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 <div class="title">Multinomial Logistic Regression</div>  </div>
 <div class="ingroups"><a class="el" href="group__grp__suplearn.html">Supervised Learning</a></div></div>
 <div class="contents">
 <div id="dynsection-0" onclick="return toggleVisibility(this)" class="dynheader closed" style="cursor:pointer;">
   <img id="dynsection-0-trigger" src="closed.png" alt="+"/> Collaboration diagram for Multinomial Logistic Regression:</div>
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 </div>
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 <dl class="user"><dt><b>About:</b></dt><dd></dd></dl>
 <p>Multinomial logistic regression is a widely used regression analysis tool that models the outcomes of categorical dependent random variables (denoted \( Y \in \{ 0,1,2 \ldots k \} \)). The models assumes that the conditional mean of the dependant categorical variables is the logistic function of an affine combination of independent variables (usually denoted \( \boldsymbol x \)). That is, </p>
 <p class="formulaDsp">
 \[ E[Y \mid \boldsymbol x] = \sigma(\boldsymbol c^T \boldsymbol x) \]
 </p>
 <p> for some unknown vector of coefficients \( \boldsymbol c \) and where \( \sigma(x) = \frac{1}{1 + \exp(-x)} \) is the logistic function. Multinomial Logistic regression finds the vector of coefficients \( \boldsymbol c \) that maximizes the likelihood of the observations.</p>
 <p>Let</p>
 <ul>
 <li>\( \boldsymbol y \in \{ 0,1 \}^{n \times k} \) denote the vector of observed dependent variables, with \( n \) rows and \( k \) columns, 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>
 </ul>
 <p>By definition, </p>
 <p class="formulaDsp">
 \[ P[Y = y_i | \boldsymbol x_i] = \sigma((-1)^{y_i} \cdot \boldsymbol c^T \boldsymbol x_i) \,. \]
 </p>
 <p> Maximizing the likelihood \( \prod_{i=1}^n \Pr(Y = y_i \mid \boldsymbol x_i) \) is equivalent to maximizing the log-likelihood \( \sum_{i=1}^n \log \Pr(Y = y_i \mid \boldsymbol x_i) \), which simplifies to </p>
 <p class="formulaDsp">
 \[ l(\boldsymbol c) = -\sum_{i=1}^n \log(1 + \exp((-1)^{y_i} \cdot \boldsymbol c^T \boldsymbol x_i)) \,. \]
 </p>
 <p> The Hessian of this objective is \( H = -X^T A X \) where \( A = \text{diag}(a_1, \dots, a_n) \) is the diagonal matrix with \( a_i = \sigma(\boldsymbol c^T \boldsymbol x) \cdot \sigma(-\boldsymbol c^T \boldsymbol x) \,. \) Since \( H \) is non-positive definite, \( l(\boldsymbol c) \) is convex. There are many techniques for solving convex optimization problems. Currently, logistic regression in MADlib can use:</p>
 <ul>
 <li>Iteratively Reweighted Least Squares</li>
 </ul>
 <p>We estimate the standard error for coefficient \( i \) as </p>
 <p class="formulaDsp">
 \[ \mathit{se}(c_i) = \left( (X^T A X)^{-1} \right)_{ii} \,. \]
 </p>
 <p> The Wald z-statistic is </p>
 <p class="formulaDsp">
 \[ z_i = \frac{c_i}{\mathit{se}(c_i)} \,. \]
 </p>
 <p>The Wald \( p \)-value for coefficient \( i \) gives the probability (under the assumptions inherent in the Wald test) of seeing a value at least as extreme as the one observed, provided that the null hypothesis ( \( c_i = 0 \)) is true. Letting \( F \) denote the cumulative density function of a standard normal distribution, the Wald \( p \)-value for coefficient \( i \) is therefore </p>
 <p class="formulaDsp">
 \[ p_i = \Pr(|Z| \geq |z_i|) = 2 \cdot (1 - F( |z_i| )) \]
 </p>
 <p> where \( Z \) is a standard normally distributed random variable.</p>
 <p>The odds ratio for coefficient \( i \) is estimated as \( \exp(c_i) \).</p>
 <p>The condition number is computed as \( \kappa(X^T A X) \) during the iteration immediately <em>preceding</em> convergence (i.e., \( A \) is computed using the coefficients of the previous iteration). A large condition number (say, more than 1000) indicates the presence of significant multicollinearity.</p>
 <dl class="user"><dt><b>Input:</b></dt><dd></dd></dl>
 <p>The training data is expected to be of the following form:<br/>
  </p>
 <pre>{TABLE|VIEW} <em>sourceName</em> (
     ...
