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 <div class="header">
   <div class="headertitle">
 <div class="title">Linear Regression<div class="ingroups"><a class="el" href="group__grp__glm.html">Generalized Linear Models</a></div></div>  </div>
 </div><!--header-->
 <div class="contents">
 <div class="toc"><b>Contents</b></p>
 <ul>
 <li class="level1">
 <a href="#about">About</a> </li>
 <li class="level1">
 <a href="#train">Training Function</a> </li>
 <li class="level1">
 <a href="#output">Output Table</a> </li>
 <li class="level1">
 <a href="#predict">Prediction Function</a> </li>
 <li class="level1">
 <a href="#examples">Examples</a> </li>
 <li class="level1">
 <a href="#seealso">See Also</a> </li>
 <li class="level1">
 <a href="#background">Technical Background</a> </li>
 <li class="level1">
 <a href="#literature">Literature</a> </li>
 </ul>
 </div><p><a class="anchor" id="about"></a></p>
 <dl class="section user"><dt>About</dt><dd>Ordinary Least Squares Regression, also called Linear Regression, is a statistical model used to fit linear models.</dd></dl>
 <p>It models a linear relationship of a scalar dependent variable \( y \) to one or more explanatory independent variables \( x \) to build a model of coefficients.</p>
 <p><a class="anchor" id="train"></a></p>
 <dl class="section user"><dt>Training Function</dt><dd><pre class="fragment">linregr_train(
     source_table,
     out_table,
     dependent_varname,
     independent_varname,
     input_group_cols,
     heteroskedasticity_option)
 </pre> <b>Arguments</b> <dl class="arglist">
 <dt>source_table </dt>
 <dd><p class="startdd">TEXT. The name of the table containing the training data.</p>
 <p class="enddd"></p>
 </dd>
 <dt>out_table </dt>
 <dd><p class="startdd">TEXT. Name of the generated table containing the output model.</p>
 <p class="enddd"></p>
 </dd>
 <dt>dependent_varname </dt>
 <dd><p class="startdd">TEXT. Expression to evaluate for the dependent variable.</p>
 <p class="enddd"></p>
 </dd>
 <dt>independent_varname </dt>
 <dd><p class="startdd">TEXT. Expression list to evaluate for the independent variables. An intercept variable is not assumed. It is common to provide an explicit intercept term by including a single constant <code>1</code> term in the independent variable list.</p>
 <p class="enddd"></p>
 </dd>
 <dt>input_group_cols (optional) </dt>
 <dd><p class="startdd">TEXT, default: NULL. An expression list used to group the input dataset into discrete groups, running one regression per group. Similar to the SQL <code>GROUP BY</code> clause. When this value is null, no grouping is used and a single result model is generated.</p>
 <p class="enddd"></p>
 </dd>
 <dt>heteroskedasticity_option (optional) </dt>
 <dd>BOOLEAN, default: FALSE. When TRUE, the heteroskedasticity of the model is also calculated and returned with the results. </dd>
 </dl>
 </dd></dl>
 <p><a class="anchor" id="notes"></a></p>
 <dl class="section note"><dt>Note</dt><dd>All table names can be optionally schema qualified (current_schemas() would be searched if a schema name is not provided) and all table and column names should follow case-sensitivity and quoting rules per the database. (For instance, 'mytable' and 'MyTable' both resolve to the same entity, i.e. 'mytable'. If mixed-case or multi-byte characters are desired for entity names then the string should be double-quoted; in this case the input would be '"MyTable"').</dd></dl>
 <p><a class="anchor" id="output"></a></p>
