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| <a href="multilogistic_8sql__in.html">Go to the documentation of this file.</a><div class="fragment"><pre class="fragment"><a name="l00001"></a>00001 <span class="comment">/* ----------------------------------------------------------------------- */</span><span class="comment">/**</span> |
| <a name="l00002"></a>00002 <span class="comment"> *</span> |
| <a name="l00003"></a>00003 <span class="comment"> * @file multilogistic.sql_in</span> |
| <a name="l00004"></a>00004 <span class="comment"> *</span> |
| <a name="l00005"></a>00005 <span class="comment"> * @brief SQL functions for multinomial logistic regression</span> |
| <a name="l00006"></a>00006 <span class="comment"> * @date July 2012</span> |
| <a name="l00007"></a>00007 <span class="comment"> *</span> |
| <a name="l00008"></a>00008 <span class="comment"> * @sa For a brief introduction to multinomial logistic regression, see the</span> |
| <a name="l00009"></a>00009 <span class="comment"> * module description \ref grp_mlogreg.</span> |
| <a name="l00010"></a>00010 <span class="comment"> *</span> |
| <a name="l00011"></a>00011 <span class="comment"> */</span><span class="comment">/* ----------------------------------------------------------------------- */</span> |
| <a name="l00012"></a>00012 |
| <a name="l00013"></a>00013 m4_include(`SQLCommon.m4<span class="stringliteral">')</span> |
| <a name="l00014"></a>00014 <span class="stringliteral"></span><span class="comment"></span> |
| <a name="l00015"></a>00015 <span class="comment">/**</span> |
| <a name="l00016"></a>00016 <span class="comment">@addtogroup grp_mlogreg</span> |
| <a name="l00017"></a>00017 <span class="comment"></span> |
| <a name="l00018"></a>00018 <span class="comment">@about</span> |
| <a name="l00019"></a>00019 <span class="comment"></span> |
| <a name="l00020"></a>00020 <span class="comment">Multinomial logistic regression is a widely used regression analysis tool that models the outcomes of categorical dependent</span> |
| <a name="l00021"></a>00021 <span class="comment">random variables (denoted \f$ Y \in \{ 0,1,2 \ldots k \} \f$). The models assumes that the conditional mean of the</span> |
| <a name="l00022"></a>00022 <span class="comment">dependant categorical variables is the logistic function of an affine combination of independent</span> |
| <a name="l00023"></a>00023 <span class="comment">variables (usually denoted \f$ \boldsymbol x \f$). That is,</span> |
| <a name="l00024"></a>00024 <span class="comment">\f[</span> |
| <a name="l00025"></a>00025 <span class="comment"> E[Y \mid \boldsymbol x] = \sigma(\boldsymbol c^T \boldsymbol x)</span> |
| <a name="l00026"></a>00026 <span class="comment">\f]</span> |
| <a name="l00027"></a>00027 <span class="comment">for some unknown vector of coefficients \f$ \boldsymbol c \f$ and where</span> |
| <a name="l00028"></a>00028 <span class="comment">\f$ \sigma(x) = \frac{1}{1 + \exp(-x)} \f$ is the logistic function. Multinomial Logistic</span> |
| <a name="l00029"></a>00029 <span class="comment">regression finds the vector of coefficients \f$ \boldsymbol c \f$ that maximizes</span> |
| <a name="l00030"></a>00030 <span class="comment">the likelihood of the observations.</span> |
| <a name="l00031"></a>00031 <span class="comment"></span> |
| <a name="l00032"></a>00032 <span class="comment">Let</span> |
| <a name="l00033"></a>00033 <span class="comment">- \f$ \boldsymbol y \in \{ 0,1 \}^{n \times k} \f$ denote the vector of observed dependent</span> |
| <a name="l00034"></a>00034 <span class="comment"> variables, with \f$ n \f$ rows and \f$ k \f$ columns, containing the observed values of the</span> |
| <a name="l00035"></a>00035 <span class="comment"> dependent variable,</span> |
| <a name="l00036"></a>00036 <span class="comment">- \f$ X \in \mathbf R^{n \times k} \f$ denote the design matrix with \f$ k \f$</span> |
| <a name="l00037"></a>00037 <span class="comment"> columns and \f$ n \f$ rows, containing all observed vectors of independent</span> |
| <a name="l00038"></a>00038 <span class="comment"> variables \f$ \boldsymbol x_i \f$ as rows.</span> |
| <a name="l00039"></a>00039 <span class="comment"></span> |
| <a name="l00040"></a>00040 <span class="comment">By definition,</span> |
| <a name="l00041"></a>00041 <span class="comment">\f[</span> |
| <a name="l00042"></a>00042 <span class="comment"> P[Y = y_i | \boldsymbol x_i]</span> |
| <a name="l00043"></a>00043 <span class="comment"> = \sigma((-1)^{y_i} \cdot \boldsymbol c^T \boldsymbol x_i)</span> |
| <a name="l00044"></a>00044 <span class="comment"> \,.</span> |
| <a name="l00045"></a>00045 <span class="comment">\f]</span> |
| <a name="l00046"></a>00046 <span class="comment">Maximizing the likelihood</span> |
| <a name="l00047"></a>00047 <span class="comment">\f$ \prod_{i=1}^n \Pr(Y = y_i \mid \boldsymbol x_i) \f$</span> |
| <a name="l00048"></a>00048 <span class="comment">is equivalent to maximizing the log-likelihood</span> |
| <a name="l00049"></a>00049 <span class="comment">\f$ \sum_{i=1}^n \log \Pr(Y = y_i \mid \boldsymbol x_i) \f$, which simplifies to</span> |
| <a name="l00050"></a>00050 <span class="comment">\f[</span> |
| <a name="l00051"></a>00051 <span class="comment"> l(\boldsymbol c) =</span> |