     <em>dependentVariable</em> INTEGER,
     <em>numCategories</em> INTEGER,
     <em>independentVariables</em> FLOAT8[],
     ...
 )</pre><dl class="user"><dt><b>Usage:</b></dt><dd><ul>
 <li>The number of independent variables cannot exceed 65535.</li>
 <li>Get vector of coefficients \( \boldsymbol c \) and all diagnostic statistics:<br/>
  <pre>SELECT * FROM <a class="el" href="multilogistic_8sql__in.html#a8037cc95b1349de7f8910768be3262c8">mlogregr</a>(
     '<em>sourceName</em>', '<em>dependentVariable</em>', '<em>numCategories</em>' , '<em>independentVariables</em>'
     [, <em>numberOfIterations</em> [, '<em>optimizer</em>' [, <em>precision</em> ] ] ]
 );</pre> Output: <pre>coef | log_likelihood | std_err | z_stats | p_values | odds_ratios | condition_no | num_iterations
 -----+----------------+---------+---------+----------+-------------+--------------+---------------
                                                ...
 </pre></li>
 <li>Get vector of coefficients \( \boldsymbol c \):<br/>
  <pre>SELECT (<a class="el" href="multilogistic_8sql__in.html#a8037cc95b1349de7f8910768be3262c8">mlogregr</a>('<em>sourceName</em>', '<em>dependentVariable</em>', '<em>numCategories</em>', '<em>independentVariables</em>')).coef;</pre></li>
 <li>Get a subset of the output columns, e.g., only the array of coefficients \( \boldsymbol c \), the log-likelihood of determination \( l(\boldsymbol c) \), and the array of p-values \( \boldsymbol p \): <pre>SELECT coef, log_likelihood, p_values
 FROM <a class="el" href="multilogistic_8sql__in.html#a8037cc95b1349de7f8910768be3262c8">mlogregr</a>('<em>sourceName</em>', '<em>dependentVariable</em>', '<em>numCategories</em>',  '<em>independentVariables</em>');</pre></li>
 </ul>
 </dd></dl>
 <p>Note that the categories are encoded as integers with values from {0, 1, 2,...numCategories} </p>
 <dl class="user"><dt><b>Examples:</b></dt><dd></dd></dl>
 <ol type="1">
 <li>Create the sample data set: <div class="fragment"><pre class="fragment">
 sql&gt; SELECT * FROM data;
                   r1                      | val
 ---------------------------------------------+-----
  {1,3.01789340097457,0.454183579888195}   | 1
  {1,-2.59380532894284,0.602678326424211}  | 0
  {1,-1.30643094424158,0.151587064377964}  | 1
  {1,3.60722299199551,0.963550757616758}   | 1
  {1,-1.52197745628655,0.0782248834148049} | 1
  {1,-4.8746574902907,0.345104880165309}   | 0
 ...