 <dl class="section user"><dt>Output Table</dt><dd>The output table produced by the linear regression training function contains the following columns. <dl class="arglist">
 <dt>&lt;...&gt; </dt>
 <dd><p class="startdd">Any grouping columns provided during training. Present only if the grouping option is used.</p>
 <p class="enddd"></p>
 </dd>
 <dt>coef </dt>
 <dd><p class="startdd">FLOAT8[]. Vector of the coefficients of the regression.</p>
 <p class="enddd"></p>
 </dd>
 <dt>r2 </dt>
 <dd><p class="startdd">FLOAT8. R-squared coefficient of determination of the model.</p>
 <p class="enddd"></p>
 </dd>
 <dt>std_err </dt>
 <dd><p class="startdd">FLOAT8[]. Vector of the standard error of the coefficients.</p>
 <p class="enddd"></p>
 </dd>
 <dt>t_stats </dt>
 <dd><p class="startdd">FLOAT8[]. Vector of the t-statistics of the coefficients.</p>
 <p class="enddd"></p>
 </dd>
 <dt>p_values </dt>
 <dd><p class="startdd">FLOAT8[]. Vector of the p-values of the coefficients.</p>
 <p class="enddd"></p>
 </dd>
 <dt>condition_no </dt>
 <dd><p class="startdd">FLOAT8 array. The condition number of the \(X^{*}X\) matrix. A high condition number is usually an indication that there may be some numeric instability in the result yielding a less reliable model. A high condition number often results when there is a significant amount of colinearity in the underlying design matrix, in which case other regression techniques, such as elastic net regression, may be more appropriate.</p>
 <p class="enddd"></p>
 </dd>
 <dt>bp_stats </dt>
 <dd><p class="startdd">FLOAT8. The Breush-Pagan statistic of heteroskedacity. Present only if the heteroskedacity argument was set to True when the model was trained.</p>
 <p class="enddd"></p>
 </dd>
 <dt>bp_p_value </dt>
 <dd>FLOAT8. The Breush-Pagan calculated p-value. Present only if the heteroskedacity parameter was set to True when the model was trained. </dd>
 </dl>
 </dd></dl>
 <p><a class="anchor" id="predict"></a></p>
 <dl class="section user"><dt>Prediction Function</dt><dd><pre class="fragment">linregr_predict(
     coef,
     col_ind
 )
 </pre> <b>Arguments</b> <dl class="arglist">
 <dt>coef </dt>
 <dd><p class="startdd">FLOAT8[]. Vector of the coefficients of regression.</p>
 <p class="enddd"></p>
 </dd>
 <dt>col_ind </dt>
 <dd><p class="startdd">FLOAT8[]. An array containing the independent variable column names. </p>
 <p class="enddd"><a class="anchor" id="examples"></a></p>
 </dd>
 </dl>
 </dd></dl>
 <dl class="section user"><dt>Examples</dt><dd><ol type="1">
 <li>Create an input data set. <pre class="fragment">CREATE TABLE houses (id INT, tax INT, bedroom INT, bath FLOAT, price INT,
             size INT, lot INT);
 COPY houses FROM STDIN WITH DELIMITER '|';
   1 |  590 |       2 |    1 |  50000 |  770 | 22100
   2 | 1050 |       3 |    2 |  85000 | 1410 | 12000
   3 |   20 |       3 |    1 |  22500 | 1060 |  3500
   4 |  870 |       2 |    2 |  90000 | 1300 | 17500
   5 | 1320 |       3 |    2 | 133000 | 1500 | 30000
   6 | 1350 |       2 |    1 |  90500 |  820 | 25700
   7 | 2790 |       3 |  2.5 | 260000 | 2130 | 25000
   8 |  680 |       2 |    1 | 142500 | 1170 | 22000
   9 | 1840 |       3 |    2 | 160000 | 1500 | 19000
  10 | 3680 |       4 |    2 | 240000 | 2790 | 20000
  11 | 1660 |       3 |    1 |  87000 | 1030 | 17500
  12 | 1620 |       3 |    2 | 118600 | 1250 | 20000
  13 | 3100 |       3 |    2 | 140000 | 1760 | 38000
  14 | 2070 |       2 |    3 | 148000 | 1550 | 14000
  15 |  650 |       3 |  1.5 |  65000 | 1450 | 12000
 \.
