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[24/51] [abbrv] [partial] madlib-site git commit: Additional updates
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+ <div id="projectbrief">User Documentation for MADlib</div>
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+<div class="title">Marginal Effects<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="#margins">Marginal Effects with Interaction Terms</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="#literature">Literature</a> </li>
+<li>
+<a href="#related">Related Topics</a> </li>
+</ul>
+</div><p>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.</p>
+<p>MADlib provides marginal effects regression functions for linear, logistic and multinomial logistic regressions.</p>
+<dl class="section warning"><dt>Warning</dt><dd>The <a class="el" href="marginal_8sql__in.html#a9517d679ee4209126895445cbed51fe3">margins_logregr()</a> and <a class="el" href="marginal_8sql__in.html#ae39ad0e1beca060fd153dba35901a4e7">margins_mlogregr()</a> functions have been deprecated in favor of the <a class="el" href="marginal_8sql__in.html#a36fcae5245ca31517723fce38b183c90" title="Marginal effects with default variable_names. ">margins()</a> function.</dd></dl>
+<p><a class="anchor" id="margins"></a></p><dl class="section user"><dt>Marginal Effects with Interaction Terms</dt><dd><pre class="syntax">
+margins( model_table,
+ output_table,
+ x_design,
+ source_table,
+ marginal_vars
+ )
+</pre> <b>Arguments</b> <dl class="arglist">
+<dt>model_table </dt>
+<dd>VARCHAR. The name of the model table, which is the output of <a class="el" href="logistic_8sql__in.html#a74210a7ef513dfcbdfdd9f3b37bfe428" title="Compute logistic-regression coefficients and diagnostic statistics. ">logregr_train()</a> or <a class="el" href="multilogistic_8sql__in.html#aedc13474e6abbc88451d120ad97e44d4" title="Compute multinomial logistic regression coefficients. ">mlogregr_train()</a>. </dd>
+<dt>output_table </dt>
+<dd>VARCHAR. The name of the result table. The output table has the following columns. <table class="output">
+<tr>
+<th>variables </th><td>INTEGER[]. The indices of the basis variables. </td></tr>
+<tr>
+<th>margins </th><td>DOUBLE PRECISION[]. The marginal effects. </td></tr>
+<tr>
+<th>std_err </th><td>DOUBLE PRECISION[]. An array of the standard errors, computed using the delta method. </td></tr>
+<tr>
+<th>z_stats </th><td>DOUBLE PRECISION[]. An array of the z-stats of the marginal effects. </td></tr>
+<tr>
+<th>p_values </th><td>DOUBLE PRECISION[]. An array of the Wald p-values of the marginal effects. </td></tr>
+</table>
+</dd>
+<dt>x_design (optional) </dt>
+<dd><p class="startdd">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.</p>
+<p>Example: The <em>independent_varname</em> in the regression method can be specified in either of the following ways:</p><ul>
+<li><code> ‘array[1, color_blue, color_green, gender_female, gpa, gpa^2, gender_female*gpa, gender_female*gpa^2, weight]’ </code></li>
+<li><code> ‘x’ </code></li>
+</ul>
+<p>In the second version, the column <em>x</em> is an array containing data identical to that expressed in the first version, computed in a prior data preparation step. Supply an <em>x_design argument</em> to the <a class="el" href="marginal_8sql__in.html#a36fcae5245ca31517723fce38b183c90" title="Marginal effects with default variable_names. ">margins()</a> function in the following way:</p><ul>
+<li><code> ‘1, i.color_blue.color, i.color_green.color, i.gender_female, gpa, gpa^2, gender_female*gpa, gender_female*gpa^2, weight’</code></li>
+</ul>
+<p>The variable names (<em>'gpa', 'weight', ...</em>), referred to here as <em>identifiers</em>, should be unique for each basis variable and need not be the same as the original variable name in <em>independent_varname</em>. They should, however, be in the same order as the corresponding variables in <em>independent_varname</em>. The length of <em>x_design</em> is expected to be the same as the length of <em>independent_varname</em>. Each <em>identifier</em> name can contain only alphanumeric characters and the underscore.</p>
+<p>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.'.</p>
+<p>An identifier may contain alphanumeric characters and underscores. To include other characters, the string must be double-quoted. Escape-characters are not currently supported. </p>
+<p class="enddd"></p>
+</dd>
+<dt>source_table (optional) </dt>
+<dd><p class="startdd">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.</p>
+<p class="enddd"></p>
+</dd>
+<dt>marginal_vars (optional) </dt>
+<dd>VARCHAR, default: NULL. Comma-separated string containing specific variable identifiers to calculate marginal effects on. When NULL, marginal effects for all variables are returned. </dd>
+</dl>
+</dd></dl>
+<dl class="section note"><dt>Note</dt><dd>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 <em>marginal_vars</em>, the <a class="el" href="marginal_8sql__in.html#a36fcae5245ca31517723fce38b183c90" title="Marginal effects with default variable_names. ">margins()</a> 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.</dd></dl>
+<p><a class="anchor" id="logregr_train"></a></p><dl class="section user"><dt>Marginal Effects for Logistic Regression</dt><dd></dd></dl>
+<dl class="section warning"><dt>Warning</dt><dd>This function has been deprecated in favor of the <a class="el" href="marginal_8sql__in.html#a36fcae5245ca31517723fce38b183c90" title="Marginal effects with default variable_names. ">margins()</a> function.</dd></dl>
+<pre class="syntax">
+margins_logregr( source_table,
+ output_table,
+ dependent_variable,
+ independent_variable,
+ grouping_cols,
+ marginal_vars,
+ max_iter,
+ optimizer,
+ tolerance,
+ verbose_mode
+ )
+</pre><p> <b>Arguments</b> </p><dl class="arglist">
+<dt>source_table </dt>
+<dd>VARCHAR. The name of the data table. </dd>
+<dt>output_table </dt>
+<dd><p class="startdd">VARCHAR. The name of the result table. The output table has the following columns. </p><table class="output">
+<tr>
+<th>margins </th><td>DOUBLE PRECISION[]. The marginal effects. </td></tr>
+<tr>
+<th>std_err </th><td>DOUBLE PRECISION[]. An array of the standard errors, using the delta method. </td></tr>
+<tr>
+<th>z_stats </th><td>DOUBLE PRECISION[]. An array of the z-stats of the marginal effects. </td></tr>
+<tr>
+<th>p_values </th><td>DOUBLE PRECISION[]. An array of the Wald p-values of the marginal effects. </td></tr>
+</table>
+<p>A summary table named <output_table>_summary is also created, which is the same as the summary table created by <a class="el" href="logistic_8sql__in.html#a74210a7ef513dfcbdfdd9f3b37bfe428" title="Compute logistic-regression coefficients and diagnostic statistics. ">logregr_train()</a> function. Refer to the documentation for logistic regression for details.</p>
