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[32/51] [abbrv] [partial] madlib-site git commit: Additional updates
for 1.13 release
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+<title>MADlib: Conjugate Gradient</title>
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+ <div id="projectname">
+ <span id="projectnumber">1.13</span>
+ </div>
+ <div id="projectbrief">User Documentation for MADlib</div>
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+ <div class="headertitle">
+<div class="title">Conjugate Gradient<div class="ingroups"><a class="el" href="group__grp__early__stage.html">Early Stage Development</a></div></div> </div>
+</div><!--header-->
+<div class="contents">
+<div class="toc"><b>Contents</b> <ul>
+<li>
+<a href="#syntax">Function Syntax</a> </li>
+<li>
+<a href="#examples">Examples</a> </li>
+<li>
+<a href="#related">Related Topics</a> </li>
+</ul>
+</div><dl class="section warning"><dt>Warning</dt><dd><em> This MADlib method is still in early stage development. There may be some issues that will be addressed in a future version. Interface and implementation is subject to change. </em></dd></dl>
+<p>This function uses the iterative conjugate gradient method [1] to find a solution to the function: </p><p class="formulaDsp">
+\[ \boldsymbol Ax = \boldsymbol b \]
+</p>
+<p> where \( \boldsymbol A \) is a symmetric, positive definite matrix and \(x\) and \( \boldsymbol b \) are vectors.</p>
+<p><a class="anchor" id="syntax"></a></p><dl class="section user"><dt>Function Syntax</dt><dd>Conjugate gradient returns x as an array. It has the following syntax.</dd></dl>
+<pre class="syntax">
+conjugate_gradient( table_name,
+ name_of_row_values_col,
+ name_of_row_number_col,
+ aray_of_b_values,
+ desired_precision
+ )
+</pre><p>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: </p><pre>{TABLE|VIEW} <em>matrix_A</em> (
+ <em>row_number</em> FLOAT,
+ <em>row_values</em> FLOAT[],
+)</pre><p> The number of elements in each row should be the same.</p>
+<p>\( \boldsymbol b \) is passed as a FLOAT[] to the function.</p>
+<p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd><ol type="1">
+<li>Construct matrix A according to structure. <pre class="example">
+SELECT * FROM data;
+</pre> Result: <pre class="result">
+ row_num | row_val
+ --------+---------
+ 1 | {2,1}
+ 2 | {1,4}
+(2 rows)
+</pre></li>
+<li>Call the conjugate gradient function. <pre class="example">
+SELECT conjugate_gradient( 'data',
+ 'row_val',
+ 'row_num',
+ '{2,1}',
+ 1E-6,1
+ );
+</pre> <pre class="result">
+INFO: COMPUTE RESIDUAL ERROR 14.5655661859659
+INFO: ERROR 0.144934004246004
+INFO: ERROR 3.12963615962926e-31
+INFO: TEST FINAL ERROR 2.90029642185163e-29
+ conjugate_gradient
+ --------------------------
+ {1,-1.31838984174237e-15}
+(1 row)
+</pre></li>
+</ol>
+</dd></dl>
+<p><a class="anchor" id="literature"></a></p><dl class="section user"><dt>Literature</dt><dd>[1] "Conjugate gradient method" Wikipedia - <a href="http://en.wikipedia.org/wiki/Conjugate_gradient_method">http://en.wikipedia.org/wiki/Conjugate_gradient_method</a></dd></dl>
+<p><a class="anchor" id="related"></a></p><dl class="section user"><dt>Related Topics</dt><dd>File <a class="el" href="conjugate__gradient_8sql__in.html" title="SQL function computing Conjugate Gradient. ">conjugate_gradient.sql_in</a> documenting the SQL function. </dd></dl>
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+ <img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.13 </li>
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+<title>MADlib: Clustered Variance</title>
+<link href="tabs.css" rel="stylesheet" type="text/css"/>
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+<div class="header">
+ <div class="headertitle">
+<div class="title">Clustered Variance<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>
+<a href="#train_linregr">Clustered Variance Linear Regression Training Function</a> </li>