 </pre></li>
 <li>Train a regression model. <pre class="fragment">-- A single regression for all the data
 SELECT madlib.linregr_train(
     'houses',
     'houses_linregr',
     'price',
     'ARRAY[1, tax, bath, size]');

 -- 3 output models, one for each value of "bedroom"
 SELECT madlib.linregr_train(
     'houses',
     'houses_linregr_bedroom',
     'price',
     'ARRAY[1, tax, bath, size]',
     'bedroom');
 </pre></li>
 <li>Examine the resulting models. <pre class="fragment">-- Set extended display on for easier reading of output
 \x on
 SELECT * from houses_linregr;
 SELECT * FROM houses_linregr_bedroom;

 -- Alternatively you can unnest the results for easier reading of output
 \x off
 SELECT unnest(ARRAY['intercept','tax','bath','size']) as attribute,
        unnest(coef) as coefficient,
        unnest(std_err) as standard_error,
        unnest(t_stats) as t_stat,
        unnest(p_values) as pvalue
 FROM houses_linregr;
 </pre></li>
 <li>Use the prediction function to evaluate residuals. <pre class="fragment">SELECT houses.*,
        madlib.linregr_predict(
           ARRAY[1,tax,bath,size],
           m.coef) as predict,
         price -
           madlib.linregr_predict(
               ARRAY[1,tax,bath,size],
               m.coef) as residual
 FROM houses, houses_linregr m;
 </pre></li>
 </ol>
 </dd></dl>
 <p><a class="anchor" id="seealso"></a></p>
 <dl class="section see"><dt>See Also</dt><dd>File <a class="el" href="linear_8sql__in.html" title="SQL functions for linear regression. ">linear.sql_in</a>, documenting the SQL training functions </dd>
 <dd>
 <a class="el" href="linear_8sql__in.html#a71d8295a18e93619b3331cefabe6e79b" title="Compute linear regression coefficients and diagnostic statistics. ">linregr()</a> </dd>
 <dd>
 <a class="el" href="logistic_8sql__in.html#a32880a39de2e36b6c6be72691a6a4a40" title="Compute logistic-regression coefficients and diagnostic statistics. ">logregr_train()</a> </dd>
 <dd>
 <a class="el" href="elastic__net_8sql__in.html#a735038a5090c112505c740a90a203e83" title="Interface for elastic net. ">elastic_net_train()</a> </dd>
 <dd>
 <a class="el" href="group__grp__robust.html">Huber White Variance</a> </dd>
 <dd>
 <a class="el" href="group__grp__clustered__errors.html">Clustered Variance</a> </dd>
 <dd>
 <a class="el" href="group__grp__validation.html">Cross Validation</a></dd></dl>
 <p><a class="anchor" id="background"></a></p>
 <dl class="section user"><dt>Technical Background</dt><dd></dd></dl>
 <p>Ordinary least-squares (OLS) linear regression refers to a stochastic model in which the conditional mean of the dependent variable (usually denoted \( Y \)) is an affine function of the vector of independent variables (usually denoted \( \boldsymbol x \)). That is, </p>
 <p class="formulaDsp">
 \[ E[Y \mid \boldsymbol x] = \boldsymbol c^T \boldsymbol x \]
 </p>
 <p> for some unknown vector of coefficients \( \boldsymbol c \). The assumption is that the residuals are i.i.d. distributed Gaussians. That is, the (conditional) probability density of \( Y \) is given by </p>
 <p class="formulaDsp">
 \[ f(y \mid \boldsymbol x) = \frac{1}{\sqrt{2 \pi \sigma^2}} \cdot \exp\left(-\frac{1}{2 \sigma^2} \cdot (y - \boldsymbol x^T \boldsymbol c)^2 \right) \,. \]
 </p>
 <p> OLS linear regression finds the vector of coefficients \( \boldsymbol c \) that maximizes the likelihood of the observations.</p>
 <p>Let</p>
 <ul>
 <li>\( \boldsymbol y \in \mathbf R^n \) denote the vector of observed dependent variables, with \( n \) rows, containing the observed values of the dependent variable,</li>
 <li>\( X \in \mathbf R^{n \times k} \) denote the design matrix with \( k \) columns and \( n \) rows, containing all observed vectors of independent variables. \( \boldsymbol x_i \) as rows,</li>