| <a name="l00052"></a>00052 <span class="comment"> -\sum_{i=1}^n \log(1 + \exp((-1)^{y_i}</span> |
| <a name="l00053"></a>00053 <span class="comment"> \cdot \boldsymbol c^T \boldsymbol x_i))</span> |
| <a name="l00054"></a>00054 <span class="comment"> \,.</span> |
| <a name="l00055"></a>00055 <span class="comment">\f]</span> |
| <a name="l00056"></a>00056 <span class="comment">The Hessian of this objective is \f$ H = -X^T A X \f$ where</span> |
| <a name="l00057"></a>00057 <span class="comment">\f$ A = \text{diag}(a_1, \dots, a_n) \f$ is the diagonal matrix with</span> |
| <a name="l00058"></a>00058 <span class="comment">\f$</span> |
| <a name="l00059"></a>00059 <span class="comment"> a_i = \sigma(\boldsymbol c^T \boldsymbol x)</span> |
| <a name="l00060"></a>00060 <span class="comment"> \cdot</span> |
| <a name="l00061"></a>00061 <span class="comment"> \sigma(-\boldsymbol c^T \boldsymbol x)</span> |
| <a name="l00062"></a>00062 <span class="comment"> \,.</span> |
| <a name="l00063"></a>00063 <span class="comment">\f$</span> |
| <a name="l00064"></a>00064 <span class="comment">Since \f$ H \f$ is non-positive definite, \f$ l(\boldsymbol c) \f$ is convex.</span> |
| <a name="l00065"></a>00065 <span class="comment">There are many techniques for solving convex optimization problems. Currently,</span> |
| <a name="l00066"></a>00066 <span class="comment">logistic regression in MADlib can use:</span> |
| <a name="l00067"></a>00067 <span class="comment">- Iteratively Reweighted Least Squares</span> |
| <a name="l00068"></a>00068 <span class="comment"></span> |
| <a name="l00069"></a>00069 <span class="comment">We estimate the standard error for coefficient \f$ i \f$ as</span> |
| <a name="l00070"></a>00070 <span class="comment">\f[</span> |
| <a name="l00071"></a>00071 <span class="comment"> \mathit{se}(c_i) = \left( (X^T A X)^{-1} \right)_{ii}</span> |
| <a name="l00072"></a>00072 <span class="comment"> \,.</span> |
| <a name="l00073"></a>00073 <span class="comment">\f]</span> |
| <a name="l00074"></a>00074 <span class="comment">The Wald z-statistic is</span> |
| <a name="l00075"></a>00075 <span class="comment">\f[</span> |
| <a name="l00076"></a>00076 <span class="comment"> z_i = \frac{c_i}{\mathit{se}(c_i)}</span> |
| <a name="l00077"></a>00077 <span class="comment"> \,.</span> |
| <a name="l00078"></a>00078 <span class="comment">\f]</span> |
| <a name="l00079"></a>00079 <span class="comment"></span> |
| <a name="l00080"></a>00080 <span class="comment">The Wald \f$ p \f$-value for coefficient \f$ i \f$ gives the probability (under</span> |
| <a name="l00081"></a>00081 <span class="comment">the assumptions inherent in the Wald test) of seeing a value at least as extreme</span> |
| <a name="l00082"></a>00082 <span class="comment">as the one observed, provided that the null hypothesis (\f$ c_i = 0 \f$) is</span> |
| <a name="l00083"></a>00083 <span class="comment">true. Letting \f$ F \f$ denote the cumulative density function of a standard</span> |
| <a name="l00084"></a>00084 <span class="comment">normal distribution, the Wald \f$ p \f$-value for coefficient \f$ i \f$ is</span> |
| <a name="l00085"></a>00085 <span class="comment">therefore</span> |
| <a name="l00086"></a>00086 <span class="comment">\f[</span> |
| <a name="l00087"></a>00087 <span class="comment"> p_i = \Pr(|Z| \geq |z_i|) = 2 \cdot (1 - F( |z_i| ))</span> |
| <a name="l00088"></a>00088 <span class="comment">\f]</span> |
| <a name="l00089"></a>00089 <span class="comment">where \f$ Z \f$ is a standard normally distributed random variable.</span> |
| <a name="l00090"></a>00090 <span class="comment"></span> |
| <a name="l00091"></a>00091 <span class="comment">The odds ratio for coefficient \f$ i \f$ is estimated as \f$ \exp(c_i) \f$.</span> |
| <a name="l00092"></a>00092 <span class="comment"></span> |
| <a name="l00093"></a>00093 <span class="comment">The condition number is computed as \f$ \kappa(X^T A X) \f$ during the iteration</span> |
| <a name="l00094"></a>00094 <span class="comment">immediately <em>preceding</em> convergence (i.e., \f$ A \f$ is computed using</span> |
| <a name="l00095"></a>00095 <span class="comment">the coefficients of the previous iteration). A large condition number (say, more</span> |
| <a name="l00096"></a>00096 <span class="comment">than 1000) indicates the presence of significant multicollinearity.</span> |
| <a name="l00097"></a>00097 <span class="comment"></span> |
| <a name="l00098"></a>00098 <span class="comment"></span> |
| <a name="l00099"></a>00099 <span class="comment">@input</span> |
| <a name="l00100"></a>00100 <span class="comment"></span> |
| <a name="l00101"></a>00101 <span class="comment">The training data is expected to be of the following form:\n</span> |
| <a name="l00102"></a>00102 <span class="comment"><pre>{TABLE|VIEW} <em>sourceName</em> (</span> |
| <a name="l00103"></a>00103 <span class="comment"> ...</span> |
| <a name="l00104"></a>00104 <span class="comment"> <em>dependentVariable</em> INTEGER,</span> |
| <a name="l00105"></a>00105 <span class="comment"> <em>numCategories</em> INTEGER,</span> |
| <a name="l00106"></a>00106 <span class="comment"> <em>independentVariables</em> FLOAT8[],</span> |