+<li>
+<a href="#train_logregr">Clustered Variance Logistic Regression Training Function</a> </li>
+<li>
+<a href="#train_mlogregr">Clustered Variance Multinomial Logistic Regression Training Function</a> </li>
+<li>
+<a href="#train_cox">Clustered Variance for Cox Proportional Hazards model</a> </li>
+<li>
+<a href="#examples">Examples</a> </li>
+<li>
+<a href="#notes">Notes</a> </li>
+<li>
+<a href="#background">Technical Background</a> </li>
+<li>
+<a href="#related">Related Topics</a> </li>
+</ul>
+</div><p>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.</p>
+<p>The MADlib Clustered Variance module includes functions to calculate linear, logistic, and multinomial logistic regression problems.</p>
+<p><a class="anchor" id="train_linregr"></a></p><dl class="section user"><dt>Clustered Variance Linear Regression Training Function</dt><dd></dd></dl>
+<p>The clustered variance linear regression training function has the following syntax. </p><pre class="syntax">
+clustered_variance_linregr ( source_table,
+ out_table,
+ dependent_varname,
+ independent_varname,
+ clustervar,
+ grouping_cols
+ )
+</pre><p> <b>Arguments</b> </p><dl class="arglist">
+<dt>source_table </dt>
+<dd><p class="startdd">TEXT. The name of the table containing the input data.</p>
+<p class="enddd"></p>
+</dd>
+<dt>out_table </dt>
+<dd><p class="startdd">VARCHAR. Name of the generated table containing the output model. The output table contains the following columns. </p><table class="output">
+<tr>
+<th>coef </th><td>DOUBLE PRECISION[]. Vector of the coefficients of the regression. </td></tr>
+<tr>
+<th>std_err </th><td>DOUBLE PRECISION[]. Vector of the standard error of the coefficients. </td></tr>
+<tr>
+<th>t_stats </th><td>DOUBLE PRECISION[]. Vector of the t-stats of the coefficients. </td></tr>
+<tr>
+<th>p_values </th><td>DOUBLE PRECISION[]. Vector of the p-values of the coefficients. </td></tr>
+</table>
+<p>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.</p>
+<p></p>
+<p class="enddd"></p>
+</dd>
+<dt>dependent_varname </dt>
+<dd>TEXT. An expression to evaluate for the dependent variable. </dd>
+<dt>independent_varname </dt>
+<dd>TEXT. An Expression to evalue for the independent variables. </dd>
+<dt>clustervar </dt>
+<dd>TEXT. A comma-separated list of the columns to use as cluster variables. </dd>
+<dt>grouping_cols (optional) </dt>
+<dd>TEXT, default: NULL. <em>Not currently implemented. Any non-NULL value is ignored.</em> 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. </dd>
+</dl>
+<p><a class="anchor" id="train_logregr"></a></p><dl class="section user"><dt>Clustered Variance Logistic Regression Training Function</dt><dd></dd></dl>
+<p>The clustered variance logistic regression training function has the following syntax. </p><pre class="syntax">
+clustered_variance_logregr( source_table,
+ out_table,
+ dependent_varname,
+ independent_varname,
+ clustervar,
+ grouping_cols,
+ max_iter,
+ optimizer,
+ tolerance,
+ verbose_mode
+ )
+</pre><p> <b>Arguments</b> </p><dl class="arglist">
+<dt>source_table </dt>
+<dd>TEXT. The name of the table containing the input data. </dd>
+<dt>out_table </dt>
+<dd><p class="startdd">VARCHAR. Name of the generated table containing the output model. The output table has the following columns: </p><table class="output">
+<tr>
+<th>coef </th><td>Vector of the coefficients of the regression. </td></tr>
+<tr>
+<th>std_err </th><td>Vector of the standard error of the coefficients. </td></tr>
+<tr>
+<th>z_stats </th><td>Vector of the z-stats of the coefficients. </td></tr>
+<tr>
+<th>p_values </th><td>Vector of the p-values of the coefficients. </td></tr>
+</table>
+<p>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.</p>
+<p class="enddd"></p>
+</dd>
+<dt>dependent_varname </dt>
+<dd>TEXT. An expression to evaluate for the dependent variable. </dd>