 <li>\( X^T \) denote the transpose of \( X \),</li>
 <li>\( X^+ \) denote the pseudo-inverse of \( X \).</li>
 </ul>
 <p>Maximizing the likelihood is equivalent to maximizing the log-likelihood \( \sum_{i=1}^n \log f(y_i \mid \boldsymbol x_i) \), which simplifies to minimizing the <b>residual sum of squares</b> \( RSS \) (also called sum of squared residuals or sum of squared errors of prediction), </p>
 <p class="formulaDsp">
 \[ RSS = \sum_{i=1}^n ( y_i - \boldsymbol c^T \boldsymbol x_i )^2 = (\boldsymbol y - X \boldsymbol c)^T (\boldsymbol y - X \boldsymbol c) \,. \]
 </p>
 <p> The first-order conditions yield that the \( RSS \) is minimized at </p>
 <p class="formulaDsp">
 \[ \boldsymbol c = (X^T X)^+ X^T \boldsymbol y \,. \]
 </p>
 <p>Computing the <b>total sum of squares</b> \( TSS \), the <b>explained sum of squares</b> \( ESS \) (also called the regression sum of squares), and the <b>coefficient of determination</b> \( R^2 \) is done according to the following formulas: </p>
 <p class="formulaDsp">
 \begin{align*} ESS &amp; = \boldsymbol y^T X \boldsymbol c - \frac{ \| y \|_1^2 }{n} \\ TSS &amp; = \sum_{i=1}^n y_i^2 - \frac{ \| y \|_1^2 }{n} \\ R^2 &amp; = \frac{ESS}{TSS} \end{align*}
 </p>
 <p> Note: The last equality follows from the definition \( R^2 = 1 - \frac{RSS}{TSS} \) and the fact that for linear regression \( TSS = RSS + ESS \). A proof of the latter can be found, e.g., at: <a href="http://en.wikipedia.org/wiki/Sum_of_squares">http://en.wikipedia.org/wiki/Sum_of_squares</a></p>
 <p>We estimate the variance \( Var[Y - \boldsymbol c^T \boldsymbol x \mid \boldsymbol x] \) as </p>
 <p class="formulaDsp">
 \[ \sigma^2 = \frac{RSS}{n - k} \]
 </p>
 <p> and compute the t-statistic for coefficient \( i \) as </p>
 <p class="formulaDsp">
 \[ t_i = \frac{c_i}{\sqrt{\sigma^2 \cdot \left( (X^T X)^{-1} \right)_{ii} }} \,. \]
 </p>
 <p>The \( p \)-value for coefficient \( i \) gives the probability of seeing a value at least as extreme as the one observed, provided that the null hypothesis ( \( c_i = 0 \)) is true. Letting \( F_\nu \) denote the cumulative density function of student-t with \( \nu \) degrees of freedom, the \( p \)-value for coefficient \( i \) is therefore </p>
 <p class="formulaDsp">
 \[ p_i = \Pr(|T| \geq |t_i|) = 2 \cdot (1 - F_{n - k}( |t_i| )) \]
 </p>
 <p> where \( T \) is a student-t distributed random variable with mean 0.</p>
 <p>The condition number [2] \( \kappa(X) = \|X\|_2\cdot\|X^{-1}\|_2\) is computed as the product of two spectral norms [3]. The spectral norm of a matrix \(X\) is the largest singular value of \(X\) i.e. the square root of the largest eigenvalue of the positive-semidefinite matrix \(X^{*}X\):</p>
 <p class="formulaDsp">
 \[ \|X\|_2 = \sqrt{\lambda_{\max}\left(X^{*}X\right)}\ , \]
 </p>
 <p> where \(X^{*}\) is the conjugate transpose of \(X\). The condition number of a linear regression problem is a worst-case measure of how sensitive the result is to small perturbations of the input. A large condition number (say, more than 1000) indicates the presence of significant multicollinearity.</p>
 <p><a class="anchor" id="literature"></a></p>
 <dl class="section user"><dt>Literature</dt><dd></dd></dl>
 <p>[1] Cosma Shalizi: Statistics 36-350: Data Mining, Lecture Notes, 21 October 2009, <a href="http://www.stat.cmu.edu/~cshalizi/350/lectures/17/lecture-17.pdf">http://www.stat.cmu.edu/~cshalizi/350/lectures/17/lecture-17.pdf</a></p>
 <p>[2] Wikipedia: Condition Number, <a href="http://en.wikipedia.org/wiki/Condition_number">http://en.wikipedia.org/wiki/Condition_number</a>.</p>