 </pre></div></li>
 <li>Run the multi-logistic regression function: <div class="fragment"><pre class="fragment">
 sql&gt; \x on
 Expanded display is off.
 sql&gt; SELECT * FROM mlogregr('data', 'val', '2', 'r1', 100, 'irls', 0.001);
 -[ RECORD 1 ]--+--------------------------------------------------------------
 coef           | {5.59049410898112,2.11077546770772,-0.237276684606453}
 log_likelihood | -467.214718489873
 std_err        | {0.318943457652178,0.101518723785383,0.294509929481773}
 z_stats        | {17.5281667482197,20.7919819024719,-0.805666162169712}
 p_values       | {8.73403463417837e-69,5.11539430631541e-96,0.420435365338518}
 odds_ratios    | {267.867942976278,8.2546400100702,0.788773016471171}
 condition_no   | 179.186118573205
 num_iterations | 9

 </pre></div></li>
 </ol>
 <dl class="user"><dt><b>Literature:</b></dt><dd></dd></dl>
 <p>A collection of nice write-ups, with valuable pointers into further literature:</p>
 <p>[1] Annette J . Dobson: An Introduction to Generalized Linear Models, Second Edition. Nov 2001</p>
 <p>[2] Cosma Shalizi: Statistics 36-350: Data Mining, Lecture Notes, 18 November 2009, <a href="http://www.stat.cmu.edu/~cshalizi/350/lectures/26/lecture-26.pdf">http://www.stat.cmu.edu/~cshalizi/350/lectures/26/lecture-26.pdf</a></p>
 <p>[3] Srikrishna Sridhar, Mark Wellons, Caleb Welton: Multilogistic Regression: Notes and References, Jul 12 2012, <a href="http://www.cs.wisc.edu/~srikris/mlogit.pdf">http://www.cs.wisc.edu/~srikris/mlogit.pdf</a></p>
 <p>[4] Scott A. Czepiel: Maximum Likelihood Estimation of Logistic Regression Models: Theory and Implementation, Retrieved Jul 12 2012, <a href="http://czep.net/stat/mlelr.pdf">http://czep.net/stat/mlelr.pdf</a></p>
 <dl class="see"><dt><b>See also:</b></dt><dd>File <a class="el" href="multilogistic_8sql__in.html" title="SQL functions for multinomial logistic regression.">multilogistic.sql_in</a> (documenting the SQL functions) </dd></dl>
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	<div class="title">Multinomial Logistic Regression</div> </div>
	<div class="ingroups"><a class="el" href="group__grp__suplearn.html">Supervised Learning</a></div></div>
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	<dl class="user"><dt><b>About:</b></dt><dd></dd></dl>
	<p>Multinomial logistic regression is a widely used regression analysis tool that models the outcomes of categorical dependent random variables (denoted \( Y \in \{ 0,1,2 \ldots k \} \)). The models assumes that the conditional mean of the dependant categorical variables is the logistic function of an affine combination of independent variables (usually denoted \( \boldsymbol x \)). That is, </p>
	<p class="formulaDsp">
	\[ E[Y \mid \boldsymbol x] = \sigma(\boldsymbol c^T \boldsymbol x) \]
	</p>
	<p> for some unknown vector of coefficients \( \boldsymbol c \) and where \( \sigma(x) = \frac{1}{1 + \exp(-x)} \) is the logistic function. Multinomial Logistic regression finds the vector of coefficients \( \boldsymbol c \) that maximizes the likelihood of the observations.</p>
	<p>Let</p>
	<ul>
	<li>\( \boldsymbol y \in \{ 0,1 \}^{n \times k} \) denote the vector of observed dependent variables, with \( n \) rows and \( k \) columns, 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>