| <a name="l00107"></a>00107 <span class="comment"> ...</span> |
| <a name="l00108"></a>00108 <span class="comment">)</pre></span> |
| <a name="l00109"></a>00109 <span class="comment"></span> |
| <a name="l00110"></a>00110 <span class="comment">@usage</span> |
| <a name="l00111"></a>00111 <span class="comment">- The number of independent variables cannot exceed 65535.</span> |
| <a name="l00112"></a>00112 <span class="comment">- Get vector of coefficients \f$ \boldsymbol c \f$ and all diagnostic</span> |
| <a name="l00113"></a>00113 <span class="comment"> statistics:\n</span> |
| <a name="l00114"></a>00114 <span class="comment"> <pre>SELECT * FROM \ref mlogregr(</span> |
| <a name="l00115"></a>00115 <span class="comment"> '<em>sourceName</em>', '<em>dependentVariable</em>', '<em>numCategories</em>' , '<em>independentVariables</em>'</span> |
| <a name="l00116"></a>00116 <span class="comment"> [, <em>numberOfIterations</em> [, '<em>optimizer</em>' [, <em>precision</em> ] ] ]</span> |
| <a name="l00117"></a>00117 <span class="comment">);</pre></span> |
| <a name="l00118"></a>00118 <span class="comment"> Output:</span> |
| <a name="l00119"></a>00119 <span class="comment"> <pre>coef | log_likelihood | std_err | z_stats | p_values | odds_ratios | condition_no | num_iterations</span> |
| <a name="l00120"></a>00120 <span class="comment">-----+----------------+---------+---------+----------+-------------+--------------+---------------</span> |
| <a name="l00121"></a>00121 <span class="comment"> ...</span> |
| <a name="l00122"></a>00122 <span class="comment"></pre></span> |
| <a name="l00123"></a>00123 <span class="comment">- Get vector of coefficients \f$ \boldsymbol c \f$:\n</span> |
| <a name="l00124"></a>00124 <span class="comment"> <pre>SELECT (\ref mlogregr('<em>sourceName</em>', '<em>dependentVariable</em>', '<em>numCategories</em>', '<em>independentVariables</em>')).coef;</pre></span> |
| <a name="l00125"></a>00125 <span class="comment">- Get a subset of the output columns, e.g., only the array of coefficients</span> |
| <a name="l00126"></a>00126 <span class="comment"> \f$ \boldsymbol c \f$, the log-likelihood of determination</span> |
| <a name="l00127"></a>00127 <span class="comment"> \f$ l(\boldsymbol c) \f$, and the array of p-values \f$ \boldsymbol p \f$:</span> |
| <a name="l00128"></a>00128 <span class="comment"> <pre>SELECT coef, log_likelihood, p_values</span> |
| <a name="l00129"></a>00129 <span class="comment">FROM \ref mlogregr('<em>sourceName</em>', '<em>dependentVariable</em>', '<em>numCategories</em>', '<em>independentVariables</em>');</pre></span> |
| <a name="l00130"></a>00130 <span class="comment"></span> |
| <a name="l00131"></a>00131 <span class="comment">Note that the categories are encoded as integers with values from {0, 1, 2,...numCategories}</span> |
| <a name="l00132"></a>00132 <span class="comment">@examp</span> |
| <a name="l00133"></a>00133 <span class="comment"></span> |
| <a name="l00134"></a>00134 <span class="comment">-# Create the sample data set:</span> |
| <a name="l00135"></a>00135 <span class="comment">@verbatim</span> |
| <a name="l00136"></a>00136 <span class="comment">sql> SELECT * FROM data;</span> |
| <a name="l00137"></a>00137 <span class="comment"> r1 | val</span> |
| <a name="l00138"></a>00138 <span class="comment">---------------------------------------------+-----</span> |
| <a name="l00139"></a>00139 <span class="comment"> {1,3.01789340097457,0.454183579888195} | 1</span> |
| <a name="l00140"></a>00140 <span class="comment"> {1,-2.59380532894284,0.602678326424211} | 0</span> |
| <a name="l00141"></a>00141 <span class="comment"> {1,-1.30643094424158,0.151587064377964} | 1</span> |
| <a name="l00142"></a>00142 <span class="comment"> {1,3.60722299199551,0.963550757616758} | 1</span> |
| <a name="l00143"></a>00143 <span class="comment"> {1,-1.52197745628655,0.0782248834148049} | 1</span> |
| <a name="l00144"></a>00144 <span class="comment"> {1,-4.8746574902907,0.345104880165309} | 0</span> |
| <a name="l00145"></a>00145 <span class="comment">...</span> |
| <a name="l00146"></a>00146 <span class="comment">@endverbatim</span> |
| <a name="l00147"></a>00147 <span class="comment">-# Run the multi-logistic regression function:</span> |
| <a name="l00148"></a>00148 <span class="comment">@verbatim</span> |
| <a name="l00149"></a>00149 <span class="comment">sql> \x on</span> |
| <a name="l00150"></a>00150 <span class="comment">Expanded display is off.</span> |
| <a name="l00151"></a>00151 <span class="comment">sql> SELECT * FROM mlogregr('data', 'val', '2', 'r1', 100, 'irls', 0.001);</span> |
| <a name="l00152"></a>00152 <span class="comment">-[ RECORD 1 ]--+--------------------------------------------------------------</span> |
| <a name="l00153"></a>00153 <span class="comment">coef | {5.59049410898112,2.11077546770772,-0.237276684606453}</span> |
| <a name="l00154"></a>00154 <span class="comment">log_likelihood | -467.214718489873</span> |
| <a name="l00155"></a>00155 <span class="comment">std_err | {0.318943457652178,0.101518723785383,0.294509929481773}</span> |
| <a name="l00156"></a>00156 <span class="comment">z_stats | {17.5281667482197,20.7919819024719,-0.805666162169712}</span> |
| <a name="l00157"></a>00157 <span class="comment">p_values | {8.73403463417837e-69,5.11539430631541e-96,0.420435365338518}</span> |