 <p>[3] Wikipedia: Spectral Norm, <a href="http://en.wikipedia.org/wiki/Spectral_norm#Spectral_norm">http://en.wikipedia.org/wiki/Spectral_norm#Spectral_norm</a></p>
 <p>[4] Wikipedia: Breusch–Pagan test, <a href="http://en.wikipedia.org/wiki/Breusch%E2%80%93Pagan_test">http://en.wikipedia.org/wiki/Breusch%E2%80%93Pagan_test</a></p>
 <p>[5] Wikipedia: Heteroscedasticity-consistent standard errors, <a href="http://en.wikipedia.org/wiki/Heteroscedasticity-consistent_standard_errors">http://en.wikipedia.org/wiki/Heteroscedasticity-consistent_standard_errors</a> </p>
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	<div class="header">
	<div class="headertitle">
	<div class="title">Linear Regression<div class="ingroups"><a class="el" href="group__grp__glm.html">Generalized Linear Models</a></div></div> </div>
	</div><!--header-->
	<div class="contents">
	<div class="toc"><b>Contents</b></p>
	<ul>
	<li class="level1">
	<a href="#about">About</a> </li>
	<li class="level1">
	<a href="#train">Training Function</a> </li>
	<li class="level1">
	<a href="#output">Output Table</a> </li>
	<li class="level1">
	<a href="#predict">Prediction Function</a> </li>
	<li class="level1">
	<a href="#examples">Examples</a> </li>
	<li class="level1">
	<a href="#seealso">See Also</a> </li>
	<li class="level1">
	<a href="#background">Technical Background</a> </li>
	<li class="level1">
	<a href="#literature">Literature</a> </li>
	</ul>
	</div><p><a class="anchor" id="about"></a></p>
	<dl class="section user"><dt>About</dt><dd>Ordinary Least Squares Regression, also called Linear Regression, is a statistical model used to fit linear models.</dd></dl>
	<p>It models a linear relationship of a scalar dependent variable \( y \) to one or more explanatory independent variables \( x \) to build a model of coefficients.</p>
	<p><a class="anchor" id="train"></a></p>
	<dl class="section user"><dt>Training Function</dt><dd><pre class="fragment">linregr_train(
	source_table,
	out_table,
	dependent_varname,
	independent_varname,
	input_group_cols,
	heteroskedasticity_option)
	</pre> <b>Arguments</b> <dl class="arglist">
	<dt>source_table </dt>
	<dd><p class="startdd">TEXT. The name of the table containing the training data.</p>
	<p class="enddd"></p>
	</dd>
	<dt>out_table </dt>
	<dd><p class="startdd">TEXT. Name of the generated table containing the output model.</p>
	<p class="enddd"></p>
	</dd>
	<dt>dependent_varname </dt>
	<dd><p class="startdd">TEXT. Expression to evaluate for the dependent variable.</p>
	<p class="enddd"></p>
	</dd>
	<dt>independent_varname </dt>
	<dd><p class="startdd">TEXT. Expression list to evaluate for the independent variables. An intercept variable is not assumed. It is common to provide an explicit intercept term by including a single constant <code>1</code> term in the independent variable list.</p>
	<p class="enddd"></p>
	</dd>
	<dt>input_group_cols (optional) </dt>
	<dd><p class="startdd">TEXT, default: NULL. An expression list used to group the input dataset into discrete groups, running one regression per group. Similar to the SQL <code>GROUP BY</code> clause. When this value is null, no grouping is used and a single result model is generated.</p>
	<p class="enddd"></p>
	</dd>
	<dt>heteroskedasticity_option (optional) </dt>
	<dd>BOOLEAN, default: FALSE. When TRUE, the heteroskedasticity of the model is also calculated and returned with the results. </dd>
	</dl>
	</dd></dl>
	<p><a class="anchor" id="notes"></a></p>
	<dl class="section note"><dt>Note</dt><dd>All table names can be optionally schema qualified (current_schemas() would be searched if a schema name is not provided) and all table and column names should follow case-sensitivity and quoting rules per the database. (For instance, 'mytable' and 'MyTable' both resolve to the same entity, i.e. 'mytable'. If mixed-case or multi-byte characters are desired for entity names then the string should be double-quoted; in this case the input would be '"MyTable"').</dd></dl>