+<p class="enddd"></p>
+</dd>
+<dt>dependent_variable </dt>
+<dd>VARCHAR. The name of the column for dependent variables. </dd>
+<dt>independent_variable </dt>
+<dd>VARCHAR. The name of the column for independent variables. Can be any SQL expression that evaluates to an array. </dd>
+<dt>grouping_cols (optional) </dt>
+<dd>VARCHAR, 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>
+<dt>marginal_vars (optional) </dt>
+<dd>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. </dd>
+<dt>max_iter (optional) </dt>
+<dd>INTEGER, default: 20. The maximum number of iterations for the logistic regression. </dd>
+<dt>optimizer (optional) </dt>
+<dd>VARCHAR, default: 'irls'. The optimizer to use for the logistic regression: newton/irls, cg, or igd. </dd>
+<dt>tolerance (optional) </dt>
+<dd>DOUBLE PRECISION, default: 1e-4. Termination criterion for logistic regression (relative). </dd>
+<dt>verbose_mode (optional) </dt>
+<dd>BOOLEAN, default FALSE. When TRUE, provides verbose output of the results of training. </dd>
+</dl>
+<p><a class="anchor" id="mlogregr_train"></a></p><dl class="section user"><dt>Marginal Effects for Multinomial Logistic Regression</dt><dd></dd></dl>
+<dl class="section warning"><dt>Warning</dt><dd>This function has been deprecated in favor of the <a class="el" href="marginal_8sql__in.html#a36fcae5245ca31517723fce38b183c90" title="Marginal effects with default variable_names. ">margins()</a> function.</dd></dl>
+<pre class="syntax">
+margins_mlogregr( source_table,
+ out_table,
+ dependent_varname,
+ independent_varname,
+ ref_category,
+ grouping_cols,
+ marginal_vars,
+ optimizer_params,
+ verbose_mode
+ )
+</pre><p> <b>Arguments</b> </p><dl class="arglist">
+<dt>source_table </dt>
+<dd>VARCHAR. The name of data table. </dd>
+<dt>out_table </dt>
+<dd><p class="startdd">VARCHAR. The name of result table. 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>margins </th><td>DOUBLE PRECISION[]. The marginal effects. </td></tr>
+<tr>
+<th>std_err </th><td>DOUBLE PRECISION[]. An array of the standard errors, using the delta method. </td></tr>
+<tr>
+<th>z_stats </th><td>DOUBLE PRECISION[]. An array of the z-stats of the marginal effects. </td></tr>
+<tr>
+<th>p_values </th><td>DOUBLE PRECISION[]. An array of the Wald p-values of the marginal effects. </td></tr>
+</table>
+<p>A summary table named <out_table>_summary is also created, which is the same as the summary table created by <a class="el" href="multilogistic_8sql__in.html#aedc13474e6abbc88451d120ad97e44d4" title="Compute multinomial logistic regression coefficients. ">mlogregr_train()</a> function. Refer to the documentation for multinomial logistic regression for details.</p>
+<p class="enddd"></p>
+</dd>
+<dt>dependent_varname </dt>
+<dd>VARCHAR. The name of the column for dependent variables. </dd>
+<dt>independent_varname </dt>
+<dd>VARCHAR. The name of the column for independent variables. Can be any SQL expression that evaluates to an array. </dd>
+<dt>ref_category (optional) </dt>
+<dd>INTEGER, default: 0. Reference category for the multinomial logistic regression. </dd>
+<dt>grouping_cols (optional) </dt>
+<dd>VARCHAR, 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>
+<dt>marginal_vars(optional) </dt>
+<dd>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. </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 'key=value' pairs separated by commas. </dd>
+<dt>verbose_mode (optional) </dt>
+<dd>BOOLEAN, default FALSE. When TRUE, provides verbose output of the results of training. </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 marginal effects function. <pre class="example">
+SELECT madlib.margins();
+</pre></li>
+<li>Create the sample data set. Use the <code>patients</code> dataset from the <a href="group__grp__logreg.html#examples">Logistic Regression examples</a>. <pre class="example">
+SELECT * FROM patients;
+</pre> Result: <pre class="result">
+ id | second_attack | treatment | trait_anxiety
+ ---+---------------+-----------+---------------
+ 1 | 1 | 1 | 70
+ 3 | 1 | 1 | 50
+ 5 | 1 | 0 | 40
+ 7 | 1 | 0 | 75
+ 9 | 1 | 0 | 70
+ 11 | 0 | 1 | 65
+ 13 | 0 | 1 | 45
+ 15 | 0 | 1 | 40
+ 17 | 0 | 0 | 55
+ 19 | 0 | 0 | 50
+ 2 | 1 | 1 | 80
+ 4 | 1 | 0 | 60
+ 6 | 1 | 0 | 65
+ 8 | 1 | 0 | 80
+ 10 | 1 | 0 | 60
+ 12 | 0 | 1 | 50
+ 14 | 0 | 1 | 35
+ 16 | 0 | 1 | 50
+ 18 | 0 | 0 | 45
+ 20 | 0 | 0 | 60
+</pre></li>
+<li>Run logistic regression to get the model, compute the marginal effects of all variables, and view the results. <pre class="example">
+DROP TABLE IF EXISTS model_table;
+DROP TABLE IF EXISTS model_table_summary;
+DROP TABLE IF EXISTS margins_table;
+SELECT madlib.logregr_train( 'patients',
+ 'model_table',
+ 'second_attack',
+ 'ARRAY[1, treatment, trait_anxiety, treatment^2, treatment * trait_anxiety]'
+ );
+SELECT madlib.margins( 'model_table',
+ 'margins_table',
+ 'intercept, treatment, trait_anxiety, treatment^2, treatment*trait_anxiety',
+ NULL,
+ NULL
+ );
+\x ON
+SELECT * FROM margins_table;
+</pre> Result: <pre class="result">
+variables | {intercept, treatment, trait_anxiety}
+margins | {-0.876046514609573,-0.0648833521465306,0.0177196513589633}
+std_err | {0.551714275062467,0.373592457067442,0.00458001207971933}
+z_stats | {-1.58786269307674,-0.173674149247659,3.86890930646828}
+p_values | {0.112317391159946,0.862121554662231,0.000109323294026272}
+</pre></li>
+<li>Compute the marginal effects of the first variable using the previous model and view the results (using different names in 'x_design'). <pre class="example">
+DROP TABLE IF EXISTS result_table;
+SELECT madlib.margins( 'model_table',
+ 'result_table',
+ 'i, tre, tra, tre^2, tre*tra',
+ NULL,
+ 'tre'
+ );
+SELECT * FROM result_table;
+</pre> Result: <pre class="result">
+-[ RECORD 1 ]-------------------
+variables | {tre}
+margins | {-0.110453283517281}
+std_err | {0.228981529064089}
+z_stats | {-0.482367656329023}
+p_values | {0.629544793219806}
+</pre></li>
+<li>Create a sample data set for multinomial logistic regression. (The full dataset has three categories.) Use the dataset from the <a href="group__grp__mlogreg.html#examples">Multinomial Regression example</a>. <pre class="example">
+\x OFF
+SELECT * FROM test3;
+</pre> Result: <pre class="result">
+ feat1 | feat2 | cat
+-------+-------+-----
+ 2 | 33 | 0
+ 2 | 31 | 1
+ 2 | 36 | 1
+ 2 | 31 | 1
+ 2 | 41 | 1
+ 2 | 37 | 1
+ 2 | 44 | 1
+ 2 | 46 | 1
+ 2 | 46 | 2
+ 2 | 39 | 0
+ 2 | 44 | 1
+ 2 | 44 | 0
+ 2 | 67 | 2
+ 2 | 59 | 2
+ 2 | 59 | 0
+...