+<dt>independent_varname </dt>
+<dd>TEXT. An expression to evaluate for the independent variable. </dd>
+<dt>clustervar </dt>
+<dd>TEXT. A comma-separated list of columns to use as cluster variables. </dd>
+<dt>grouping_cols (optional) </dt>
+<dd>TEXT, default: NULL. <em>Not yet implemented. Any non-NULL values are ignored.</em> 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. </dd>
+<dt>max_iter (optional) </dt>
+<dd>INTEGER, default: 20. The maximum number of iterations that are allowed. </dd>
+<dt>optimizer (optional) </dt>
+<dd>TEXT, default: 'irls'. The name of the optimizer to use: <ul>
+<li>
+'newton' or 'irls': Iteratively reweighted least squares </li>
+<li>
+'cg': conjugate gradient </li>
+<li>
+'igd': incremental gradient descent. </li>
+</ul>
+</dd>
+<dt>tolerance (optional) </dt>
+<dd>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 <em>n</em> Iterations have completed. </dd>
+<dt>verbose_mode (optional) </dt>
+<dd>BOOLEAN, default FALSE. Provides verbose_mode output of the results of training. </dd>
+</dl>
+<p><a class="anchor" id="train_mlogregr"></a></p><dl class="section user"><dt>Clustered Variance Multinomial Logistic Regression Training Function</dt><dd></dd></dl>
+<pre class="syntax">
+clustered_variance_mlogregr( source_table,
+ out_table,
+ dependent_varname,
+ independent_varname,
+ cluster_varname,
+ ref_category,
+ grouping_cols,
+ optimizer_params,
+ verbose_mode
+ )
+</pre><p> <b>Arguments</b> </p><dl class="arglist">
+<dt>source_table </dt>
+<dd>TEXT. The name of the table containing the input data. </dd>
+<dt>out_table </dt>
+<dd><p class="startdd">TEXT. The name of the table where the regression model will be stored. The output table has the following columns: </p><table class="output">
+<tr>
+<th>category </th><td>The category. </td></tr>
+<tr>
+<th>ref_category </th><td>The refererence category used for modeling. </td></tr>
+<tr>
+<th>coef </th><td>Vector of the coefficients of the regression. </td></tr>
+<tr>
+<th>std_err </th><td>Vector of the standard error of the coefficients. </td></tr>
+<tr>
+<th>z_stats </th><td>Vector of the z-stats of the coefficients. </td></tr>
+<tr>
+<th>p_values </th><td>Vector of the p-values of the coefficients. </td></tr>
+</table>
+<p class="enddd">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. </p>
+</dd>
+<dt>dependent_varname </dt>
+<dd>TEXT. An expression to evaluate for the dependent variable. </dd>
+<dt>independent_varname </dt>
+<dd>TEXT. An expression to evaluate for the independent variable. </dd>
+<dt>cluster_varname </dt>
+<dd>TEXT. A comma-separated list of columns to use as cluster variables. </dd>
+<dt>ref_category (optional) </dt>
+<dd>INTEGER. Reference category in the range [0, num_category). </dd>
+<dt>groupingvarng_cols (optional) </dt>
+<dd>TEXT, default: NULL. <em>Not yet implemented. Any non-NULL values are ignored.</em> A comma-separated list of columns to use as grouping variables. </dd>
+<dt>optimizer_params (optional) </dt>
+<dd>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. </dd>
+<dt>verbose_mode (optional) </dt>
+<dd>BOOLEAN, default FALSE. If TRUE, detailed information is printed when computing logistic regression. </dd>
+</dl>
+<p><a class="anchor" id="train_cox"></a></p><dl class="section user"><dt>Clustered Variance for Cox Proportional Hazards model</dt><dd></dd></dl>
+<p>The clustered robust variance estimator function for the Cox Proportional Hazards model has the following syntax. </p><pre class="syntax">
+clustered_variance_coxph(model_table, output_table, clustervar)
+</pre><p><b>Arguments</b> </p><dl class="arglist">
+<dt>model_table </dt>
+<dd>TEXT. The name of the model table, which is exactaly the same as the 'output_table' parameter of <a class="el" href="cox__prop__hazards_8sql__in.html#a737450bbfe0f10204b0074a9d45b0cef" title="Compute cox-regression coefficients and diagnostic statistics. ">coxph_train()</a> function. </dd>