	</ul>
	<p>By definition, </p>
	<p class="formulaDsp">
	\[ P[Y = y_i \| \boldsymbol x_i] = \sigma((-1)^{y_i} \cdot \boldsymbol c^T \boldsymbol x_i) \,. \]
	</p>
	<p> Maximizing the likelihood \( \prod_{i=1}^n \Pr(Y = y_i \mid \boldsymbol x_i) \) is equivalent to maximizing the log-likelihood \( \sum_{i=1}^n \log \Pr(Y = y_i \mid \boldsymbol x_i) \), which simplifies to </p>
	<p class="formulaDsp">
	\[ l(\boldsymbol c) = -\sum_{i=1}^n \log(1 + \exp((-1)^{y_i} \cdot \boldsymbol c^T \boldsymbol x_i)) \,. \]
	</p>
	<p> The Hessian of this objective is \( H = -X^T A X \) where \( A = \text{diag}(a_1, \dots, a_n) \) is the diagonal matrix with \( a_i = \sigma(\boldsymbol c^T \boldsymbol x) \cdot \sigma(-\boldsymbol c^T \boldsymbol x) \,. \) Since \( H \) is non-positive definite, \( l(\boldsymbol c) \) is convex. There are many techniques for solving convex optimization problems. Currently, logistic regression in MADlib can use:</p>
	<ul>
	<li>Iteratively Reweighted Least Squares</li>
	</ul>
	<p>We estimate the standard error for coefficient \( i \) as </p>
	<p class="formulaDsp">
	\[ \mathit{se}(c_i) = \left( (X^T A X)^{-1} \right)_{ii} \,. \]
	</p>
	<p> The Wald z-statistic is </p>
	<p class="formulaDsp">
	\[ z_i = \frac{c_i}{\mathit{se}(c_i)} \,. \]
	</p>
	<p>The Wald \( p \)-value for coefficient \( i \) gives the probability (under the assumptions inherent in the Wald test) of seeing a value at least as extreme as the one observed, provided that the null hypothesis ( \( c_i = 0 \)) is true. Letting \( F \) denote the cumulative density function of a standard normal distribution, the Wald \( p \)-value for coefficient \( i \) is therefore </p>
	<p class="formulaDsp">
	\[ p_i = \Pr(\|Z\| \geq \|z_i\|) = 2 \cdot (1 - F( \|z_i\| )) \]
	</p>
	<p> where \( Z \) is a standard normally distributed random variable.</p>
	<p>The odds ratio for coefficient \( i \) is estimated as \( \exp(c_i) \).</p>
	<p>The condition number is computed as \( \kappa(X^T A X) \) during the iteration immediately <em>preceding</em> convergence (i.e., \( A \) is computed using the coefficients of the previous iteration). A large condition number (say, more than 1000) indicates the presence of significant multicollinearity.</p>
	<dl class="user"><dt><b>Input:</b></dt><dd></dd></dl>
	<p>The training data is expected to be of the following form:<br/>
	</p>
	<pre>{TABLE\|VIEW} <em>sourceName</em> (
	...
	<em>dependentVariable</em> INTEGER,
	<em>numCategories</em> INTEGER,
	<em>independentVariables</em> FLOAT8[],
	...
	)</pre><dl class="user"><dt><b>Usage:</b></dt><dd><ul>
	<li>The number of independent variables cannot exceed 65535.</li>
	<li>Get vector of coefficients \( \boldsymbol c \) and all diagnostic statistics:<br/>
	<pre>SELECT * FROM <a class="el" href="multilogistic_8sql__in.html#a8037cc95b1349de7f8910768be3262c8">mlogregr</a>(
	'<em>sourceName</em>', '<em>dependentVariable</em>', '<em>numCategories</em>' , '<em>independentVariables</em>'
	[, <em>numberOfIterations</em> [, '<em>optimizer</em>' [, <em>precision</em> ] ] ]
	);</pre> Output: <pre>coef \| log_likelihood \| std_err \| z_stats \| p_values \| odds_ratios \| condition_no \| num_iterations
	-----+----------------+---------+---------+----------+-------------+--------------+---------------
	...
