.835664388
+ 4 | 90000 | 104675.17768007 | -14675.17768007
+ 5 | 133000 | 125887.70644358 | 7112.29355642
+ 6 | 90500 | 78601.843595366 | 11898.156404634
+ 7 | 260000 | 199257.358231079 | 60742.641768921
+ 8 | 142500 | 82514.559377081 | 59985.440622919
+ 9 | 160000 | 137735.93215082 | 22264.06784918
+ 10 | 240000 | 250347.627648647 | -10347.627648647
+ 11 | 87000 | 97172.428263539 | -10172.428263539
+ 12 | 118600 | 119024.150628605 | -424.150628604999
+ 13 | 140000 | 180692.127913358 | -40692.127913358
+ 14 | 148000 | 156424.249824545 | -8424.249824545
+ 15 | 65000 | 102527.938104575 | -37527.938104575
+ 16 | 91000 | 102396.67273637 | -11396.67273637
+ 17 | 132300 | 123609.20149988 | 8690.79850012
+ 18 | 91100 | 74044.833707966 | 17055.166292034
+ 19 | 260011 | 196978.853287379 | 63032.146712621
+ 20 | 141800 | 84793.064320781 | 57006.935679219
+ 21 | 160900 | 139330.88561141 | 21569.11438859
+ 22 | 239000 | 248524.823693687 | -9524.82369368701
+ 23 | 81010 | 95805.325297319 | -14795.325297319
+ 24 | 117910 | 118340.599145495 | -430.599145494998
+ 25 | 141100 | 182970.632857058 | -41870.632857058
+ 26 | 148011 | 160981.259711945 | -12970.259711945
+ 27 | 66000 | 104806.443048275 | -38806.443048275
+</pre></li>
+</ol>
+<h4>Example with Grouping</h4>
+<ol type="1">
+<li>Reuse the houses table above and train the model by grouping on zip code: <pre class="example">
+DROP TABLE IF EXISTS houses_en1, houses_en1_summary;
+SELECT madlib.elastic_net_train( 'houses', -- Source table
+ 'houses_en1', -- Result table
+ 'price', -- Dependent variable
+ 'array[tax, bath, size]', -- Independent variable
+ 'gaussian', -- Regression family
+ 0.5, -- Alpha value
+ 0.1, -- Lambda value
+ TRUE, -- Standardize
+ 'zipcode', -- Grouping column(s)
+ 'fista', -- Optimizer
+ '', -- Optimizer parameters
+ NULL, -- Excluded columns
+ 10000, -- Maximum iterations
+ 1e-6 -- Tolerance value
+ );
+</pre></li>
+<li>View the resulting model with a separate model for each group: <pre class="example">
+-- Turn on expanded display to make it easier to read results.
+\x on
+SELECT * FROM houses_en1;
+</pre> Result: <pre class="result">
+-[ RECORD 1 ]-----+--------------------------------------------
+zipcode | 94301
+family | gaussian
+features | {tax,bath,size}
+features_selected | {tax,bath,size}
+coef_nonzero | {27.0542096962,12351.5244083,47.5833289771}
+coef_all | {27.0542096962,12351.5244083,47.5833289771}
+intercept | -7191.19791597
+log_likelihood | -519199964.967
+standardize | t
+iteration_run | 10000
+-[ RECORD 2 ]-----+--------------------------------------------
+zipcode | 76010
+family | gaussian
+features | {tax,bath,size}
+features_selected | {tax,bath,size}
+coef_nonzero | {15.6325953499,10166.6608469,57.8689916035}
+coef_all | {15.6325953499,10166.6608469,57.8689916035}
+intercept | 513.912201627
+log_likelihood | -538806528.45
+standardize | t
+iteration_run | 10000
+</pre></li>
+<li>Use the prediction function to evaluate residuals: <pre class="example">
+\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;
+</pre></li>
+</ol>
+<p><a class="anchor" id="additional_example"></a></p><h4>Example where coef_nonzero is different from coef_all</h4>
+<ol type="1">
+<li>Reuse the <a href="#examples">houses</a> table above and train the model with alpha=1 (L1) and a large lambda value (30000). <pre class="example">
+DROP TABLE IF EXISTS houses_en2, houses_en2_summary;
+SELECT madlib.elastic_net_train( 'houses', -- Source table
+ 'houses_en2', -- Result table
+ 'price', -- Dependent variable
+ 'array[tax, bath, size]', -- Independent variable
+ 'gaussian', -- Regression family
+ 1, -- Alpha value
+ 30000, -- Lambda value
+ TRUE, -- Standardize
+ NULL, -- Grouping column(s)
+ 'fista', -- Optimizer
+ '', -- Optimizer parameters
+ NULL, -- Excluded columns
+ 10000, -- Maximum iterations
+ 1e-6 -- Tolerance value
+ );
+</pre></li>
+<li>View the resulting model and see coef_nonzero is different from coef_all: <pre class="example">
+-- Turn on expanded display to make it easier to read results.