	<p><a class="anchor" id="output"></a></p>
	<dl class="section user"><dt>Output Table</dt><dd>The output table produced by the linear regression training function contains the following columns. <dl class="arglist">
	<dt><...> </dt>
	<dd><p class="startdd">Any grouping columns provided during training. Present only if the grouping option is used.</p>
	<p class="enddd"></p>
	</dd>
	<dt>coef </dt>
	<dd><p class="startdd">FLOAT8[]. Vector of the coefficients of the regression.</p>
	<p class="enddd"></p>
	</dd>
	<dt>r2 </dt>
	<dd><p class="startdd">FLOAT8. R-squared coefficient of determination of the model.</p>
	<p class="enddd"></p>
	</dd>
	<dt>std_err </dt>
	<dd><p class="startdd">FLOAT8[]. Vector of the standard error of the coefficients.</p>
	<p class="enddd"></p>
	</dd>
	<dt>t_stats </dt>
	<dd><p class="startdd">FLOAT8[]. Vector of the t-statistics of the coefficients.</p>
	<p class="enddd"></p>
	</dd>
	<dt>p_values </dt>
	<dd><p class="startdd">FLOAT8[]. Vector of the p-values of the coefficients.</p>
	<p class="enddd"></p>
	</dd>
	<dt>condition_no </dt>
	<dd><p class="startdd">FLOAT8 array. The condition number of the \(X^{*}X\) matrix. A high condition number is usually an indication that there may be some numeric instability in the result yielding a less reliable model. A high condition number often results when there is a significant amount of colinearity in the underlying design matrix, in which case other regression techniques, such as elastic net regression, may be more appropriate.</p>
	<p class="enddd"></p>
	</dd>
	<dt>bp_stats </dt>
	<dd><p class="startdd">FLOAT8. The Breush-Pagan statistic of heteroskedacity. Present only if the heteroskedacity argument was set to True when the model was trained.</p>
	<p class="enddd"></p>
	</dd>
	<dt>bp_p_value </dt>
	<dd>FLOAT8. The Breush-Pagan calculated p-value. Present only if the heteroskedacity parameter was set to True when the model was trained. </dd>
	</dl>
	</dd></dl>
	<p><a class="anchor" id="predict"></a></p>
	<dl class="section user"><dt>Prediction Function</dt><dd><pre class="fragment">linregr_predict(
	coef,
	col_ind
	)
	</pre> <b>Arguments</b> <dl class="arglist">
	<dt>coef </dt>
	<dd><p class="startdd">FLOAT8[]. Vector of the coefficients of regression.</p>
	<p class="enddd"></p>
	</dd>
	<dt>col_ind </dt>
	<dd><p class="startdd">FLOAT8[]. An array containing the independent variable column names. </p>
	<p class="enddd"><a class="anchor" id="examples"></a></p>
	</dd>
	</dl>
	</dd></dl>
	<dl class="section user"><dt>Examples</dt><dd><ol type="1">
	<li>Create an input data set. <pre class="fragment">CREATE TABLE houses (id INT, tax INT, bedroom INT, bath FLOAT, price INT,
	size INT, lot INT);
	COPY houses FROM STDIN WITH DELIMITER '\|';
	1 \| 590 \| 2 \| 1 \| 50000 \| 770 \| 22100
	2 \| 1050 \| 3 \| 2 \| 85000 \| 1410 \| 12000
	3 \| 20 \| 3 \| 1 \| 22500 \| 1060 \| 3500
	4 \| 870 \| 2 \| 2 \| 90000 \| 1300 \| 17500
	5 \| 1320 \| 3 \| 2 \| 133000 \| 1500 \| 30000
	6 \| 1350 \| 2 \| 1 \| 90500 \| 820 \| 25700
	7 \| 2790 \| 3 \| 2.5 \| 260000 \| 2130 \| 25000
	8 \| 680 \| 2 \| 1 \| 142500 \| 1170 \| 22000
	9 \| 1840 \| 3 \| 2 \| 160000 \| 1500 \| 19000
	10 \| 3680 \| 4 \| 2 \| 240000 \| 2790 \| 20000
	11 \| 1660 \| 3 \| 1 \| 87000 \| 1030 \| 17500
	12 \| 1620 \| 3 \| 2 \| 118600 \| 1250 \| 20000
	13 \| 3100 \| 3 \| 2 \| 140000 \| 1760 \| 38000
	14 \| 2070 \| 2 \| 3 \| 148000 \| 1550 \| 14000
	15 \| 650 \| 3 \| 1.5 \| 65000 \| 1450 \| 12000
	\.