	</pre></li>
	<li>Get vector of coefficients \( \boldsymbol c \):<br/>
	<pre>SELECT (<a class="el" href="multilogistic_8sql__in.html#a8037cc95b1349de7f8910768be3262c8">mlogregr</a>('<em>sourceName</em>', '<em>dependentVariable</em>', '<em>numCategories</em>', '<em>independentVariables</em>')).coef;</pre></li>
	<li>Get a subset of the output columns, e.g., only the array of coefficients \( \boldsymbol c \), the log-likelihood of determination \( l(\boldsymbol c) \), and the array of p-values \( \boldsymbol p \): <pre>SELECT coef, log_likelihood, p_values
	FROM <a class="el" href="multilogistic_8sql__in.html#a8037cc95b1349de7f8910768be3262c8">mlogregr</a>('<em>sourceName</em>', '<em>dependentVariable</em>', '<em>numCategories</em>', '<em>independentVariables</em>');</pre></li>
	</ul>
	</dd></dl>
	<p>Note that the categories are encoded as integers with values from {0, 1, 2,...numCategories} </p>
	<dl class="user"><dt><b>Examples:</b></dt><dd></dd></dl>
	<ol type="1">
	<li>Create the sample data set: <div class="fragment"><pre class="fragment">
	sql> SELECT * FROM data;
	r1 \| val
	---------------------------------------------+-----
	{1,3.01789340097457,0.454183579888195} \| 1
	{1,-2.59380532894284,0.602678326424211} \| 0
	{1,-1.30643094424158,0.151587064377964} \| 1
	{1,3.60722299199551,0.963550757616758} \| 1
	{1,-1.52197745628655,0.0782248834148049} \| 1
	{1,-4.8746574902907,0.345104880165309} \| 0
	...
	</pre></div></li>
	<li>Run the multi-logistic regression function: <div class="fragment"><pre class="fragment">
	sql> \x on
	Expanded display is off.
	sql> SELECT * FROM mlogregr('data', 'val', '2', 'r1', 100, 'irls', 0.001);
	-[ RECORD 1 ]--+--------------------------------------------------------------
	coef \| {5.59049410898112,2.11077546770772,-0.237276684606453}
	log_likelihood \| -467.214718489873
	std_err \| {0.318943457652178,0.101518723785383,0.294509929481773}
	z_stats \| {17.5281667482197,20.7919819024719,-0.805666162169712}
	p_values \| {8.73403463417837e-69,5.11539430631541e-96,0.420435365338518}
	odds_ratios \| {267.867942976278,8.2546400100702,0.788773016471171}
	condition_no \| 179.186118573205
	num_iterations \| 9

	</pre></div></li>
	</ol>
	<dl class="user"><dt><b>Literature:</b></dt><dd></dd></dl>
	<p>A collection of nice write-ups, with valuable pointers into further literature:</p>
	<p>[1] Annette J . Dobson: An Introduction to Generalized Linear Models, Second Edition. Nov 2001</p>
	<p>[2] Cosma Shalizi: Statistics 36-350: Data Mining, Lecture Notes, 18 November 2009, <a href="http://www.stat.cmu.edu/~cshalizi/350/lectures/26/lecture-26.pdf">http://www.stat.cmu.edu/~cshalizi/350/lectures/26/lecture-26.pdf</a></p>
	<p>[3] Srikrishna Sridhar, Mark Wellons, Caleb Welton: Multilogistic Regression: Notes and References, Jul 12 2012, <a href="http://www.cs.wisc.edu/~srikris/mlogit.pdf">http://www.cs.wisc.edu/~srikris/mlogit.pdf</a></p>
	<p>[4] Scott A. Czepiel: Maximum Likelihood Estimation of Logistic Regression Models: Theory and Implementation, Retrieved Jul 12 2012, <a href="http://czep.net/stat/mlelr.pdf">http://czep.net/stat/mlelr.pdf</a></p>
	<dl class="see"><dt><b>See also:</b></dt><dd>File <a class="el" href="multilogistic_8sql__in.html" title="SQL functions for multinomial logistic regression.">multilogistic.sql_in</a> (documenting the SQL functions) </dd></dl>
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