| <a name="l00158"></a>00158 <span class="comment">odds_ratios | {267.867942976278,8.2546400100702,0.788773016471171}</span> |
| <a name="l00159"></a>00159 <span class="comment">condition_no | 179.186118573205</span> |
| <a name="l00160"></a>00160 <span class="comment">num_iterations | 9</span> |
| <a name="l00161"></a>00161 <span class="comment"></span> |
| <a name="l00162"></a>00162 <span class="comment">@endverbatim</span> |
| <a name="l00163"></a>00163 <span class="comment"></span> |
| <a name="l00164"></a>00164 <span class="comment">@literature</span> |
| <a name="l00165"></a>00165 <span class="comment"></span> |
| <a name="l00166"></a>00166 <span class="comment">A collection of nice write-ups, with valuable pointers into</span> |
| <a name="l00167"></a>00167 <span class="comment">further literature:</span> |
| <a name="l00168"></a>00168 <span class="comment"></span> |
| <a name="l00169"></a>00169 <span class="comment">[1] Annette J . Dobson: An Introduction to Generalized Linear Models, Second Edition. Nov 2001</span> |
| <a name="l00170"></a>00170 <span class="comment"></span> |
| <a name="l00171"></a>00171 <span class="comment">[2] Cosma Shalizi: Statistics 36-350: Data Mining, Lecture Notes, 18 November</span> |
| <a name="l00172"></a>00172 <span class="comment"> 2009, http://www.stat.cmu.edu/~cshalizi/350/lectures/26/lecture-26.pdf</span> |
| <a name="l00173"></a>00173 <span class="comment"></span> |
| <a name="l00174"></a>00174 <span class="comment">[3] Srikrishna Sridhar, Mark Wellons, Caleb Welton: Multilogistic Regression:</span> |
| <a name="l00175"></a>00175 <span class="comment"> Notes and References, Jul 12 2012, http://www.cs.wisc.edu/~srikris/mlogit.pdf</span> |
| <a name="l00176"></a>00176 <span class="comment"></span> |
| <a name="l00177"></a>00177 <span class="comment">[4] Scott A. Czepiel: Maximum Likelihood Estimation</span> |
| <a name="l00178"></a>00178 <span class="comment"> of Logistic Regression Models: Theory and Implementation,</span> |
| <a name="l00179"></a>00179 <span class="comment"> Retrieved Jul 12 2012, http://czep.net/stat/mlelr.pdf</span> |
| <a name="l00180"></a>00180 <span class="comment"></span> |
| <a name="l00181"></a>00181 <span class="comment"></span> |
| <a name="l00182"></a>00182 <span class="comment"></span> |
| <a name="l00183"></a>00183 <span class="comment">@sa File multilogistic.sql_in (documenting the SQL functions)</span> |
| <a name="l00184"></a>00184 <span class="comment"></span> |
| <a name="l00185"></a>00185 <span class="comment">@internal</span> |
| <a name="l00186"></a>00186 <span class="comment">@sa Namespace multilogistic (documenting the driver/outer loop implemented in</span> |
| <a name="l00187"></a>00187 <span class="comment"> Python), Namespace</span> |
| <a name="l00188"></a>00188 <span class="comment"> \ref madlib::modules::regress documenting the implementation in C++</span> |
| <a name="l00189"></a>00189 <span class="comment">@endinternal</span> |
| <a name="l00190"></a>00190 <span class="comment"></span> |
| <a name="l00191"></a>00191 <span class="comment">*/</span> |
| <a name="l00192"></a>00192 |
| <a name="l00193"></a>00193 |
| <a name="l00194"></a>00194 DROP TYPE IF EXISTS MADLIB_SCHEMA.mlogregr_result; |
| <a name="l00195"></a>00195 CREATE TYPE MADLIB_SCHEMA.mlogregr_result AS ( |
| <a name="l00196"></a>00196 coef DOUBLE PRECISION[], |
| <a name="l00197"></a>00197 log_likelihood DOUBLE PRECISION, |
| <a name="l00198"></a>00198 std_err DOUBLE PRECISION[], |
| <a name="l00199"></a>00199 z_stats DOUBLE PRECISION[], |
| <a name="l00200"></a>00200 p_values DOUBLE PRECISION[], |
| <a name="l00201"></a>00201 odds_ratios DOUBLE PRECISION[], |
| <a name="l00202"></a>00202 condition_no DOUBLE PRECISION, |
| <a name="l00203"></a>00203 num_iterations INTEGER |
| <a name="l00204"></a>00204 ); |
| <a name="l00205"></a>00205 |
| <a name="l00206"></a>00206 |
| <a name="l00207"></a>00207 CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlogregr_irls_step_transition( |
| <a name="l00208"></a>00208 DOUBLE PRECISION[], |
| <a name="l00209"></a>00209 INTEGER, |
| <a name="l00210"></a>00210 INTEGER, |
| <a name="l00211"></a>00211 DOUBLE PRECISION[], |
| <a name="l00212"></a>00212 DOUBLE PRECISION[]) |
| <a name="l00213"></a>00213 RETURNS DOUBLE PRECISION[] |
| <a name="l00214"></a>00214 AS 'MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00215"></a>00215 <span class="stringliteral">LANGUAGE C IMMUTABLE;</span> |
| <a name="l00216"></a>00216 <span class="stringliteral"></span> |
| <a name="l00217"></a>00217 <span class="stringliteral"></span> |
| <a name="l00218"></a>00218 <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlogregr_irls_step_merge_states(</span> |
| <a name="l00219"></a>00219 <span class="stringliteral"> state1 DOUBLE PRECISION[],</span> |
| <a name="l00220"></a>00220 <span class="stringliteral"> state2 DOUBLE PRECISION[])</span> |
| <a name="l00221"></a>00221 <span class="stringliteral">RETURNS DOUBLE PRECISION[]</span> |
| <a name="l00222"></a>00222 <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00223"></a>00223 <span class="stringliteral">LANGUAGE C IMMUTABLE STRICT;</span> |