	</pre></li>
	<li>Train a regression model. <pre class="fragment">-- A single regression for all the data
	SELECT madlib.linregr_train(
	'houses',
	'houses_linregr',
	'price',
	'ARRAY[1, tax, bath, size]');

	-- 3 output models, one for each value of "bedroom"
	SELECT madlib.linregr_train(
	'houses',
	'houses_linregr_bedroom',
	'price',
	'ARRAY[1, tax, bath, size]',
	'bedroom');
	</pre></li>
	<li>Examine the resulting models. <pre class="fragment">-- Set extended display on for easier reading of output
	\x on
	SELECT * from houses_linregr;
	SELECT * FROM houses_linregr_bedroom;

	-- Alternatively you can unnest the results for easier reading of output
	\x off
	SELECT unnest(ARRAY['intercept','tax','bath','size']) as attribute,
	unnest(coef) as coefficient,
	unnest(std_err) as standard_error,
	unnest(t_stats) as t_stat,
	unnest(p_values) as pvalue
	FROM houses_linregr;
	</pre></li>
	<li>Use the prediction function to evaluate residuals. <pre class="fragment">SELECT houses.*,
	madlib.linregr_predict(
	ARRAY[1,tax,bath,size],
	m.coef) as predict,
	price -
	madlib.linregr_predict(
	ARRAY[1,tax,bath,size],
	m.coef) as residual
	FROM houses, houses_linregr m;
	</pre></li>
	</ol>
	</dd></dl>
	<p><a class="anchor" id="seealso"></a></p>
	<dl class="section see"><dt>See Also</dt><dd>File <a class="el" href="linear_8sql__in.html" title="SQL functions for linear regression. ">linear.sql_in</a>, documenting the SQL training functions </dd>
	<dd>
	<a class="el" href="linear_8sql__in.html#a71d8295a18e93619b3331cefabe6e79b" title="Compute linear regression coefficients and diagnostic statistics. ">linregr()</a> </dd>
	<dd>
	<a class="el" href="logistic_8sql__in.html#a32880a39de2e36b6c6be72691a6a4a40" title="Compute logistic-regression coefficients and diagnostic statistics. ">logregr_train()</a> </dd>
	<dd>
	<a class="el" href="elastic__net_8sql__in.html#a735038a5090c112505c740a90a203e83" title="Interface for elastic net. ">elastic_net_train()</a> </dd>
	<dd>
	<a class="el" href="group__grp__robust.html">Huber White Variance</a> </dd>
	<dd>
	<a class="el" href="group__grp__clustered__errors.html">Clustered Variance</a> </dd>
	<dd>
	<a class="el" href="group__grp__validation.html">Cross Validation</a></dd></dl>
	<p><a class="anchor" id="background"></a></p>
	<dl class="section user"><dt>Technical Background</dt><dd></dd></dl>
	<p>Ordinary least-squares (OLS) linear regression refers to a stochastic model in which the conditional mean of the dependent variable (usually denoted \( Y \)) is an affine function of the vector of independent variables (usually denoted \( \boldsymbol x \)). That is, </p>
	<p class="formulaDsp">
	\[ E[Y \mid \boldsymbol x] = \boldsymbol c^T \boldsymbol x \]
	</p>
	<p> for some unknown vector of coefficients \( \boldsymbol c \). The assumption is that the residuals are i.i.d. distributed Gaussians. That is, the (conditional) probability density of \( Y \) is given by </p>
	<p class="formulaDsp">
	\[ f(y \mid \boldsymbol x) = \frac{1}{\sqrt{2 \pi \sigma^2}} \cdot \exp\left(-\frac{1}{2 \sigma^2} \cdot (y - \boldsymbol x^T \boldsymbol c)^2 \right) \,. \]
	</p>
	<p> OLS linear regression finds the vector of coefficients \( \boldsymbol c \) that maximizes the likelihood of the observations.</p>
	<p>Let</p>
	<ul>
	<li>\( \boldsymbol y \in \mathbf R^n \) denote the vector of observed dependent variables, with \( n \) rows, containing the observed values of the dependent variable,</li>
	<li>\( X \in \mathbf R^{n \times k} \) denote the design matrix with \( k \) columns and \( n \) rows, containing all observed vectors of independent variables. \( \boldsymbol x_i \) as rows,</li>
	<li>\( X^T \) denote the transpose of \( X \),</li>
	<li>\( X^+ \) denote the pseudo-inverse of \( X \).</li>
	</ul>
	<p>Maximizing the likelihood is equivalent to maximizing the log-likelihood \( \sum_{i=1}^n \log f(y_i \mid \boldsymbol x_i) \), which simplifies to minimizing the <b>residual sum of squares</b> \( RSS \) (also called sum of squared residuals or sum of squared errors of prediction), </p>
