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<title>MADlib: Linear Regression</title>
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<div class="title">Linear Regression<div class="ingroups"><a class="el" href="group__grp__glm.html">Generalized Linear Models</a></div></div> </div>
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<div class="contents">
<div class="toc"><b>Contents</b> </p>
<ul>
<li class="level1">
<a href="#train">Training Function</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="#background">Technical Background</a> </li>
<li class="level1">
<a href="#literature">Literature</a> </li>
<li class="level1">
<a href="#related">Related Topics</a> </li>
</ul>
</div><p>Linear regression 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></dd></dl>
<p>The linear regression training function has the following syntax. </p>
<pre class="syntax">
linregr_train( source_table,
out_table,
dependent_varname,
independent_varname,
grouping_cols,
heteroskedasticity_option
)
</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 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>The output table contains the following columns. </p>
<table class="output">
<tr>
<th>&lt;...&gt; </th><td>Any grouping columns provided during training. Present only if the grouping option is used. </td></tr>
<tr>
<th>coef </th><td>FLOAT8[]. Vector of the coefficients of the regression. </td></tr>
<tr>
<th>r2 </th><td>FLOAT8. R-squared coefficient of determination of the model. </td></tr>
<tr>
<th>std_err </th><td>FLOAT8[]. Vector of the standard error of the coefficients. </td></tr>
<tr>
<th>t_stats </th><td>FLOAT8[]. Vector of the t-statistics of the coefficients. </td></tr>
<tr>
<th>p_values </th><td>FLOAT8[]. Vector of the p-values of the coefficients. </td></tr>
<tr>
<th>condition_no </th><td>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. </td></tr>
<tr>
<th>bp_stats </th><td>FLOAT8. The Breush-Pagan statistic of heteroskedacity. Present only if the heteroskedacity argument was set to True when the model was trained. </td></tr>
<tr>
<th>bp_p_value </th><td>FLOAT8. The Breush-Pagan calculated p-value. Present only if the heteroskedacity parameter was set to True when the model was trained. </td></tr>
<tr>
<th>num_rows_processed </th><td>INTEGER. The number of rows that are actually used in each group. </td></tr>
<tr>
<th>num_missing_rows_skipped </th><td>INTEGER. The number of rows that have NULL values in the dependent and independent variables, and were skipped in the computation for each group. </td></tr>
</table>
<p>A summary table named &lt;out_table&gt;_summary is created together with the output table. It has the following columns: </p>
<table class="output">
<tr>
<th>source_table </th><td>The data source table name </td></tr>
<tr>
<th>out_table </th><td>The output table name </td></tr>
<tr>
<th>dependent_varname </th><td>The dependent variable </td></tr>
<tr>
<th>independent_varname </th><td>The independent variables </td></tr>
<tr>
<th>num_rows_processed </th><td>The total number of rows that were used in the computation. </td></tr>
<tr>
<th>num_missing_rows_skipped </th><td>The total number of rows that were skipped because of NULL values in them. </td></tr>
</table>
<dl class="section note"><dt>Note</dt><dd>For p-values, we just return the computation result directly. Other statistical packages, like 'R', produce the same result, but on printing the result to screen, another format function is used and any p-value that is smaller than the machine epsilon (the smallest positive floating-point number 'x' such that '1 + x != 1') will be printed on screen as "&lt; xxx" (xxx is the value of the machine epsilon). Although the result may look different, they are in fact the same. </dd></dl>
</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>grouping_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>
<p><a class="anchor" id="warning"></a></p>
<dl class="section warning"><dt>Warning</dt><dd>The aggregate 'linregr' has been deprecated in favor of the function 'linregr_train'. If the aggregate 'linregr' is used to output the results of linear regression to a table, it is recommended to follow the general pattern shown below (replace text within '&lt;...&gt;' with the appropriate variable names). <pre class="syntax">
CREATE TABLE &lt;output table&gt; AS
SELECT (r).*
FROM (
SELECT linregr(&lt;dependent variable&gt;, &lt;independent variable&gt;) as r
FROM &lt;source table&gt;
) q;
</pre></dd></dl>
<p><a class="anchor" id="predict"></a></p>
<dl class="section user"><dt>Prediction Function</dt><dd><pre class="syntax">
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="example">
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. First, a single regression for all the data. <pre class="example">
SELECT madlib.linregr_train( 'houses',
'houses_linregr',
'price',
'ARRAY[1, tax, bath, size]'
);
</pre></li>
<li>Generate three output models, one for each value of "bedroom". <pre class="example">
SELECT madlib.linregr_train( 'houses',
'houses_linregr_bedroom',
'price',
'ARRAY[1, tax, bath, size]',
'bedroom'
);
</pre></li>
<li>Examine the resulting models. <pre class="example">
-- Set extended display on for easier reading of output
\x ON
SELECT * FROM houses_linregr;
</pre> Result: <pre class="result">
-[ RECORD 1 ]+---------------------------------------------------------------------------
coef | {-12849.4168959872,28.9613922651765,10181.6290712648,50.516894915354}
r2 | 0.768577580597443
std_err | {33453.0344331391,15.8992104963997,19437.7710925923,32.928023174087}
t_stats | {-0.38410317968819,1.82156166004184,0.523806408809133,1.53416118083605}
p_values | {0.708223134615422,0.0958005827189772,0.610804093526536,0.153235085548186}
condition_no | 9002.50457085737
</pre></li>
<li>View the results grouped by bedroom. <pre class="example">
SELECT * FROM houses_linregr_bedroom;
</pre> Result: <pre class="result">
-[ RECORD 1 ]+--------------------------------------------------------------------------
bedroom | 2
coef | {-84242.0345406597,55.4430144648696,-78966.9753675319,225.611910021192}
r2 | 0.968809546465313
std_err | {35018.9991665742,19.5731125320686,23036.8071292552,49.0448678148784}
t_stats | {-2.40560942761235,2.83261103077151,-3.42786111480046,4.60011251070697}
p_values | {0.250804617665239,0.21605133377602,0.180704400437373,0.136272031474122}
condition_no | 10086.1048721726
-[ RECORD 2 ]+--------------------------------------------------------------------------
bedroom | 4
coef | {0.0112536020318378,41.4132554771633,0.0225072040636757,31.3975496688276}
r2 | 1
std_err | {0,0,0,0}
t_stats | {Infinity,Infinity,Infinity,Infinity}
p_values |
condition_no | Infinity
-[ RECORD 3 ]+--------------------------------------------------------------------------
bedroom | 3
coef | {-88155.8292501601,27.1966436294429,41404.0293363612,62.637521075324}
r2 | 0.841699901311252
std_err | {57867.9999702625,17.8272309154689,43643.1321511114,70.8506824863954}
t_stats | {-1.52339512849005,1.52556747362508,0.948695185143966,0.884077878676067}
p_values | {0.188161432894871,0.187636685729869,0.386340032374927,0.417132778705789}
condition_no | 11722.6225642147
</pre> Alternatively you can unnest the results for easier reading of output. <pre class="example">
\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="example">
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="notes"></a></p>
<dl class="section user"><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="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>
<p><a class="anchor" id="related"></a></p>
<dl class="section user"><dt>Related Topics</dt><dd></dd></dl>
<p><a class="el" href="group__grp__robust.html">Robust Variance</a></p>
<p><a class="el" href="group__grp__clustered__errors.html">Clustered Variance</a></p>
<p><a class="el" href="group__grp__validation.html">Cross Validation</a></p>
<p>File <a class="el" href="linear_8sql__in.html" title="SQL functions for linear regression. ">linear.sql_in</a>, source file for the SQL functions </p>
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