+<dt>output_table </dt>
+<dd>TEXT. The name of the table where the output is saved. It has the following columns: <table class="output">
+<tr>
+<th>coef </th><td>FLOAT8[]. Vector of the coefficients. </td></tr>
+<tr>
+<th>loglikelihood </th><td>FLOAT8. Log-likelihood value of the MLE estimate. </td></tr>
+<tr>
+<th>std_err </th><td>FLOAT8[]. Vector of the standard error of the coefficients. </td></tr>
+<tr>
+<th>clustervar </th><td>TEXT. A comma-separated list of columns to use as cluster variables. </td></tr>
+<tr>
+<th>clustered_se </th><td>FLOAT8[]. Vector of the robust standard errors of the coefficients. </td></tr>
+<tr>
+<th>clustered_z </th><td>FLOAT8[]. Vector of the robust z-stats of the coefficients. </td></tr>
+<tr>
+<th>clustered_p </th><td>FLOAT8[]. Vector of the robust p-values of the coefficients. </td></tr>
+<tr>
+<th>hessian </th><td>FLOAT8[]. The Hessian matrix. </td></tr>
+</table>
+</dd>
+<dt>clustervar </dt>
+<dd>TEXT. A comma-separated list of columns to use as cluster variables. </dd>
+</dl>
+<p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl>
+<ol type="1">
+<li>View online help for the clustered variance linear regression function. <pre class="example">
+SELECT madlib.clustered_variance_linregr();
+</pre></li>
+<li>Run the linear regression function and view the results. <pre class="example">
+DROP TABLE IF EXISTS out_table;
+SELECT madlib.clustered_variance_linregr( 'abalone',
+ 'out_table',
+ 'rings',
+ 'ARRAY[1, diameter, length, width]',
+ 'sex',
+ NULL
+ );
+SELECT * FROM out_table;
+</pre></li>
+<li>View online help for the clustered variance logistic regression function. <pre class="example">
+SELECT madlib.clustered_variance_logregr();
+</pre></li>
+<li>Run the logistic regression function and view the results. <pre class="example">
+DROP TABLE IF EXISTS out_table;
+SELECT madlib.clustered_variance_logregr( 'abalone',
+ 'out_table',
+ 'rings < 10',
+ 'ARRAY[1, diameter, length, width]',
+ 'sex'
+ );
+SELECT * FROM out_table;
+</pre></li>
+<li>View online help for the clustered variance multinomial logistic regression function. <pre class="example">
+SELECT madlib.clustered_variance_mlogregr();
+</pre></li>
+<li>Run the multinomial logistic regression and view the results. <pre class="example">
+DROP TABLE IF EXISTS out_table;
+SELECT madlib.clustered_variance_mlogregr( 'abalone',
+ 'out_table',
+ 'CASE WHEN rings < 10 THEN 1 ELSE 0 END',
+ 'ARRAY[1, diameter, length, width]',
+ 'sex',
+ 0
+ );
+SELECT * FROM out_table;
+</pre></li>
+<li>Run the Cox Proportional Hazards regression and compute the clustered robust estimator. <pre class="example">
+DROP TABLE IF EXISTS lung_cl_out;
+DROP TABLE IF EXISTS lung_out;
+DROP TABLE IF EXISTS lung_out_summary;
+SELECT madlib.coxph_train('lung',
+ 'lung_out',
+ 'time',
+ 'array[age, "ph.ecog"]',
+ 'TRUE',
+ NULL,
+ NULL);
+SELECT madlib.clustered_variance_coxph('lung_out',
+ 'lung_cl_out',
+ '"ph.karno"');
+SELECT * FROM lung_cl_out;
+</pre></li>
+</ol>
+<p><a class="anchor" id="notes"></a></p><dl class="section user"><dt>Notes</dt><dd></dd></dl>
+<ul>
+<li>Note that we need to manually include an intercept term in the independent variable expression. The NULL value of <em>groupingvar</em> means that there is no grouping in the calculation.</li>
+</ul>
+<p><a class="anchor" id="background"></a></p><dl class="section user"><dt>Technical Background</dt><dd></dd></dl>
+<p>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.</p>
+<p>The estimator has a similar form to the usual sandwich estimator </p><p class="formulaDsp">
+\[ S(\vec{c}) = B(\vec{c}) M(\vec{c}) B(\vec{c}) \]
+</p>
+<p>The bread part is the same as Huber-White sandwich estimator </p><p class="formulaDsp">
+\begin{eqnarray} B(\vec{c}) & = & \left(-\sum_{i=1}^{n} H(y_i, \vec{x}_i, \vec{c})\right)^{-1}\\ & = & \left(-\sum_{i=1}^{n}\frac{\partial^2 l(y_i, \vec{x}_i, \vec{c})}{\partial c_\alpha \partial c_\beta}\right)^{-1} \end{eqnarray}