+</pre></li>
+<li>Run the regression function and then compute the marginal effects of all variables in the regression. <pre class="example">
+DROP TABLE IF EXISTS model_table;
+DROP TABLE IF EXISTS model_table_summary;
+DROP TABLE IF EXISTS result_table;
+SELECT madlib.mlogregr_train('test3', 'model_table', 'cat',
+ 'ARRAY[1, feat1, feat2, feat1*feat2]',
+ 0);
+SELECT madlib.margins('model_table',
+ 'result_table',
+ 'intercept, feat1, feat2, feat1*feat2');
+\x ON
+SELECT * FROM result_table;
+</pre> Result: <pre class="result">
+-[ RECORD 1 ]+-------------------------------------------------------------
+category | 1
+ref_category | 0
+variables | {intercept,feat1,feat2}
+margins | {2.38176571752675,-0.0545733108729351,-0.0147264917310351}
+std_err | {0.851299967007829,0.0697049196489632,0.00374946341567828}
+z_stats | {2.79779843748643,-0.782919070099622,-3.92762646235104}
+p_values | {0.00514522099923651,0.43367463815468,8.57883141882439e-05}
+-[ RECORD 2 ]+-------------------------------------------------------------
+category | 2
+ref_category | 0
+variables | {intercept,feat1,feat2}
+margins | {-1.99279068434949,0.0922540608068343,0.0168049205501686}
+std_err | {0.742790306495022,0.0690712705200096,0.00202015384479213}
+z_stats | {-2.68284422524683,1.33563578767686,8.31863404536785}
+p_values | {0.00729989838349161,0.181668346802398,8.89828265128986e-17}
+</pre></li>
+</ol>
+<p><a class="anchor" id="notes"></a> </p><dl class="section note"><dt>Note</dt><dd>The <em>marginal_vars</em> 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 <em>marginal_vars</em> must be the indices (base 1) of variables in 'independent_varname'. Use <em>NULL</em> to use all independent variables. It is important to note that the <em>independent_varname</em> 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.</dd></dl>
+<p><a class="anchor" id="background"></a></p><dl class="section user"><dt>Technical Background</dt><dd></dd></dl>
+<p>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, </p><p class="formulaDsp">
+\[ y = G(X' \beta), \]
+</p>
+<p>where G is the specified binomial distribution. For logistic regression, the function \( G \) represents the inverse logit function.</p>
+<p>In logistic regression: </p><p class="formulaDsp">
+\[ P = \frac{1}{1 + e^{-(\beta_0 + \beta_1 x_1 + \dots \beta_j x_j)}} = \frac{1}{1 + e^{-z}} \implies \frac{\partial P}{\partial X_k} = \beta_k \cdot \frac{1}{1 + e^{-z}} \cdot \frac{e^{-z}}{1 + e^{-z}} \\ = \beta_k \cdot P \cdot (1-P) \]
+</p>
+<p>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.</p>
+<p>This is calculated as follows: </p><p class="formulaDsp">
+\[ \frac{\partial y}{\partial x_k} = \beta_k \frac{\sum_{i=1}^n P(y_i = 1)(1-P(y_i = 1))}{n}, \\ \text{where}, P(y_i=1) = g(X^{(i)}\beta) \]
+</p>
+<p>We use the delta method for calculating standard errors on the marginal effects.</p>
+<p><a class="anchor" id="literature"></a></p><dl class="section user"><dt>Literature</dt><dd></dd></dl>
+<p>[1] mfx function in STATA: <a href="http://www.stata.com/help.cgi?mfx_option">http://www.stata.com/help.cgi?mfx_option</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="marginal_8sql__in.html" title="SQL functions for linear regression. ">marginal.sql_in</a> documenting the SQL functions.</p>
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+<div class="title">Matrix Operations<div class="ingroups"><a class="el" href="group__grp__datatrans.html">Data Types and Transformations</a> » <a class="el" href="group__grp__arraysmatrix.html">Arrays and Matrices</a></div></div> </div>
+</div><!--header-->
+<div class="contents">
+<div class="toc"><b>Contents</b> <ul>
+<li class="level1">
+<a href="#description">Description</a> </li>
+<li class="level1">
+<a href="#operations">Matrix Operations</a> </li>
+<li class="level1">
+<a href="#glossary">Glossary of arguments</a> </li>
+<li class="level1">
+<a href="#examples">Examples</a> </li>
+<li class="level1">
+<a href="#related">Related Topics</a> </li>
+</ul>
+</div><p><a class="anchor" id="description"></a>This module provides a set of basic matrix operations for matrices that are too big to fit in memory. We provide two storage formats for a matrix:</p>
+<ul>
+<li>Dense: The matrix is represented as a distributed collection of 1-D arrays. An example 3x10 matrix would be the table below: <pre>
+ row_id | row_vec
+--------+-------------------------
+ 1 | {9,6,5,8,5,6,6,3,10,8}
+ 2 | {8,2,2,6,6,10,2,1,9,9}
+ 3 | {3,9,9,9,8,6,3,9,5,6}
+</pre></li>
+</ul>
+<p>A '<em>row</em>' column (called <em>row_id</em> above) provides the row number of each row and a '<em>val</em>' column (called <em>row_vec</em> above) provides each row as an array. <b>The <em>row</em> column should contain a series of integers from 1 to <em>N</em> with no duplicates, where <em>N</em> is the row dimensionality</b>.</p>
+<ul>
+<li>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 <em>row</em> and <em>col</em> 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. <pre>
+ row_id | col_id | value
+--------+--------+-------
+ 1 | 1 | 9
+ 1 | 5 | 6
+ 1 | 6 | 6
+ 2 | 1 | 8
+ 3 | 1 | 3
+ 3 | 2 | 9
+ 4 | 7 | 0
+(6 rows)
+</pre></li>
+</ul>
+<p><b>For sparse matrices, the <em>row</em> and <em>col</em> columns together should not contain a duplicate entry and the <em>val</em> column should be of scalar (non-array) data type</b>. <br />
+ 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 <em>row</em> and <em>col</em> is 4 and 7 respectively, leading to all zeros in the last row and last column.   </p><pre>
+ row_id | row_vec
+--------+-------------------------
+ 1 | {9,0,0,0,6,6,0}
+ 2 | {8,0,0,0,0,0,0}
+ 3 | {3,9,0,0,0,0,0}
+ 4 | {0,0,0,0,0,0,0}
+</pre><dl class="section note"><dt>Note</dt><dd>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).</dd></dl>