| <a name="l00224"></a>00224 <span class="stringliteral"></span> |
| <a name="l00225"></a>00225 <span class="stringliteral"></span> |
| <a name="l00226"></a>00226 <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlogregr_irls_step_final(</span> |
| <a name="l00227"></a>00227 <span class="stringliteral"> state DOUBLE PRECISION[])</span> |
| <a name="l00228"></a>00228 <span class="stringliteral">RETURNS DOUBLE PRECISION[]</span> |
| <a name="l00229"></a>00229 <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00230"></a>00230 <span class="stringliteral">LANGUAGE C IMMUTABLE STRICT;</span> |
| <a name="l00231"></a>00231 <span class="stringliteral"></span> |
| <a name="l00232"></a>00232 <span class="stringliteral"></span><span class="comment"></span> |
| <a name="l00233"></a>00233 <span class="comment">/**</span> |
| <a name="l00234"></a>00234 <span class="comment"> * @internal</span> |
| <a name="l00235"></a>00235 <span class="comment"> * @brief Perform one iteration of the iteratively-reweighted-least-squares</span> |
| <a name="l00236"></a>00236 <span class="comment"> * method for computing linear regression</span> |
| <a name="l00237"></a>00237 <span class="comment"> */</span> |
| <a name="l00238"></a>00238 CREATE AGGREGATE MADLIB_SCHEMA.mlogregr_irls_step( |
| <a name="l00239"></a>00239 /*+ y */ INTEGER, |
| <a name="l00240"></a>00240 /*+ numCategories */ INTEGER, |
| <a name="l00241"></a>00241 /*+ x */ DOUBLE PRECISION[], |
| <a name="l00242"></a>00242 /*+ previous_state */ DOUBLE PRECISION[]) ( |
| <a name="l00243"></a>00243 |
| <a name="l00244"></a>00244 STYPE=DOUBLE PRECISION[], |
| <a name="l00245"></a>00245 SFUNC=MADLIB_SCHEMA.mlogregr_irls_step_transition, |
| <a name="l00246"></a>00246 m4_ifdef(`__GREENPLUM__',`prefunc=MADLIB_SCHEMA.mlogregr_irls_step_merge_states,<span class="stringliteral">')</span> |
| <a name="l00247"></a>00247 <span class="stringliteral"> FINALFUNC=MADLIB_SCHEMA.mlogregr_irls_step_final,</span> |
| <a name="l00248"></a>00248 <span class="stringliteral"> INITCOND='</span>{0,0,0,0}<span class="stringliteral">'</span> |
| <a name="l00249"></a>00249 <span class="stringliteral">);</span> |
| <a name="l00250"></a>00250 <span class="stringliteral"></span> |
| <a name="l00251"></a>00251 <span class="stringliteral"></span> |
| <a name="l00252"></a>00252 <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.internal_mlogregr_irls_step_distance(</span> |
| <a name="l00253"></a>00253 <span class="stringliteral"> /*+ state1 */ DOUBLE PRECISION[],</span> |
| <a name="l00254"></a>00254 <span class="stringliteral"> /*+ state2 */ DOUBLE PRECISION[])</span> |
| <a name="l00255"></a>00255 <span class="stringliteral">RETURNS DOUBLE PRECISION AS</span> |
| <a name="l00256"></a>00256 <span class="stringliteral">'</span>MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00257"></a>00257 <span class="stringliteral">LANGUAGE c IMMUTABLE STRICT;</span> |
| <a name="l00258"></a>00258 <span class="stringliteral"></span> |
| <a name="l00259"></a>00259 <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.internal_mlogregr_irls_result(</span> |
| <a name="l00260"></a>00260 <span class="stringliteral"> /*+ state */ DOUBLE PRECISION[])</span> |
| <a name="l00261"></a>00261 <span class="stringliteral">RETURNS MADLIB_SCHEMA.mlogregr_result AS</span> |
| <a name="l00262"></a>00262 <span class="stringliteral">'</span>MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00263"></a>00263 <span class="stringliteral">LANGUAGE c IMMUTABLE STRICT;</span> |
| <a name="l00264"></a>00264 <span class="stringliteral"></span> |
| <a name="l00265"></a>00265 <span class="stringliteral"></span> |
| <a name="l00266"></a>00266 <span class="stringliteral">-- We only need to document the last one (unfortunately, in Greenplum we have to</span> |
| <a name="l00267"></a>00267 <span class="stringliteral">-- use function overloading instead of default arguments).</span> |
| <a name="l00268"></a>00268 <span class="stringliteral">CREATE FUNCTION MADLIB_SCHEMA.compute_mlogregr(</span> |
| <a name="l00269"></a>00269 <span class="stringliteral"> "source" VARCHAR,</span> |
| <a name="l00270"></a>00270 <span class="stringliteral"> "depvar" VARCHAR,</span> |
| <a name="l00271"></a>00271 <span class="stringliteral"> "numcategories" INTEGER,</span> |
| <a name="l00272"></a>00272 <span class="stringliteral"> "indepvar" VARCHAR,</span> |
| <a name="l00273"></a>00273 <span class="stringliteral"> "maxnumiterations" INTEGER,</span> |
| <a name="l00274"></a>00274 <span class="stringliteral"> "optimizer" VARCHAR,</span> |
| <a name="l00275"></a>00275 <span class="stringliteral"> "precision" DOUBLE PRECISION)</span> |
| <a name="l00276"></a>00276 <span class="stringliteral">RETURNS INTEGER</span> |
| <a name="l00277"></a>00277 <span class="stringliteral">AS $$PythonFunction(regress, multilogistic, compute_mlogregr)$$</span> |
| <a name="l00278"></a>00278 <span class="stringliteral">LANGUAGE plpythonu VOLATILE;</span> |
| <a name="l00279"></a>00279 <span class="stringliteral"></span><span class="comment"></span> |
| <a name="l00280"></a>00280 <span class="comment">/**</span> |
| <a name="l00281"></a>00281 <span class="comment"> * @brief Compute logistic-regression coefficients and diagnostic statistics</span> |