	<p class="formulaDsp">
	\[ RSS = \sum_{i=1}^n ( y_i - \boldsymbol c^T \boldsymbol x_i )^2 = (\boldsymbol y - X \boldsymbol c)^T (\boldsymbol y - X \boldsymbol c) \,. \]
	</p>
	<p> The first-order conditions yield that the \( RSS \) is minimized at </p>
	<p class="formulaDsp">
	\[ \boldsymbol c = (X^T X)^+ X^T \boldsymbol y \,. \]
	</p>
	<p>Computing the <b>total sum of squares</b> \( TSS \), the <b>explained sum of squares</b> \( ESS \) (also called the regression sum of squares), and the <b>coefficient of determination</b> \( R^2 \) is done according to the following formulas: </p>
	<p class="formulaDsp">
	\begin{align} ESS & = \boldsymbol y^T X \boldsymbol c - \frac{ \\| y \\|_1^2 }{n} \\ TSS & = \sum_{i=1}^n y_i^2 - \frac{ \\| y \\|_1^2 }{n} \\ R^2 & = \frac{ESS}{TSS} \end{align}
	</p>
	<p> Note: The last equality follows from the definition \( R^2 = 1 - \frac{RSS}{TSS} \) and the fact that for linear regression \( TSS = RSS + ESS \). A proof of the latter can be found, e.g., at: <a href="http://en.wikipedia.org/wiki/Sum_of_squares">http://en.wikipedia.org/wiki/Sum_of_squares</a></p>
	<p>We estimate the variance \( Var[Y - \boldsymbol c^T \boldsymbol x \mid \boldsymbol x] \) as </p>
	<p class="formulaDsp">
	\[ \sigma^2 = \frac{RSS}{n - k} \]
	</p>
	<p> and compute the t-statistic for coefficient \( i \) as </p>
	<p class="formulaDsp">
	\[ t_i = \frac{c_i}{\sqrt{\sigma^2 \cdot \left( (X^T X)^{-1} \right)_{ii} }} \,. \]
	</p>
	<p>The \( p \)-value for coefficient \( i \) gives the probability of seeing a value at least as extreme as the one observed, provided that the null hypothesis ( \( c_i = 0 \)) is true. Letting \( F_\nu \) denote the cumulative density function of student-t with \( \nu \) degrees of freedom, the \( p \)-value for coefficient \( i \) is therefore </p>
	<p class="formulaDsp">
	\[ p_i = \Pr(\|T\| \geq \|t_i\|) = 2 \cdot (1 - F_{n - k}( \|t_i\| )) \]
	</p>
	<p> where \( T \) is a student-t distributed random variable with mean 0.</p>
	<p>The condition number [2] \( \kappa(X) = \\|X\\|_2\cdot\\|X^{-1}\\|_2\) is computed as the product of two spectral norms [3]. The spectral norm of a matrix \(X\) is the largest singular value of \(X\) i.e. the square root of the largest eigenvalue of the positive-semidefinite matrix \(X^{*}X\):</p>
	<p class="formulaDsp">
	\[ \\|X\\|_2 = \sqrt{\lambda_{\max}\left(X^{*}X\right)}\ , \]
	</p>
	<p> where \(X^{*}\) is the conjugate transpose of \(X\). The condition number of a linear regression problem is a worst-case measure of how sensitive the result is to small perturbations of the input. A large condition number (say, more than 1000) indicates the presence of significant multicollinearity.</p>
	<p><a class="anchor" id="literature"></a></p>
	<dl class="section user"><dt>Literature</dt><dd></dd></dl>
	<p>[1] Cosma Shalizi: Statistics 36-350: Data Mining, Lecture Notes, 21 October 2009, <a href="http://www.stat.cmu.edu/~cshalizi/350/lectures/17/lecture-17.pdf">http://www.stat.cmu.edu/~cshalizi/350/lectures/17/lecture-17.pdf</a></p>
	<p>[2] Wikipedia: Condition Number, <a href="http://en.wikipedia.org/wiki/Condition_number">http://en.wikipedia.org/wiki/Condition_number</a>.</p>
	<p>[3] Wikipedia: Spectral Norm, <a href="http://en.wikipedia.org/wiki/Spectral_norm#Spectral_norm">http://en.wikipedia.org/wiki/Spectral_norm#Spectral_norm</a></p>
	<p>[4] Wikipedia: Breusch–Pagan test, <a href="http://en.wikipedia.org/wiki/Breusch%E2%80%93Pagan_test">http://en.wikipedia.org/wiki/Breusch%E2%80%93Pagan_test</a></p>
	<p>[5] Wikipedia: Heteroscedasticity-consistent standard errors, <a href="http://en.wikipedia.org/wiki/Heteroscedasticity-consistent_standard_errors">http://en.wikipedia.org/wiki/Heteroscedasticity-consistent_standard_errors</a> </p>
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