+<p><a class="anchor" id="operations"></a></p><dl class="section user"><dt>Matrix Operations</dt><dd></dd></dl>
+<p>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.</p>
+<ul>
+<li><b>Representation</b> <pre class="syntax">
+-- Convert to sparse representation
+  <b>matrix_sparsify</b>( matrix_in, in_args, matrix_out, out_args)
+ 
+-- Convert to dense representation
+  <b>matrix_densify</b>( matrix_in, in_args, matrix_out, out_args)
+ 
+-- Get dimensions of matrix
+  <b>matrix_ndims</b>( matrix_in, in_args )
+</pre></li>
+<li><b>Mathematical operations</b> <pre class="syntax">
+-- Matrix transposition
+  <b>matrix_trans</b>( matrix_in, in_args, matrix_out, out_args)
+ 
+-- Matrix addition
+  <b>matrix_add</b>( matrix_a, a_args, matrix_b, b_args, matrix_out, out_args)
+ 
+-- Matrix subtraction
+  <b>matrix_sub</b>( matrix_a, a_args, matrix_b, b_args, matrix_out, out_args)
+ 
+-- Matrix multiplication
+  <b>matrix_mult</b>( matrix_a, a_args, matrix_b, b_args, matrix_out, out_args)
+ 
+-- Element-wise matrix multiplication
+  <b>matrix_elem_mult</b>( matrix_a, a_args, matrix_b, b_args, matrix_out, out_args)
+ 
+-- Multiply matrix with scalar
+  <b>matrix_scalar_mult</b>( matrix_in, in_args, scalar, matrix_out, out_args)
+ 
+-- Multiply matrix with vector
+  <b>matrix_vec_mult</b>( matrix_in, in_args, vector)
+</pre></li>
+<li><b>Extraction/visitor methods</b> <pre class="syntax">
+-- Extract row from matrix given row index
+  <b>matrix_extract_row</b>( matrix_in, in_args, index)
+ 
+-- Extract column from matrix given column index
+  <b>matrix_extract_col</b>( matrix_in, in_args, index)
+ 
+-- Extract main (principal) diagonal elements
+  <b>matrix_extract_diag</b>( matrix_in, in_args)
+</pre></li>
+<li><b>Reduction operations (aggregate across specific dimension)</b> <pre class="syntax">
+-- Get max value along dim. Returns corresponding index if <em>fetch_index</em> = True
+  <b>matrix_max</b>( matrix_in, in_args, dim, matrix_out, fetch_index)
+ 
+-- Get min value along dim. Returns corresponding index if <em>fetch_index</em> = True
+  <b>matrix_min</b>( matrix_in, in_args, dim, matrix_out, fetch_index)
+ 
+-- Get sum value along dimension
+  <b>matrix_sum</b>( matrix_in, in_args, dim)
+ 
+-- Get mean value along dimension
+  <b>matrix_mean</b>( matrix_in, in_args, dim)
+ 
+-- Get matrix norm
+  <b>matrix_norm</b>( matrix_in, in_args, norm_type)
+</pre></li>
+<li><b>Creation methods</b> <pre class="syntax">
+-- Create a matrix initialized with ones of given row and column dimension
+  <b>matrix_ones</b>( row_dim, col_dim, matrix_out, out_args)
+ 
+-- Create a matrix initialized with zeros of given row and column dimension
+  <b>matrix_zeros</b>( row_dim, col_dim, matrix_out, out_args)
+ 
+-- Create an square identity matrix of size dim x dim
+  <b>matrix_identity</b>( dim, matrix_out, out_args)
+ 
+-- Create a diag matrix initialized with given diagonal elements
+  <b>matrix_diag</b>( diag_elements, matrix_out, out_args)
+-- Create a matrix initialized with values sampled from a distribution
+-- Supported distributions: normal, uniform, bernoulli
+  <b>matrix_random</b>( distribution, row_dim, col_dim, in_args, matrix_out, out_args )
+</pre></li>
+<li><b>Decomposition operations</b> <dl class="section note"><dt>Note</dt><dd>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.</dd></dl>
+<pre class="syntax">
+-- Matrix inverse
+  <b>matrix_inverse</b>( matrix_in, in_args, matrix_out, out_args)
+ 
+-- Matrix generic inverse
+  <b>matrix_pinv</b>( matrix_in, in_args, matrix_out, out_args)
+ 
+-- Matrix eigen extraction
+  <b>matrix_eigen</b>( matrix_in, in_args, matrix_out, out_args)
+ 
+-- Matrix Cholesky decomposition
+  <b>matrix_cholesky</b>( matrix_in, in_args, matrix_out_prefix, out_args)
+ 
+-- Matrix QR decomposition
+  <b>matrix_qr</b>( matrix_in, in_args, matrix_out_prefix, out_args)
+ 
+-- Matrix LU decomposition
+  <b>matrix_lu</b>( matrix_in, in_args, matrix_out_prefix, out_args)
+ 
+-- Matrix nuclear norm computing
+  <b>matrix_nuclear_norm</b>( matrix_in, in_args)
+ 
+-- Matrix rank computing
+  <b>matrix_rank</b>( matrix_in, in_args)
+</pre></li>
+</ul>
+<p><a class="anchor" id="glossary"></a><b>Glossary</b> </p>
+<p>The table below provides a glossary of the terms used in the matrix operations.</p>
+<dl class="arglist">
+<dt>matrix_in, matrix_a, matrix_b </dt>
+<dd><p class="startdd">TEXT. Name of the table containing the input matrix.</p><ul>
+<li>For functions accepting one matrix, <em>matrix_in</em> denotes the input matrix.</li>
+<li>For functions accepting two matrices, <em>matrix_a</em> denotes the first matrix and <em>matrix_b</em> denotes the second matrix. These two matrices can <b>independently</b> be in either dense or sparse format. </li>
+</ul>
+<p class="enddd"></p>
+</dd>
+<dt>in_args, a_args, b_args </dt>
+<dd><p class="startdd">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.</p>
+<p>The following parameters are supported for this string argument: </p><table class="output">
+<tr>
+<th>row </th><td>(Default: 'row_num') Name of the column containing row index of the matrix. </td></tr>
+<tr>
+<th>col </th><td>(Default: 'col_num') Name of the column containing column index of the matrix. </td></tr>
+<tr>
+<th>val </th><td>(Default: 'val') Name of the column containing the entries of the matrix. </td></tr>
+<tr>
+<th>trans </th><td>(Default: False) Boolean flag to indicate if the matrix should be transposed before the operation. This is currently functional only for <em>matrix_mult</em>. </td></tr>
+</table>
+<p>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 <em>NULL</em> or be blank ('') if default values are to be used. </p>
+<p class="enddd"></p>