+</p>
+<p> where \(H\) is the hessian matrix, which is the second derivative of the target function </p><p class="formulaDsp">
+\[ L(\vec{c}) = \sum_{i=1}^n l(y_i, \vec{x}_i, \vec{c})\ . \]
+</p>
+<p>The meat part is different </p><p class="formulaDsp">
+\[ M(\vec{c}) = \bf{A}^T\bf{A} \]
+</p>
+<p> where the \(m\)-th row of \(\bf{A}\) is </p><p class="formulaDsp">
+\[ A_m = \sum_{i\in G_m}\frac{\partial l(y_i,\vec{x}_i,\vec{c})}{\partial \vec{c}} \]
+</p>
+<p> where \(G_m\) is the set of rows that belong to the same cluster.</p>
+<p>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.</p>
+<p>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 <em>marginal_mlogregr</em>. This is deliberate because the interfaces of all multinomial regressions (robust, clustered, ...) will be moved to match that used in marginal.</p>
+<p><a class="anchor" id="literature"></a></p><dl class="section user"><dt>Literature</dt><dd></dd></dl>
+<p>[1] Standard, Robust, and Clustered Standard Errors Computed in R, <a href="http://diffuseprior.wordpress.com/2012/06/15/standard-robust-and-clustered-standard-errors-computed-in-r/">http://diffuseprior.wordpress.com/2012/06/15/standard-robust-and-clustered-standard-errors-computed-in-r/</a></p>
+<p><a class="anchor" id="related"></a></p><dl class="section user"><dt>Related Topics</dt><dd>File <a class="el" href="clustered__variance_8sql__in.html">clustered_variance.sql_in</a> documenting the clustered variance SQL functions.</dd></dl>
+<p>File <a class="el" href="clustered__variance__coxph_8sql__in.html" title="SQL functions for clustered robust cox proportional hazards regression. ">clustered_variance_coxph.sql_in</a> documenting the clustered variance for Cox proportional hazards SQL functions.</p>
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+
+<div class="header">
+ <div class="summary">
+<a href="#groups">Modules</a> </div>
+ <div class="headertitle">
+<div class="title">Clustering<div class="ingroups"><a class="el" href="group__grp__unsupervised.html">Unsupervised Learning</a></div></div> </div>
+</div><!--header-->
+<div class="contents">
+<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
+<p>A collection of methods for clustering data </p>
+<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__kmeans"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__kmeans.html">k-Means Clustering</a></td></tr>
+<tr class="memdesc:group__grp__kmeans"><td class="mdescLeft"> </td><td class="mdescRight">Partitions a set of observations into clusters by finding centroids that minimize the sum of observations' distances from their closest centroid. <br /></td></tr>
+<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
+</table>
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+<div class="title">Pearson's Correlation<div class="ingroups"><a class="el" href="group__grp__stats.html">Statistics</a> » <a class="el" href="group__grp__desc__stats.html">Descriptive Statistics</a></div></div> </div>
+</div><!--header-->
+<div class="contents">
+<div class="toc"><b>Contents</b> <ul>
+<li>
+<a href="#usage">Correlation Function</a> </li>
+<li>
+<a href="#examples">Examples</a> </li>
+<li>
+<a href="#literature">Literature</a> </li>
+<li>
+<a href="#related">Related Topics</a> </li>
+</ul>
+</div><p>A correlation function is the degree and direction of association of two variables—how well one random variable can be predicted from the other. The coefficient of correlation varies from -1 to 1. A coefficient of 1 implies perfect correlation, 0 means no correlation, and -1 means perfect anti-correlation.</p>
+<p>This function provides a cross-correlation matrix for all pairs of numeric columns in a <em>source_table</em>. A correlation matrix describes correlation among \( M \) variables. It is a square symmetrical \( M \)x \(M \) matrix with the \( (ij) \)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.</p>