+\x on
+SELECT * FROM houses_en2;
+</pre> Result: <pre class="result">
+-[ RECORD 1 ]-----+--------------------------------
+family | gaussian
+features | {tax,bath,size}
+features_selected | {tax,size}
+coef_nonzero | {6.94744249834,29.7137297658}
+coef_all | {6.94744249834,0,29.7137297658}
+intercept | 74445.7039382
+log_likelihood | -1635348585.07
+standardize | t
+iteration_run | 151
+</pre></li>
+<li>We can still use the prediction function with <em>coef_all</em> to evaluate residuals: <pre class="example">
+\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;
+</pre></li>
+<li>We can speed up the prediction function with <em>coef_nonzero</em> to evaluate residuals. This requires the user to examine the <em>feature_selected</em> column in the result table to construct the correct set of independent variables to provide to the prediction function: <pre class="example">
+\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;
+</pre> The two queries above will result in same residuals: <pre class="result">
+ id | price | predict | residual
+----+--------+------------------+-------------------
+ 1 | 50000 | 101424.266931887 | -51424.2669318866
+ 2 | 85000 | 123636.877531235 | -38636.877531235
+ 3 | 22500 | 106081.206339915 | -83581.2063399148
+ 4 | 90000 | 119117.827607296 | -29117.8276072958
+ 5 | 133000 | 128186.922684709 | 4813.0773152912
+ 6 | 90500 | 108190.009718915 | -17690.009718915
+ 7 | 260000 | 157119.312909723 | 102880.687090277
+ 8 | 142500 | 113935.028663057 | 28564.9713369428
+ 9 | 160000 | 131799.592783846 | 28200.4072161544
+ 10 | 240000 | 182913.598378673 | 57086.4016213268
+ 11 | 87000 | 116583.600144218 | -29583.6001442184
+ 12 | 118600 | 122842.722992761 | -4242.7229927608
+ 13 | 140000 | 148278.940070862 | -8278.94007086201
+ 14 | 148000 | 134883.191046754 | 13116.8089532462
+ 15 | 65000 | 122046.449722531 | -57046.449722531
+ 16 | 91000 | 118423.083357462 | -27423.0833574618
+ 17 | 132300 | 127492.178434875 | 4807.8215651252
+ 18 | 91100 | 106800.521219247 | -15700.521219247
+ 19 | 260011 | 156424.568659889 | 103586.431340111
+ 20 | 141800 | 114629.772912891 | 27170.2270871088
+ 21 | 160900 | 132285.913758729 | 28614.0862412706
+ 22 | 239000 | 182357.802978806 | 56642.197021194
+ 23 | 81010 | 116166.753594318 | -35156.753594318
+ 24 | 117910 | 122634.299717811 | -4724.29971781059
+ 25 | 141100 | 148973.684320696 | -7873.68432069599
+ 26 | 148011 | 136272.679546422 | 11738.3204535782
+ 27 | 66000 | 122741.193972365 | -56741.193972365
+(27 rows)
+</pre></li>
+</ol>
+<h4>Example with Cross Validation</h4>
+<ol type="1">
+<li>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.) <pre class="example">
+DROP TABLE IF EXISTS houses_en3, houses_en3_summary, houses_en3_cv;
+SELECT madlib.elastic_net_train( 'houses', -- Source table
+ 'houses_en3', -- Result table
+ 'price', -- Dependent variable
+ 'array[tax, bath, size]', -- Independent variable
+ 'gaussian', -- Regression family
+ 0.5, -- Alpha value
+ 0.1, -- Lambda value
+ TRUE, -- Standardize
+ NULL, -- Grouping column(s)
+ 'fista', -- Optimizer
+ $$ n_folds = 3, -- Cross validation parameters
+ validation_result=houses_en3_cv,
+ n_lambdas = 3,
+ alpha = {0, 0.1, 1}
+ $$,
+ NULL, -- Excluded columns
+ 10000, -- Maximum iterations
+ 1e-6 -- Tolerance value
+ );
+SELECT * FROM houses_en3;
+</pre> <pre class="result">
+-[ RECORD 1 ]-----+--------------------------------------------
+family | gaussian
+features | {tax,bath,size}
+features_selected | {tax,bath,size}
+coef_nonzero | {22.4584783679,11657.0825871,52.1622899664}
+coef_all | {22.4584783679,11657.0825871,52.1622899664}
+intercept | -5067.27288499
+log_likelihood | -543193170.15
+standardize | t
+iteration_run | 392
+</pre></li>
+<li>Details of the cross validation: <pre class="example">
+SELECT * FROM houses_en3_cv ORDER BY lambda_value DESC, alpha ASC;
+</pre> <pre class="result">
+alpha | lambda_value | mean | std