+</dd>
+<dt>matrix_out </dt>
+<dd><p class="startdd">TEXT. Name of the table to store the result matrix.</p>
+<p>For Cholesky, QR and LU decompositions, a prefix (<em>matrix_out_prefix</em>) is used as a basis to build the names of the various output tables.</p>
+<p>For Cholesky decomposition ( \( PA = LDL* \)), the following suffixes are added to <em>matrix_out_prefix</em>:</p><ul>
+<li><em>_p</em> for row permutation matrix P</li>
+<li><em>_l</em> for lower triangular factor L</li>
+<li><em>_d</em> for diagonal matrix D</li>
+</ul>
+<p>For QR decomposition ( \( A = QR \)) the following suffixes are added to <em>matrix_out_prefix</em>:</p><ul>
+<li><em>_q</em> for orthogonal matrix Q</li>
+<li><em>_r</em> for upper triangular factor R</li>
+</ul>
+<p>For LU decomposition with full pivoting ( \( PAQ = LU \)), the following suffixes are added to <em>matrix_out_prefix</em>:</p><ul>
+<li><em>_p</em> for row permutation matrix P</li>
+<li><em>_q</em> for column permutation matrix Q</li>
+<li><em>_l</em> for lower triangular factor L</li>
+<li><em>_u</em> for upper triangular factor U </li>
+</ul>
+<p class="enddd"></p>
+</dd>
+<dt>out_args </dt>
+<dd><p class="startdd">TEXT. A comma-delimited string containing named arguments of the form "name=value". This is an <b>optional parameter</b> and the default value is set as follows:</p><ul>
+<li>For functions with one input matrix, default <em>out_args</em> will the be same as specified in <em>in_args</em>.</li>
+<li>For functions with two input matrices, default <em>out_args</em> will be the same as specified in <em>a_args</em>.</li>
+</ul>
+<p>The following parameters are supported for this string argument: </p><table class="output">
+<tr>
+<th>row </th><td>Name of the column containing row index of the matrix. </td></tr>
+<tr>
+<th>col </th><td>Name of the column containing column index of the matrix. </td></tr>
+<tr>
+<th>val </th><td>Name of the column containing the entries of the matrix. </td></tr>
+<tr>
+<th>fmt </th><td>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. </td></tr>
+</table>
+<dl class="section note"><dt>Note</dt><dd>One exception is for <em>matrix_eigen</em>: the default output column name is <em>’eigen_values’</em> in the format <em>(real, imaginary)</em>.</dd></dl>
+</dd>
+<dt>index </dt>
+<dd><p class="startdd">INTEGER. An integer representing a row or column index of the matrix. Should be a number from 1 to <em>N</em>, where <em>N</em> is the maximum size of the dimension.</p>
+<p class="enddd"></p>
+</dd>
+<dt>dim </dt>
+<dd><p class="startdd">INTEGER. Should either be 1 or 2. This value indicates the dimension to operate along for the reduction/aggregation operations. <b>The value of <em>dim</em> should be interpreted as the dimension to be flattened i.e. whose length reduces to 1 in the result.</b></p>
+<p>For any reduction function applied on an <em>N x M</em> matrix:</p>
+<ul>
+<li>if <em>dim=1</em>, operation is applied on successive elements in each column; return value is a single vector with <em>M</em> elements (i.e. matrix with 1 row and <em>M</em> columns). <br />
+</li>
+<li>if <em>dim=2</em>, operation is applied on successive elements in each row; return value is a single vector with <em>N</em> elements (i.e. matrix with <em>N</em> rows and 1 column). </li>
+</ul>
+<p class="enddd"></p>
+</dd>
+<dt>row_dim </dt>
+<dd><p class="startdd">INTEGER. Should be greater than 0. This value indicates the row dimension of result matrix. </p>
+<p class="enddd"></p>
+</dd>
+<dt>col_dim </dt>
+<dd><p class="startdd">INTEGER. Should be greater than 0. This value indicates the column dimension of result matrix. </p>
+<p class="enddd"></p>
+</dd>
+<dt>diag_elements </dt>
+<dd><p class="startdd">ARRAY OF FLOAT. Should not be empty. This value indicates the float array which is used to generate diag elements of result output matrix. </p>
+<p class="enddd"></p>
+</dd>
+<dt>norm_type </dt>
+<dd><p class="startdd">TEXT. Optional parameter. Defaults to Frobenius norm. Other supported norms for this string argument:</p>
+<table class="output">
+<tr>
+<th>'one' or 'o' </th><td>1 norm </td></tr>
+<tr>
+<th>float > 0 </th><td>Element-wise norm </td></tr>
+<tr>
+<th>'inf' or 'i' </th><td>Infinite norm </td></tr>
+<tr>
+<th>'max' or 'm' </th><td>Max absolute value norm </td></tr>
+<tr>
+<th>'fro' or 'f' </th><td>Frobenius norm (default) </td></tr>
+</table>
+<p class="enddd"></p>
+</dd>
+</dl>
+<p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl>
+<p>Here are some examples of matrix operations in dense format. Later on this page we will show examples of matrix operations in sparse format.</p>
+<ul>
+<li>First let’s create example data tables in dense format: <pre class="syntax">
+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}');
+</pre></li>
+<li>Transpose a matrix <pre class="syntax">
+SELECT madlib.matrix_trans('"mat_B"', 'row=row_id, val=vector',
+ 'mat_r');
+SELECT * FROM mat_r ORDER BY row_id;
+</pre> <pre class="result">
+-- 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)
+</pre></li>
+<li>Extract main diagonal of a matrix <pre class="syntax">
+SELECT madlib.matrix_extract_diag('"mat_B"', 'row=row_id, val=vector');
+</pre> <pre class="result">
+-- Note the result is an array
+ matrix_extract_diag
+---------------------------------
+{9,3,2,4,0,3,10,6,0,5}
+(1 row)
+</pre></li>
+<li>Add two matrices <pre class="syntax">
+SELECT madlib.matrix_add('"mat_A"', 'row=row_id, val=row_vec',
+ '"mat_B"', 'row=row_id, val=vector',
+ 'mat_r', 'val=vector, fmt=dense');
+SELECT * FROM mat_r ORDER BY row_id;
+</pre> <pre class="result">
+ row_id | vector
+--------+-------------------------------
+ 1 | {18,16,7,12,11,11,9,10,15,14}
+ 2 | {13,5,7,8,14,16,11,8,16,15}