+<p><a class="anchor" id="usage"></a></p><dl class="section user"><dt>Correlation Function</dt><dd></dd></dl>
+<p>The correlation function has the following syntax: </p><pre class="syntax">
+correlation( source_table,
+ output_table,
+ target_cols,
+ verbose
+ )
+</pre><p>The covariance function, with a similar syntax, can be used to compute the covariance between features. </p><pre class="syntax">
+covariance( source_table,
+ output_table,
+ target_cols,
+ verbose
+ )
+</pre><dl class="arglist">
+<dt>source_table </dt>
+<dd><p class="startdd">TEXT. The name of the data containing the input data.</p>
+<p class="enddd"></p>
+</dd>
+<dt>output_table </dt>
+<dd><p class="startdd">TEXT. The name of the table where the cross-correlation matrix will be saved. The output is a table with N+2 columns and N rows, where N is the number of target columns. It contains the following columns. </p><table class="output">
+<tr>
+<th>column_position </th><td>The first column is a sequential counter indicating the position of the variable in the '<em>output_table</em>'. </td></tr>
+<tr>
+<th>variable </th><td>The second column contains the row-header for the variables. </td></tr>
+<tr>
+<th><...> </th><td>The remainder of the table is the NxN correlation matrix for the pairs of numeric 'source_table' columns. </td></tr>
+</table>
+<p>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 '<em>output_table</em>' in this matrix format ensure to order the elements using the '<em>column_position</em>', as shown in the example below. </p><pre class="example">
+SELECT * FROM output_table ORDER BY column_position;
+</pre><p>In addition to output table, a summary table named <output_table>_summary is also created at the same time, which has the following columns: </p><table class="output">
+<tr>
+<th>method</th><td>'correlation' </td></tr>
+<tr>
+<th>source_table</th><td>VARCHAR. The data source table name. </td></tr>
+<tr>
+<th>output_table</th><td>VARCHAR. The output table name. </td></tr>
+<tr>
+<th>column_names</th><td>VARCHAR. Column names used for correlation computation, comma-separated string. </td></tr>
+<tr>
+<th>mean_vector</th><td>FLOAT8[]. Vector where each is the mean of a column. </td></tr>
+<tr>
+<th>total_rows_processed </th><td>BIGINT. Total numbers of rows processed. </td></tr>
+</table>
+<p class="enddd"></p>
+</dd>
+<dt>target_cols (optional) </dt>
+<dd><p class="startdd">TEXT, default: '*'. A comma-separated list of the columns to correlate. If NULL or <code>'*'</code>, results are produced for all numeric columns.</p>
+<p class="enddd"></p>
+</dd>
+<dt>verbose (optional) </dt>
+<dd><p class="startdd">BOOLEAN, default: FALSE. Print verbose debugging information if TRUE.</p>
+<p class="enddd"></p>
+</dd>
+</dl>
+<p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl>
+<ol type="1">
+<li>View online help for the correlation function. <pre class="example">
+SELECT madlib.correlation();
+</pre></li>
+<li>Create an input data set. <pre class="example">
+DROP TABLE IF EXISTS example_data;
+CREATE TABLE example_data(
+ id SERIAL, outlook TEXT,
+ temperature FLOAT8, humidity FLOAT8,
+ windy TEXT, class TEXT);
+INSERT INTO example_data VALUES
+(1, 'sunny', 85, 85, 'false', 'Dont Play'),
+(2, 'sunny', 80, 90, 'true', 'Dont Play'),
+(3, 'overcast', 83, 78, 'false', 'Play'),
+(4, 'rain', 70, 96, 'false', 'Play'),
+(5, 'rain', 68, 80, 'false', 'Play'),
+(6, 'rain', 65, 70, 'true', 'Dont Play'),
+(7, 'overcast', 64, 65, 'true', 'Play'),
+(8, 'sunny', 72, 95, 'false', 'Dont Play'),
+(9, 'sunny', 69, 70, 'false', 'Play'),
+(10, 'rain', 75, 80, 'false', 'Play'),
+(11, 'sunny', 75, 70, 'true', 'Play'),
+(12, 'overcast', 72, 90, 'true', 'Play'),
+(13, 'overcast', 81, 75, 'false', 'Play'),
+(14, 'rain', 71, 80, 'true', 'Dont Play'),
+(15, NULL, 100, 100, 'true', NULL),
+(16, NULL, 110, 100, 'true', NULL);