| <a name="l00282"></a>00282 <span class="comment"> *</span> |
| <a name="l00283"></a>00283 <span class="comment"> * To include an intercept in the model, set one coordinate in the</span> |
| <a name="l00284"></a>00284 <span class="comment"> * <tt>independentVariables</tt> array to 1.</span> |
| <a name="l00285"></a>00285 <span class="comment"> *</span> |
| <a name="l00286"></a>00286 <span class="comment"> * @param source Name of the source relation containing the training data</span> |
| <a name="l00287"></a>00287 <span class="comment"> * @param depvar Name of the dependent column (of type INTEGER < numcategories - 1)</span> |
| <a name="l00288"></a>00288 <span class="comment"> * @param numcategories Number of categories for the dependant variables (</span> |
| <a name="l00289"></a>00289 <span class="comment"> of type INTEGER)</span> |
| <a name="l00290"></a>00290 <span class="comment"> * @param indepvar Name of the independent column (of type DOUBLE</span> |
| <a name="l00291"></a>00291 <span class="comment"> * PRECISION[])</span> |
| <a name="l00292"></a>00292 <span class="comment"> * @param maxnumiterations The maximum number of iterations</span> |
| <a name="l00293"></a>00293 <span class="comment"> * @param optimizer The optimizer to use (</span> |
| <a name="l00294"></a>00294 <span class="comment"> * <tt>'irls'</tt>/<tt>'newton'</tt> for iteratively reweighted least</span> |
| <a name="l00295"></a>00295 <span class="comment"> * squares)</span> |
| <a name="l00296"></a>00296 <span class="comment"> * @param precision The difference between log-likelihood values in successive</span> |
| <a name="l00297"></a>00297 <span class="comment"> * iterations that should indicate convergence. Note that a non-positive</span> |
| <a name="l00298"></a>00298 <span class="comment"> * value here disables the convergence criterion, and execution will only</span> |
| <a name="l00299"></a>00299 <span class="comment"> * stop after \ maxNumIterations iterations.</span> |
| <a name="l00300"></a>00300 <span class="comment"> *</span> |
| <a name="l00301"></a>00301 <span class="comment"> * @return A composite value:</span> |
| <a name="l00302"></a>00302 <span class="comment"> * - <tt>coef FLOAT8[]</tt> - Array of coefficients, \f$ \boldsymbol c \f$</span> |
| <a name="l00303"></a>00303 <span class="comment"> * - <tt>log_likelihood FLOAT8</tt> - Log-likelihood \f$ l(\boldsymbol c) \f$</span> |
| <a name="l00304"></a>00304 <span class="comment"> * - <tt>std_err FLOAT8[]</tt> - Array of standard errors,</span> |
| <a name="l00305"></a>00305 <span class="comment"> * \f$ \mathit{se}(c_1), \dots, \mathit{se}(c_k) \f$</span> |
| <a name="l00306"></a>00306 <span class="comment"> * - <tt>z_stats FLOAT8[]</tt> - Array of Wald z-statistics, \f$ \boldsymbol z \f$</span> |
| <a name="l00307"></a>00307 <span class="comment"> * - <tt>p_values FLOAT8[]</tt> - Array of Wald p-values, \f$ \boldsymbol p \f$</span> |
| <a name="l00308"></a>00308 <span class="comment"> * - <tt>odds_ratios FLOAT8[]</tt>: Array of odds ratios,</span> |
| <a name="l00309"></a>00309 <span class="comment"> * \f$ \mathit{odds}(c_1), \dots, \mathit{odds}(c_k) \f$</span> |
| <a name="l00310"></a>00310 <span class="comment"> * - <tt>condition_no FLOAT8</tt> - The condition number of matrix</span> |
| <a name="l00311"></a>00311 <span class="comment"> * \f$ X^T A X \f$ during the iteration immediately <em>preceding</em></span> |
| <a name="l00312"></a>00312 <span class="comment"> * convergence (i.e., \f$ A \f$ is computed using the coefficients of the</span> |
| <a name="l00313"></a>00313 <span class="comment"> * previous iteration)</span> |
| <a name="l00314"></a>00314 <span class="comment"> * - <tt>num_iterations INTEGER</tt> - The number of iterations before the</span> |
| <a name="l00315"></a>00315 <span class="comment"> * algorithm terminated</span> |
| <a name="l00316"></a>00316 <span class="comment"> *</span> |
| <a name="l00317"></a>00317 <span class="comment"> * @usage</span> |
| <a name="l00318"></a>00318 <span class="comment"> * - Get vector of coefficients \f$ \boldsymbol c \f$ and all diagnostic</span> |
| <a name="l00319"></a>00319 <span class="comment"> * statistics:\n</span> |
| <a name="l00320"></a>00320 <span class="comment"> * <pre>SELECT * FROM mlogregr('<em>sourceName</em>', '<em>dependentVariable</em>',</span> |
| <a name="l00321"></a>00321 <span class="comment"> * '<em>numCategories</em>', '<em>independentVariables</em>');</pre></span> |
| <a name="l00322"></a>00322 <span class="comment"> * - Get vector of coefficients \f$ \boldsymbol c \f$:\n</span> |
| <a name="l00323"></a>00323 <span class="comment"> * <pre>SELECT (mlogregr('<em>sourceName</em>', '<em>dependentVariable</em>',</span> |
| <a name="l00324"></a>00324 <span class="comment"> * '<em>numCategories</em>', '<em>independentVariables</em>')).coef;</pre></span> |
| <a name="l00325"></a>00325 <span class="comment"> * - Get a subset of the output columns, e.g., only the array of coefficients</span> |
| <a name="l00326"></a>00326 <span class="comment"> * \f$ \boldsymbol c \f$, the log-likelihood of determination</span> |
| <a name="l00327"></a>00327 <span class="comment"> * \f$ l(\boldsymbol c) \f$, and the array of p-values \f$ \boldsymbol p \f$:</span> |