+ 3 | {3,10,11,12,10,13,10,12,15,7}
+ 4 | {8,13,2,6,5,13,16,14,3,11}
+ 5 | {9,16,16,16,4,11,12,14,9,17}
+ 6 | {9,13,8,10,15,8,14,5,9,8}
+ 7 | {12,18,11,14,3,16,17,9,11,10}
+ 8 | {11,10,5,7,5,9,7,7,13,16}
+ 9 | {14,13,9,12,4,13,19,7,8,9}
+ 10 | {5,10,7,6,14,9,3,10,8,13}
+(10 rows)
+</pre></li>
+<li>Multiply two matrices <pre class="syntax">
+SELECT madlib.matrix_mult('"mat_A"', 'row=row_id, val=row_vec',
+ '"mat_B"', 'row=row_id, val=vector, trans=true',
+ 'mat_r');
+SELECT * FROM mat_r ORDER BY row_id;
+</pre> <pre class="result">
+ row_id | row_vec
+--------+-------------------------------------------
+ 1 | {380,373,251,283,341,303,302,309,323,281}
+ 2 | {318,318,222,221,269,259,236,249,264,248}
+ 3 | {382,366,216,300,397,276,277,270,313,338}
+ 4 | {275,284,154,244,279,183,226,215,295,204}
+ 5 | {381,392,258,319,394,298,342,302,360,300}
+ 6 | {321,333,189,276,278,232,300,236,281,250}
+ 7 | {443,411,282,365,456,318,360,338,406,330}
+ 8 | {267,240,150,186,270,194,210,184,233,193}
+ 9 | {322,328,234,264,291,245,317,253,291,219}
+ 10 | {246,221,109,173,222,164,167,185,181,189}
+(10 rows)
+</pre></li>
+<li>Create a diagonal matrix <pre class="syntax">
+SELECT madlib.matrix_diag(array[9,6,3,10],
+ 'mat_r', 'row=row_id, col=col_id, val=val');
+SELECT * FROM mat_r ORDER BY row_id::bigint;
+</pre> <pre class="result">
+ row_id | col_id | val
+--------+---------+--------
+ 1 | 1 | 9
+ 2 | 2 | 6
+ 3 | 3 | 3
+ 4 | 4 | 10
+(11 rows)
+</pre></li>
+<li>Create an identity matrix <pre class="syntax">
+SELECT madlib.matrix_identity(4, 'mat_r', 'row=row_id,col=col_id,val=val');
+SELECT * FROM mat_r ORDER BY row_id;
+</pre> <pre class="result">
+ row_id | col_id | val
+--------+---------+--------
+ 1 | 1 | 1
+ 2 | 2 | 1
+ 3 | 3 | 1
+ 4 | 4 | 1
+(5 rows)
+</pre></li>
+<li>Extract row and column from a matrix by specifying index <pre class="syntax">
+SELECT madlib.matrix_extract_row('"mat_A"', 'row=row_id, val=row_vec', 2) as row,
+ madlib.matrix_extract_col('"mat_A"', 'row=row_id, val=row_vec', 3) as col;
+</pre> <pre class="result">
+ row | col
+------------------------+-----------------------
+ {8,2,2,6,6,10,2,1,9,9} | {5,2,9,2,9,7,7,5,8,3}
+(1 rows)
+</pre></li>
+<li>Get min and max values along a specific dimension, as well as the corresponding indices. Note that in this example <em>dim=2</em> implies that the min and max is computed on each row, returning a column vector i.e. the column (dim=2) is flattened. <pre class="syntax">
+SELECT madlib.matrix_max('"mat_A"', 'row=row_id, val=row_vec', 2, 'mat_max_r', true),
+ madlib.matrix_min('"mat_A"', 'row=row_id, val=row_vec', 2, 'mat_min_r', true);
+SELECT * from mat_max_r;
+SELECT * from mat_min_r;
+</pre> <pre class="result">
+ index | max
+-----------------------+---------------------------
+ {9,6,2,7,3,2,2,6,7,10} | {10,10,9,8,9,10,10,8,9,8}
+(1 rows)
+ 
+ index | min
+-----------------------+-----------------------
+ {8,8,1,9,5,8,5,7,8,7} | {3,1,3,0,4,2,1,1,1,1}
+(1 rows)
+</pre></li>
+<li>Initialize matrix with zeros in sparse format <pre class="syntax">
+SELECT madlib.matrix_zeros(5, 4, 'mat_r', 'row=row_id, col=col_id, val=entry');
+SELECT * FROM mat_r;
+</pre> <pre class="result">
+ row_id | col_id | entry
+--------+---------+--------
+ 5 | 4 | 0
+(1 rows)
+</pre></li>
+<li>Initialize matrix with zeros in dense format <pre class="syntax">
+SELECT madlib.matrix_zeros(5, 4, 'mat_r', 'fmt=dense');
+SELECT * FROM mat_r ORDER BY row;
+</pre> <pre class="result">
+ row | val
+-----+-----------
+ 1 | {0,0,0,0}
+ 2 | {0,0,0,0}
+ 3 | {0,0,0,0}
+ 4 | {0,0,0,0}
+ 5 | {0,0,0,0}
+(5 rows)
+</pre></li>
+<li>Initialize matrix with ones <pre class="syntax">
+SELECT madlib.matrix_ones(5, 4, 'mat_r', 'row=row,col=col, val=val');
+SELECT * FROM mat_r;
+</pre> <pre class="result">
+ row | col | val
+-----+-----+-----
+ 1 | 1 | 1
+ 1 | 2 | 1
+ 1 | 3 | 1
+ 1 | 4 | 1
+ 2 | 1 | 1
+ 2 | 2 | 1
+ 2 | 3 | 1
+ 2 | 4 | 1
+ 3 | 1 | 1
+ 3 | 2 | 1
+ 3 | 3 | 1
+ 3 | 4 | 1
+ 4 | 1 | 1
+ 4 | 2 | 1
+ 4 | 3 | 1
+ 4 | 4 | 1
+ 5 | 1 | 1
+ 5 | 2 | 1
+ 5 | 3 | 1
+ 5 | 4 | 1
+(20 rows)
+</pre></li>
+<li>Initialize matrix with ones in dense format <pre class="syntax">
+SELECT madlib.matrix_ones(5, 4, 'mat_r', 'fmt=dense');
+SELECT * FROM mat_r ORDER BY row;
+</pre> <pre class="result">
+ row | val
+-----+-----------
+ 1 | {1,1,1,1}
+ 2 | {1,1,1,1}
+ 3 | {1,1,1,1}
+ 4 | {1,1,1,1}
+ 5 | {1,1,1,1}
+(5 rows)
+</pre></li>
+<li>Element-wise multiplication between two matrices <pre class="syntax">
+SELECT madlib.matrix_elem_mult('"mat_A"', 'row=row_id, val=row_vec',
+ '"mat_B"', 'row=row_id, val=vector',
+ 'mat_r', 'val=vector');
+SELECT * FROM mat_r ORDER BY row_id;
+</pre> <pre class="result">
+ row_id | vector
+--------+---------------------------------
+ 1 | {81,60,10,32,30,30,18,21,50,48}
+ 2 | {40,6,10,12,48,60,18,7,63,54}
+ 3 | {0,9,18,27,16,42,21,27,50,6}
+ 4 | {12,36,0,8,6,42,64,48,0,28}
+ 5 | {18,64,63,63,0,30,27,45,14,70}
+ 6 | {20,30,7,21,54,15,45,6,18,16}
+ 7 | {32,80,28,40,2,63,70,0,24,21}
+ 8 | {28,24,0,6,6,8,6,6,36,64}
+ 9 | {48,40,8,35,4,42,90,6,0,18}
+ 10 | {4,24,12,8,48,20,2,16,15,40}
+</pre></li>
+<li>Get sum values along a dimension. In this example, the sum is computed for each row (i.e. column is flattened since dim=2). <pre class="syntax">
+SELECT madlib.matrix_sum('"mat_A"', 'row=row_id, val=row_vec', 2);
+</pre> <pre class="result">
+ matrix_sum
+---------------------------------
+ {66,55,67,46,70,56,76,46,58,39}
+(1 rows)
+</pre></li>
+<li>Get mean values along dimension <pre class="syntax">
+SELECT madlib.matrix_mean('"mat_A"', 'row=row_id, val=row_vec', 2);
+</pre> <pre class="result">
+ matrix_mean
+-----------------------------------------
+ {6.6,5.5,6.7,4.6,7,5.6,7.6,4.6,5.8,3.9}
+(1 rows)
+</pre></li>