+</pre></li>
+<li>Run the <a class="el" href="correlation_8sql__in.html#ada17a10ea8a6c4580e7413c86ae5345e">correlation()</a> function on the data set. <pre class="example">
+-- Correlate all numeric columns
+SELECT madlib.correlation( 'example_data',
+ 'example_data_output'
+ );
+-- Setting target_cols to NULL or '*' also correlates all numeric columns
+SELECT madlib.correlation( 'example_data',
+ 'example_data_output',
+ '*'
+ );
+-- Correlate only the temperature and humidity columns
+SELECT madlib.correlation( 'example_data',
+ 'example_data_output',
+ 'temperature, humidity'
+ );
+</pre></li>
+<li>View the correlation matrix. <pre class="example">
+SELECT * FROM example_data_output ORDER BY column_position;
+</pre> Result: <pre class="result">
+ column_position | variable | temperature | humidity
+-----------------+-------------+-------------------+----------
+ 1 | temperature | 1.0 |
+ 2 | humidity | 0.616876934548786 | 1.0
+(2 rows)
+</pre></li>
+<li>Compute the covariance of features in the data set. <pre class="example">
+SELECT madlib.covariance( 'example_data',
+ 'cov_output'
+ );
+</pre></li>
+<li>View the covariance matrix. <pre class="example">
+SELECT * FROM cov_output ORDER BY column_position;
+</pre> Result: <pre class="result">
+ column_position | variable | temperature | humidity
+-----------------+-------------+-------------------+----------
+ 1 | temperature | 146.25 |
+ 2 | humidity | 82.125 | 121.1875
+(2 rows)
+</pre></li>
+</ol>
+<dl class="section user"><dt>Notes</dt><dd></dd></dl>
+<p>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.</p>
+<p>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.</p>
+<p><a class="anchor" id="literature"></a></p><dl class="section user"><dt>Literature</dt><dd></dd></dl>
+<p>[1] <a href="https://en.wikipedia.org/wiki/Imputation_(statistics)">https://en.wikipedia.org/wiki/Imputation_(statistics)</a></p>
+<p>[2] <a href="https://www.iriseekhout.com/missing-data/missing-data-methods/imputation-methods/">https://www.iriseekhout.com/missing-data/missing-data-methods/imputation-methods/</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="correlation_8sql__in.html" title="SQL functions for correlation computation. ">correlation.sql_in</a> documenting the SQL functions</p>
+<p><a class="el" href="group__grp__summary.html">Summary</a> for general descriptive statistics for a table </p>
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+<div class="title">CountMin (Cormode-Muthukrishnan)<div class="ingroups"><a class="el" href="group__grp__stats.html">Statistics</a> » <a class="el" href="group__grp__desc__stats.html">Descriptive Statistics</a> » <a class="el" href="group__grp__sketches.html">Cardinality Estimators</a></div></div> </div>
+</div><!--header-->
+<div class="contents">
+<div class="toc"><b>Contents</b> <ul>
+<li>
+<a href="#syntax">Syntax</a> </li>
+<li>
+<a href="#examples">Examples</a> </li>
+<li>
+<a href="#literature">Literature</a> </li>
+<li>
+<a href="#related">Related Topics</a> </li>
+</ul>
+</div><p>This module implements Cormode-Muthukrishnan <em>CountMin</em> 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.</p>
+<p><a class="anchor" id="syntax"></a></p><dl class="section user"><dt>Syntax</dt><dd><ul>
+<li>Get a sketch of a selected column specified by <em>col_name</em>. <pre class="syntax">
+cmsketch( col_name )
+</pre></li>
+<li>Get the number of rows where <em>col_name = p</em>, computed from the sketch obtained from <code>cmsketch</code>. <pre class="syntax">
+cmsketch_count( cmsketch,
+ p )
+</pre></li>
+<li>Get the number of rows where <em>col_name</em> is between <em>m</em> and <em>n</em> inclusive. <pre class="syntax">
+cmsketch_rangecount( cmsketch,
+ m,
+ n )
+</pre></li>
+<li>Get the <em>k</em>th percentile of <em>col_name</em> where <em>count</em> specifies number of rows. <em>k</em> should be an integer between 1 to 99. <pre class="syntax">