| <a name="l00328"></a>00328 <span class="comment"> * <pre>SELECT coef, log_likelihood, p_values</span> |
| <a name="l00329"></a>00329 <span class="comment"> * FROM mlogregr('<em>sourceName</em>', '<em>dependentVariable</em>',</span> |
| <a name="l00330"></a>00330 <span class="comment"> * '<em>numCategories</em>', '<em>independentVariables</em>');</pre></span> |
| <a name="l00331"></a>00331 <span class="comment"> *</span> |
| <a name="l00332"></a>00332 <span class="comment"> * @note This function starts an iterative algorithm. It is not an aggregate</span> |
| <a name="l00333"></a>00333 <span class="comment"> * function. Source and column names have to be passed as strings (due to</span> |
| <a name="l00334"></a>00334 <span class="comment"> * limitations of the SQL syntax).</span> |
| <a name="l00335"></a>00335 <span class="comment"> *</span> |
| <a name="l00336"></a>00336 <span class="comment"> * @internal</span> |
| <a name="l00337"></a>00337 <span class="comment"> * @sa This function is a wrapper for multilogistic::compute_mlogregr(), which</span> |
| <a name="l00338"></a>00338 <span class="comment"> * sets the default values.</span> |
| <a name="l00339"></a>00339 <span class="comment"> */</span> |
| <a name="l00340"></a>00340 CREATE FUNCTION MADLIB_SCHEMA.mlogregr( |
| <a name="l00341"></a>00341 "source" VARCHAR, |
| <a name="l00342"></a>00342 "depvar" VARCHAR, |
| <a name="l00343"></a>00343 "numcategories" INTEGER, |
| <a name="l00344"></a>00344 "indepvar" VARCHAR, |
| <a name="l00345"></a>00345 "maxnumiterations" INTEGER /*+ DEFAULT 20 */, |
| <a name="l00346"></a>00346 "optimizer" VARCHAR /*+ DEFAULT 'irls<span class="stringliteral">' */,</span> |
| <a name="l00347"></a>00347 <span class="stringliteral"> "precision" DOUBLE PRECISION /*+ DEFAULT 0.0001 */)</span> |
| <a name="l00348"></a>00348 <span class="stringliteral">RETURNS MADLIB_SCHEMA.mlogregr_result AS $$</span> |
| <a name="l00349"></a>00349 <span class="stringliteral">DECLARE</span> |
| <a name="l00350"></a>00350 <span class="stringliteral"> observed_count INTEGER;</span> |
| <a name="l00351"></a>00351 <span class="stringliteral"> theIteration INTEGER;</span> |
| <a name="l00352"></a>00352 <span class="stringliteral"> fnName VARCHAR;</span> |
| <a name="l00353"></a>00353 <span class="stringliteral"> theResult MADLIB_SCHEMA.mlogregr_result;</span> |
| <a name="l00354"></a>00354 <span class="stringliteral">BEGIN</span> |
| <a name="l00355"></a>00355 <span class="stringliteral"> IF (SELECT atttypid::regtype <> '</span>INTEGER<span class="stringliteral">'::regtype</span> |
| <a name="l00356"></a>00356 <span class="stringliteral"> FROM pg_attribute</span> |
| <a name="l00357"></a>00357 <span class="stringliteral"> WHERE attrelid = source::regclass AND attname = depvar) THEN</span> |
| <a name="l00358"></a>00358 <span class="stringliteral"> RAISE EXCEPTION '</span>The dependent variable column should be of type INTEGER<span class="stringliteral">';</span> |
| <a name="l00359"></a>00359 <span class="stringliteral"> END IF;</span> |
| <a name="l00360"></a>00360 <span class="stringliteral"></span> |
| <a name="l00361"></a>00361 <span class="stringliteral"> EXECUTE $sql$ SELECT count(DISTINCT $sql$ || depvar || $sql$ )</span> |
| <a name="l00362"></a><a class="code" href="multilogistic_8sql__in.html#a8037cc95b1349de7f8910768be3262c8">00362</a> <span class="stringliteral"> FROM $sql$ || textin(regclassout(source))</span> |
| <a name="l00363"></a>00363 <span class="stringliteral"> INTO observed_count;</span> |
| <a name="l00364"></a>00364 <span class="stringliteral"> IF observed_count <> numcategories</span> |
| <a name="l00365"></a>00365 <span class="stringliteral"> THEN</span> |
| <a name="l00366"></a>00366 <span class="stringliteral"> RAISE WARNING '</span>Results will be undefined, <span class="keywordflow">if</span> <span class="stringliteral">''</span>numcategories<span class="stringliteral">''</span> is not |
| <a name="l00367"></a>00367 same as the number of distinct categories observed in the training data.<span class="stringliteral">';</span> |
| <a name="l00368"></a>00368 <span class="stringliteral"> END IF;</span> |
| <a name="l00369"></a>00369 <span class="stringliteral"></span> |
| <a name="l00370"></a>00370 <span class="stringliteral"> IF optimizer = '</span>irls<span class="stringliteral">' OR optimizer = '</span>newton<span class="stringliteral">' THEN</span> |
| <a name="l00371"></a>00371 <span class="stringliteral"> fnName := '</span>internal_mlogregr_irls_result<span class="stringliteral">';</span> |
| <a name="l00372"></a>00372 <span class="stringliteral"> ELSE</span> |
| <a name="l00373"></a>00373 <span class="stringliteral"> RAISE EXCEPTION '</span>Unknown optimizer (<span class="stringliteral">''</span>%<span class="stringliteral">''</span>)<span class="stringliteral">', optimizer;</span> |
| <a name="l00374"></a>00374 <span class="stringliteral"> END IF;</span> |
| <a name="l00375"></a>00375 <span class="stringliteral"></span> |
| <a name="l00376"></a>00376 <span class="stringliteral"> theIteration := (</span> |
| <a name="l00377"></a>00377 <span class="stringliteral"> SELECT MADLIB_SCHEMA.compute_mlogregr($1, $2, $3, $4, $5, $6, $7)</span> |
| <a name="l00378"></a>00378 <span class="stringliteral"> );</span> |