+------+--------------+---------------------+--------------------
+ 0 | 100000 | -1.41777698585e+110 | 1.80536123195e+110
+ 0.1 | 100000 | -1.19953054719e+107 | 1.72846143163e+107
+ 1 | 100000 | -4175743937.91 | 2485189261.38
+ 0 | 100 | -4054694238.18 | 2424765457.66
+ 0.1 | 100 | -4041768667.28 | 2418294966.72
+ 1 | 100 | -1458791218.11 | 483327430.802
+ 0 | 0.1 | -1442293698.38 | 426795110.876
+ 0.1 | 0.1 | -1442705511.6 | 429680202.16
+| 1 | 0.1 | -1459206061.39 | 485107796.02
+(9 rows)
+</pre></li>
+</ol>
+<p><a class="anchor" id="notes"></a></p><dl class="section user"><dt>Note</dt><dd>It is <b>strongly</b> <b>recommended</b> that you run <code><a class="el" href="elastic__net_8sql__in.html#a735038a5090c112505c740a90a203e83" title="Interface for elastic net. ">elastic_net_train()</a></code> on a subset of the data with a limited <em>max_iter</em> before applying it to the full data set with a large <em>max_iter</em>. 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.</dd></dl>
+<p><a class="anchor" id="background"></a></p><dl class="section user"><dt>Technical Background</dt><dd></dd></dl>
+<p>Elastic net regularization seeks to find a weight vector that, for any given training example set, minimizes: </p><p class="formulaDsp">
+<img class="formulaDsp" alt="\[\min_{w \in R^N} L(w) + \lambda \left(\frac{(1-\alpha)}{2} \|w\|_2^2 + \alpha \|w\|_1 \right)\]" src="form_82.png"/>
+</p>
+<p> where <img class="formulaInl" alt="$L$" src="form_83.png"/> is the metric function that the user wants to minimize. Here <img class="formulaInl" alt="$ \alpha \in [0,1] $" src="form_84.png"/> and <img class="formulaInl" alt="$ lambda \geq 0 $" src="form_85.png"/>. If <img class="formulaInl" alt="$alpha = 0$" src="form_86.png"/>, we have the ridge regularization (known also as Tikhonov regularization), and if <img class="formulaInl" alt="$\alpha = 1$" src="form_87.png"/>, we have the LASSO regularization.</p>
+<p>For the Gaussian response family (or linear model), we have </p><p class="formulaDsp">
+<img class="formulaDsp" alt="\[L(\vec{w}) = \frac{1}{2}\left[\frac{1}{M} \sum_{m=1}^M (w^{t} x_m + w_{0} - y_m)^2 \right] \]" src="form_88.png"/>
+</p>
+<p>For the Binomial response family (or logistic model), we have </p><p class="formulaDsp">
+<img class="formulaDsp" alt="\[ L(\vec{w}) = \sum_{m=1}^M\left[y_m \log\left(1 + e^{-(w_0 + \vec{w}\cdot\vec{x}_m)}\right) + (1-y_m) \log\left(1 + e^{w_0 + \vec{w}\cdot\vec{x}_m}\right)\right]\ , \]" src="form_89.png"/>
+</p>
+<p> where <img class="formulaInl" alt="$y_m \in {0,1}$" src="form_90.png"/>.</p>
+<p>To get better convergence, one can rescale the value of each element of x </p><p class="formulaDsp">
+<img class="formulaDsp" alt="\[ x' \leftarrow \frac{x - \bar{x}}{\sigma_x} \]" src="form_91.png"/>
+</p>
+<p> and for Gaussian case we also let </p><p class="formulaDsp">
+<img class="formulaDsp" alt="\[y' \leftarrow y - \bar{y} \]" src="form_92.png"/>
+</p>
+<p> 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.</p>
+<p>Note that fitting after scaling is not equivalent to directly fitting.</p>
+<p><a class="anchor" id="literature"></a></p><dl class="section user"><dt>Literature</dt><dd></dd></dl>
+<p>[1] Elastic net regularization, <a href="http://en.wikipedia.org/wiki/Elastic_net_regularization">http://en.wikipedia.org/wiki/Elastic_net_regularization</a></p>
+<p>[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.</p>
+<p>[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.</p>
+<p>[4] Stochastic gradient descent, <a href="https://en.wikipedia.org/wiki/Stochastic_gradient_descent">https://en.wikipedia.org/wiki/Stochastic_gradient_descent</a></p>
+<p><a class="anchor" id="related"></a></p><dl class="section user"><dt>Related Topics</dt><dd></dd></dl>
+<p>File <a class="el" href="elastic__net_8sql__in.html" title="SQL functions for elastic net regularization. ">elastic_net.sql_in</a> documenting the SQL functions. </p>
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