+cmsketch_centile( cmsketch,
+ k,
+ count )
+</pre></li>
+<li>Get the median of col_name where <em>count</em> specifies number of rows. This is equivalent to <code><a class="el" href="sketch_8sql__in.html#a2f2ab2fe3244515f5f73d49690e73b39">cmsketch_centile</a>(<em>cmsketch</em>,50,<em>count</em>)</code>. <pre class="syntax">
+cmsketch_median( cmsketch,
+ count )
+</pre></li>
+<li>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. <pre class="syntax">
+cmsketch_width_histogram( cmsketch,
+ min,
+ max,
+ n )
+</pre></li>
+<li>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. <pre class="syntax">
+cmsketch_depth_histogram( cmsketch,
+ n )
+</pre></li>
+</ul>
+</dd></dl>
+<dl class="section note"><dt>Note</dt><dd>This is a <a href="https://www.postgresql.org/docs/current/static/xaggr.html">User Defined Aggregate</a> 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.</dd></dl>
+<p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl>
+<ol type="1">
+<li>Generate some data. <pre class="example">
+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);
+</pre></li>
+<li>Count number of rows where a1 = 2 in each class. Store results in a table. <pre class="example">
+CREATE TABLE sketch_count AS
+SELECT class,
+ cmsketch_count( cmsketch( a1 ), 2 )
+FROM data GROUP BY data.class;
+SELECT * FROM sketch_count;
+</pre> Result: <pre class="result">
+ class | cmsketch_count
+ ------+----------------
+ 2 | 0
+ 1 | 15000
+(2 rows)
+</pre></li>
+<li>Count number of rows where a1 is between 3 and 6. <pre class="example">
+SELECT class,
+ cmsketch_rangecount( cmsketch(a1), 3, 6 )
+FROM data GROUP BY data.class;
+</pre> Result: <pre class="result">
+ class | cmsketch_rangecount
+ ------+---------------------
+ 2 | 2000
+ 1 | 10000
+(2 rows)
+</pre></li>
+<li>Compute the 90th percentile of all of a1. <pre class="example">
+SELECT cmsketch_centile( cmsketch( a1 ), 90, count(*) )
+FROM data;
+</pre> Result: <pre class="result">
+ cmsketch_centile
+ -----------------
+ 3
+(1 row)
+</pre></li>
+<li>Produce an equi-width histogram with 2 bins between 0 and 10. <pre class="example">
+SELECT cmsketch_width_histogram( cmsketch( a1 ), 0, 10, 2 )
+FROM data;
+</pre> Result: <pre class="result">
+ cmsketch_width_histogram
+ -----------------------------------
+ [[0L, 4L, 35000], [5L, 10L, 2000]]
+(1 row)
+</pre></li>
+<li>Produce an equi-depth histogram of a1 with 2 bins of approximately equal depth. <pre class="example">
+SELECT cmsketch_depth_histogram( cmsketch( a1 ), 2 )
+FROM data;
+</pre> Result: <pre class="result">
+ cmsketch_depth_histogram
+ ----------------------------------------------------------------------
+ [[-9223372036854775807L, 1, 10000], [2, 9223372036854775807L, 27000]]
+(1 row)
+</pre></li>
+</ol>
+<p><a class="anchor" id="literature"></a></p><dl class="section user"><dt>Literature</dt><dd></dd></dl>
+<p>[1] G. Cormode and S. Muthukrishnan. An improved data stream summary: The count-min sketch and its applications. LATIN 2004, J. Algorithm 55(1): 58-75 (2005) . <a href="http://dimacs.rutgers.edu/~graham/pubs/html/CormodeMuthukrishnan04CMLatin.html">http://dimacs.rutgers.edu/~graham/pubs/html/CormodeMuthukrishnan04CMLatin.html</a></p>
+<p>[2] G. Cormode. Encyclopedia entry on 'Count-Min Sketch'. In L. Liu and M. T. Ozsu, editors, Encyclopedia of Database Systems, pages 511-516. Springer, 2009. <a href="http://dimacs.rutgers.edu/~graham/pubs/html/Cormode09b.html">http://dimacs.rutgers.edu/~graham/pubs/html/Cormode09b.html</a></p>
+<p><a class="anchor" id="related"></a>File <a class="el" href="sketch_8sql__in.html" title="SQL functions for sketch-based approximations of descriptive statistics. ">sketch.sql_in</a> documenting the SQL functions.</p>
+<p>Module grp_quantile for a different implementation of quantile function. </p>
+</div><!-- contents -->
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