+<li>Compute matrix norm. In this example, we ask for the Euclidean norm: <pre class="syntax">
+SELECT madlib.matrix_norm('"mat_A"', 'row=row_id, val=row_vec', '2');
+</pre> <pre class="result">
+ matrix_norm
+---------------
+ 64.1014820421
+(1 row)
+</pre></li>
+<li>Multiply matrix with scalar <pre class="syntax">
+SELECT madlib.matrix_scalar_mult('"mat_A"', 'row=row_id, val=row_vec', 3, 'mat_r');
+SELECT * FROM mat_r ORDER BY row_id;
+</pre> <pre class="result">
+ row_id | row_vec
+--------+---------------------------------
+ 1 | {27,18,15,24,15,18,18,9,30,24}
+ 2 | {24,6,6,18,18,30,6,3,27,27}
+ 3 | {9,27,27,27,24,18,9,27,15,18}
+ 4 | {18,12,6,6,6,21,24,24,0,21}
+ 5 | {18,24,27,27,12,18,27,15,21,21}
+ 6 | {12,30,21,9,27,15,27,6,9,12}
+ 7 | {24,30,21,30,3,27,21,27,24,21}
+ 8 | {21,12,15,18,6,24,3,3,12,24}
+ 9 | {24,24,24,15,6,18,27,3,24,9}
+ 10 | {12,18,9,6,18,12,3,6,9,24}
+(10 rows)
+</pre></li>
+<li>Get the row dimension and column dimension of matrix <pre class="syntax">
+SELECT madlib.matrix_ndims('"mat_A"', 'row=row_id, val=row_vec');
+</pre> <pre class="result">
+ matrix_ndims
+--------------
+ {10,10}
+(1 row)
+</pre></li>
+<li>Multiply matrix with vector <pre class="syntax">
+SELECT madlib.matrix_vec_mult('"mat_A"', 'row=row_id, val=row_vec',
+ array[1,2,3,4,5,6,7,8,9,10]);
+</pre> <pre class="result">
+ matrix_vec_mult
+-------------------------------------------
+ {365,325,358,270,377,278,411,243,287,217}
+(10 rows)
+</pre></li>
+<li>Inverse of matrix <pre class="syntax">
+SELECT madlib.matrix_inverse('"mat_A"', 'row=row_id, val=row_vec', 'mat_r');
+SELECT row_vec FROM mat_r ORDER BY row_id;
+</pre></li>
+<li>Generic inverse of matrix <pre class="syntax">
+SELECT madlib.matrix_pinv('"mat_A"', 'row=row_id, val=row_vec', 'mat_r');
+SELECT row_vec FROM mat_r ORDER BY row_id;
+</pre></li>
+<li>Eigen values of matrix (note default column name of eigenvalues) <pre class="syntax">
+SELECT madlib.matrix_eigen('"mat_A"', 'row=row_id, val=row_vec', 'mat_r');
+SELECT eigen_values FROM mat_r ORDER BY row_id;
+</pre></li>
+<li>Cholesky decomposition of matrix <pre class="syntax">
+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;
+</pre></li>
+<li>QR decomposition of matrix <pre class="syntax">
+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;
+</pre></li>
+<li>LU decomposition of matrix <pre class="syntax">
+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;
+</pre></li>
+<li>Nuclear norm of matrix <pre class="syntax">
+SELECT madlib.matrix_nuclear_norm('"mat_A"', 'row=row_id, val=row_vec');
+</pre> <pre class="result">
+ matrix_nuclear_norm
+---------------------
+ 118.852685995
+(1 row)
+</pre></li>
+<li>Rank of matrix <pre class="syntax">
+SELECT madlib.matrix_rank('"mat_A"', 'row=row_id, val=row_vec');
+</pre> <pre class="result">
+ matrix_rank
+-------------
+ 10
+(1 row)
+</pre></li>
+</ul>
+<p>Below are some examples of matrix operations in sparse format.</p>
+<ul>
+<li>Convert a matrix from dense to sparse format <pre class="syntax">
+SELECT madlib.matrix_sparsify('"mat_B"', 'row=row_id, val=vector',
+ '"mat_B_sparse"', 'col=col_id, val=val');
+SELECT * FROM "mat_B_sparse" ORDER BY row_id, col_id;
+</pre></li>
+<li>Create a matrix in sparse format. <pre class="syntax">
+CREATE TABLE "mat_A_sparse" (
+ "rowNum" integer,
+ col_num integer,
+ entry integer
+);
+INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (1, 1, 9);
+INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (1, 2, 6);
+INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (1, 7, 3);
+INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (1, 8, 10);
+INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (1, 9, 8);
+INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (2, 1, 8);
+INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (2, 2, 2);
+INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (2, 3, 6);
+INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (3, 5, 6);
+INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (3, 6, 3);
+INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (7, 1, 7);
+INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (8, 2, 8);
+INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (8, 3, 5);
+INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (9, 1, 6);
+INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (9, 2, 3);
+INSERT INTO "mat_A_sparse" ("rowNum", col_num, entry) VALUES (10, 10, 0);
+</pre></li>
+<li>Get the row_dims and col_dims of a matrix in sparse format <pre class="syntax">
+SELECT madlib.matrix_ndims('"mat_A_sparse"', 'row="rowNum", val=entry')
+</pre> <pre class="result">
+ matrix_ndims
+--------------
+ {10,10}
+(1 row)
+</pre></li>
+<li>Transpose a matrix in sparse format <pre class="syntax">
+-- 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;
+</pre> <pre class="result">
+ rowNum | col_num | entry
+--------+---------+-------
+ 1 | 1 | 9
+ 2 | 1 | 6
+ 7 | 1 | 3
+ 8 | 1 | 10
+ 9 | 1 | 8
+ 1 | 2 | 8
+ 2 | 2 | 2
+ 3 | 2 | 6
+ 5 | 3 | 6
+ 6 | 3 | 3
+ 1 | 7 | 7
+ 2 | 8 | 8
+ 3 | 8 | 5
+ 1 | 9 | 6
+ 2 | 9 | 3
+ 10 | 10 | 0
+(16 rows)
+</pre></li>
+<li>Main diagonal of a matrix in sparse format <pre class="syntax">
+SELECT madlib.matrix_extract_diag('"mat_A_sparse"', 'row="rowNum", val=entry');
+</pre> <pre class="result">
+ matrix_extract_diag
+-----------------
+{9,2,0,0,0,0,0,0,0,0}
+(1 row)
+</pre></li>
+<li>Add two sparse matrices then convert to dense format <pre class="syntax">
+SELECT madlib.matrix_add('"mat_A_sparse"', 'row="rowNum", val=entry',
+ '"mat_B_sparse"', 'row=row_id, col=col_id, val=val',
+ 'matrix_r_sparse', 'col=col_out');
+SELECT madlib.matrix_densify('matrix_r_sparse', 'row="rowNum", col=col_out, val=entry',
+ 'matrix_r');
+SELECT * FROM matrix_r ORDER BY "rowNum";