| <a name="l00379"></a>00379 <span class="stringliteral"> -- Because of Greenplum bug MPP-10050, we have to use dynamic SQL (using</span> |
| <a name="l00380"></a>00380 <span class="stringliteral"> -- EXECUTE) in the following</span> |
| <a name="l00381"></a>00381 <span class="stringliteral"> -- Because of Greenplum bug MPP-6731, we have to hide the tuple-returning</span> |
| <a name="l00382"></a>00382 <span class="stringliteral"> -- function in a subquery</span> |
| <a name="l00383"></a>00383 <span class="stringliteral"> EXECUTE</span> |
| <a name="l00384"></a>00384 <span class="stringliteral"> $sql$</span> |
| <a name="l00385"></a>00385 <span class="stringliteral"> SELECT (result).*</span> |
| <a name="l00386"></a>00386 <span class="stringliteral"> FROM (</span> |
| <a name="l00387"></a>00387 <span class="stringliteral"> SELECT</span> |
| <a name="l00388"></a>00388 <span class="stringliteral"> MADLIB_SCHEMA.$sql$ || fnName || $sql$(_madlib_state) AS result</span> |
| <a name="l00389"></a>00389 <span class="stringliteral"> FROM _madlib_iterative_alg</span> |
| <a name="l00390"></a>00390 <span class="stringliteral"> WHERE _madlib_iteration = $sql$ || theIteration || $sql$</span> |
| <a name="l00391"></a>00391 <span class="stringliteral"> ) subq</span> |
| <a name="l00392"></a>00392 <span class="stringliteral"> $sql$</span> |
| <a name="l00393"></a>00393 <span class="stringliteral"> INTO theResult;</span> |
| <a name="l00394"></a>00394 <span class="stringliteral"> -- The number of iterations are not updated in the C++ code. We do it here.</span> |
| <a name="l00395"></a>00395 <span class="stringliteral"> IF NOT (theResult IS NULL) THEN</span> |
| <a name="l00396"></a>00396 <span class="stringliteral"> theResult.num_iterations = theIteration;</span> |
| <a name="l00397"></a>00397 <span class="stringliteral"> END IF;</span> |
| <a name="l00398"></a>00398 <span class="stringliteral"> RETURN theResult;</span> |
| <a name="l00399"></a>00399 <span class="stringliteral">END;</span> |
| <a name="l00400"></a>00400 <span class="stringliteral">$$ LANGUAGE plpgsql VOLATILE;</span> |
| <a name="l00401"></a>00401 <span class="stringliteral"></span> |
| <a name="l00402"></a>00402 <span class="stringliteral"></span> |
| <a name="l00403"></a>00403 <span class="stringliteral">CREATE FUNCTION MADLIB_SCHEMA.mlogregr(</span> |
| <a name="l00404"></a>00404 <span class="stringliteral"> "source" VARCHAR,</span> |
| <a name="l00405"></a>00405 <span class="stringliteral"> "depvar" VARCHAR,</span> |
| <a name="l00406"></a>00406 <span class="stringliteral"> "numcategories" INTEGER,</span> |
| <a name="l00407"></a>00407 <span class="stringliteral"> "indepvar" VARCHAR)</span> |
| <a name="l00408"></a>00408 <span class="stringliteral">RETURNS MADLIB_SCHEMA.mlogregr_result AS</span> |
| <a name="l00409"></a>00409 <span class="stringliteral">$$SELECT MADLIB_SCHEMA.mlogregr($1, $2, $3, $4, 20, '</span>irls<span class="stringliteral">', 0.0001);$$</span> |
| <a name="l00410"></a>00410 <span class="stringliteral">LANGUAGE sql VOLATILE;</span> |
| <a name="l00411"></a>00411 <span class="stringliteral"></span> |
| <a name="l00412"></a>00412 <span class="stringliteral">CREATE FUNCTION MADLIB_SCHEMA.mlogregr(</span> |
| <a name="l00413"></a>00413 <span class="stringliteral"> "source" VARCHAR,</span> |
| <a name="l00414"></a>00414 <span class="stringliteral"> "depvar" VARCHAR,</span> |
| <a name="l00415"></a>00415 <span class="stringliteral"> "numcategories" INTEGER,</span> |
| <a name="l00416"></a>00416 <span class="stringliteral"> "indepvar" VARCHAR,</span> |
| <a name="l00417"></a>00417 <span class="stringliteral"> "maxnumiterations" INTEGER)</span> |
| <a name="l00418"></a>00418 <span class="stringliteral">RETURNS MADLIB_SCHEMA.mlogregr_result AS</span> |
| <a name="l00419"></a>00419 <span class="stringliteral">$$SELECT MADLIB_SCHEMA.mlogregr($1, $2, $3, $4, $5, '</span>irls<span class="stringliteral">', 0.0001);$$</span> |
| <a name="l00420"></a>00420 <span class="stringliteral">LANGUAGE sql VOLATILE;</span> |
| <a name="l00421"></a>00421 <span class="stringliteral"></span> |
| <a name="l00422"></a>00422 <span class="stringliteral">CREATE FUNCTION MADLIB_SCHEMA.mlogregr(</span> |
| <a name="l00423"></a>00423 <span class="stringliteral"> "source" VARCHAR,</span> |
| <a name="l00424"></a>00424 <span class="stringliteral"> "depvar" VARCHAR,</span> |
| <a name="l00425"></a>00425 <span class="stringliteral"> "numcategories" INTEGER,</span> |
| <a name="l00426"></a>00426 <span class="stringliteral"> "indepvar" VARCHAR,</span> |
| <a name="l00427"></a>00427 <span class="stringliteral"> "maxbumiterations" INTEGER,</span> |
| <a name="l00428"></a>00428 <span class="stringliteral"> "optimizer" VARCHAR)</span> |
| <a name="l00429"></a>00429 <span class="stringliteral">RETURNS MADLIB_SCHEMA.mlogregr_result AS</span> |
| <a name="l00430"></a>00430 <span class="stringliteral">$$SELECT MADLIB_SCHEMA.mlogregr($1, $2, $3, $4, $5, $6, 0.0001);$$</span> |
| <a name="l00431"></a>00431 <span class="stringliteral">LANGUAGE sql VOLATILE;</span> |
| <a name="l00432"></a>00432 <span class="stringliteral"></span> |
| </pre></div></div> |
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