+</pre> <pre class="result">
+ rowNum | entry
+--------+---------------------------
+ 1 | {18,16,2,4,6,5,6,17,13,6}
+ 2 | {13,5,11,2,8,6,9,7,7,6}
+ 3 | {0,1,2,3,8,10,7,3,10,1}
+ 4 | {2,9,0,4,3,6,8,6,3,4}
+ 5 | {3,8,7,7,0,5,3,9,2,10}
+ 6 | {5,3,1,7,6,3,5,3,6,4}
+ 7 | {11,8,4,4,2,7,10,0,3,3}
+ 8 | {4,14,5,1,3,1,6,6,9,8}
+ 9 | {12,8,1,7,2,7,10,6,0,6}
+ 10 | {1,4,4,4,8,5,2,8,5,5}
+(10 rows)
+</pre></li>
+<li>Multiply two sparse matrices <pre class="syntax">
+SELECT madlib.matrix_mult('"mat_A_sparse"', 'row="rowNum", col=col_num, val=entry',
+ '"mat_B_sparse"', 'row=row_id, col=col_id, val=val, trans=true',
+ 'matrix_r');
+SELECT * FROM matrix_r ORDER BY "rowNum";
+</pre> <pre class="result">
+ rowNum | entry
+--------+-------------------------------------------
+ 1 | {260,216,137,180,190,156,138,222,174,159}
+ 2 | {104,76,14,34,82,52,72,44,64,40}
+ 3 | {51,66,33,36,15,45,33,21,33,63}
+ 4 | {0,0,0,0,0,0,0,0,0,0}
+ 5 | {0,0,0,0,0,0,0,0,0,0}
+ 6 | {0,0,0,0,0,0,0,0,0,0}
+ 7 | {63,35,0,14,21,35,28,28,42,7}
+ 8 | {90,49,18,72,99,29,84,48,45,52}
+ 9 | {84,39,3,39,42,39,48,42,51,18}
+ 10 | {0,0,0,0,0,0,0,0,0,0}
+(10 rows)
+</pre></li>
+<li>Initialize matrix with ones <pre class="syntax">
+SELECT madlib.matrix_ones(5, 4, 'mat_r', 'row=row,col=col, val=val');
+SELECT * FROM mat_r ORDER BY row, col;
+</pre> <pre class="result">
+ row | col | val
+-----+-----+-----
+ 1 | 1 | 1
+ 1 | 2 | 1
+ 1 | 3 | 1
+ 1 | 4 | 1
+ 2 | 1 | 1
+ 2 | 2 | 1
+ 2 | 3 | 1
+ 2 | 4 | 1
+ 3 | 1 | 1
+ 3 | 2 | 1
+ 3 | 3 | 1
+ 3 | 4 | 1
+ 4 | 1 | 1
+ 4 | 2 | 1
+ 4 | 3 | 1
+ 4 | 4 | 1
+ 5 | 1 | 1
+ 5 | 2 | 1
+ 5 | 3 | 1
+ 5 | 4 | 1
+(20 rows)
+</pre></li>
+<li>Initialize matrix with zeros in sparse format <pre class="syntax">
+SELECT madlib.matrix_zeros(5, 4, 'mat_r', 'row=row_id, col=col_id, val=entry');
+SELECT * FROM mat_r;
+</pre> <pre class="result">
+ row_id | col_id | entry
+--------+---------+--------
+ 5 | 4 | 0
+(1 rows)
+</pre></li>
+<li>Compute matrix norm on sparse matrix. In this example, we ask for the Euclidean norm: <pre class="syntax">
+SELECT madlib.matrix_norm('"mat_A_sparse"', 'row="rowNum", col=col_num, val=entry', '2');
+</pre> <pre class="result">
+ matrix_norm
+---------------
+ 24.9399278267
+(1 row)
+</pre></li>
+</ul>
+<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="array__ops_8sql__in.html" title="implementation of array operations in SQL ">array_ops.sql_in</a> documents the array operations <a class="el" href="group__grp__array.html">Array Operations</a></p>
+<p>File <a class="el" href="matrix__ops_8sql__in.html" title="Implementation of matrix operations in SQL. ">matrix_ops.sql_in</a> for list of functions and usage. </p>
+</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 Wed Dec 27 2017 19:05:57 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>
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+</html>
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new file mode 100644
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+ <div id="projectbrief">User Documentation for MADlib</div>
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+<a href="#groups">Modules</a> </div>
+ <div class="headertitle">
+<div class="title">Matrix Factorization<div class="ingroups"><a class="el" href="group__grp__datatrans.html">Data Types and Transformations</a> » <a class="el" href="group__grp__arraysmatrix.html">Arrays and Matrices</a></div></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__lmf"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__lmf.html">Low-Rank Matrix Factorization</a></td></tr>
+<tr class="memdesc:group__grp__lmf"><td class="mdescLeft"> </td><td class="mdescRight">Performs low-rank matrix factorization for an incomplete matrix. <br /></td></tr>
+<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
+<tr class="memitem:group__grp__svd"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__svd.html">Singular Value Decomposition</a></td></tr>
+<tr class="memdesc:group__grp__svd"><td class="mdescLeft"> </td><td class="mdescRight">Performs factorization of dense and sparse matrices. <br /></td></tr>
+<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
+</table>
+</div><!-- contents -->
+</div><!-- doc-content -->
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+</html>
http://git-wip-us.apache.org/repos/asf/madlib-site/blob/6c103d3e/docs/v1.13/group__grp__matrix__factorization.js
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+<title>MADlib: Model Selection</title>
+<link href="tabs.css" rel="stylesheet" type="text/css"/>
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+ <td style="padding-left: 0.5em;">
+ <div id="projectname">
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+ </div>
+ <div id="projectbrief">User Documentation for MADlib</div>
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+</div><!--header-->
+<div class="contents">
+<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
+<p>Contains functions for model selection and model evaluation. </p>
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+<tr class="memdesc:group__grp__validation"><td class="mdescLeft"> </td><td class="mdescRight">Estimates the fit of a predictive model given a data set and specifications for the training, prediction, and error estimation functions. <br /></td></tr>
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+<tr class="memitem:group__grp__pred"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__pred.html">Prediction Metrics</a></td></tr>
+<tr class="memdesc:group__grp__pred"><td class="mdescLeft"> </td><td class="mdescRight">Provides various prediction accuracy metrics. <br /></td></tr>
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+</table>
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