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| <a href="cox__prop__hazards_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 cox_prop_hazards.sql_in</span> |
| <a name="l00004"></a>00004 <span class="comment"> *</span> |
| <a name="l00005"></a>00005 <span class="comment"> * @brief SQL functions for cox proportional hazards</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 cox regression, see the</span> |
| <a name="l00009"></a>00009 <span class="comment"> * module description \ref grp_cox_prop_hazards</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_cox_prop_hazards</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">Proportional-Hazard models enable the comparison of various survival models. </span> |
| <a name="l00020"></a>00020 <span class="comment">These survival models are functions describing the probability of an one-item </span> |
| <a name="l00021"></a>00021 <span class="comment">event (prototypically, this event is death) with respect to time. </span> |
| <a name="l00022"></a>00022 <span class="comment">The interval of time before death occurs is the survival time. </span> |
| <a name="l00023"></a>00023 <span class="comment">Let T be a random variable representing the survival time, </span> |
| <a name="l00024"></a>00024 <span class="comment">with a cumulative probability function P(t). Informally, P(t) is </span> |
| <a name="l00025"></a>00025 <span class="comment">the probability that death has happened before time t.</span> |
| <a name="l00026"></a>00026 <span class="comment"></span> |
| <a name="l00027"></a>00027 <span class="comment">Generally, applications start with a list of \f$ \boldsymbol n \f$ observations, </span> |
| <a name="l00028"></a>00028 <span class="comment">each with \f$ \boldsymbol m \f$ covariates and a time of death. From this </span> |
| <a name="l00029"></a>00029 <span class="comment">\f$ \boldsymbol n \times m \f$ matrix, we would like to derive the correlation </span> |
| <a name="l00030"></a>00030 <span class="comment">between the covariates and the hazard function. This amounts to finding </span> |
| <a name="l00031"></a>00031 <span class="comment">the parameters \f$ \boldsymbol \beta \f$ that best fit the model described below.</span> |
| <a name="l00032"></a>00032 <span class="comment"></span> |
| <a name="l00033"></a>00033 <span class="comment">Let us define:</span> |
| <a name="l00034"></a>00034 <span class="comment">- \f$ \boldsymbol t \in \mathbf R^{m} \f$ denote the vector of observed dependent</span> |
| <a name="l00035"></a>00035 <span class="comment"> variables, with \f$ n \f$ rows.</span> |
| <a name="l00036"></a>00036 <span class="comment">- \f$ X \in \mathbf R^{m} \f$ denote the design matrix with \f$ m \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">- \f$ R(t_i) \f$ denote the set of observations still alive at time \f$ t_i \f$</span> |
| <a name="l00040"></a>00040 <span class="comment"></span> |
| <a name="l00041"></a>00041 <span class="comment">Note that this model <b>does not</b> include a <b>constant term</b>, and the data</span> |
| <a name="l00042"></a>00042 <span class="comment">cannot contain a column of 1s.</span> |
| <a name="l00043"></a>00043 <span class="comment"></span> |
| <a name="l00044"></a>00044 <span class="comment">By definition,</span> |
| <a name="l00045"></a>00045 <span class="comment">\f[</span> |
| <a name="l00046"></a>00046 <span class="comment"> P[T_k = t_i | \boldsymbol R(t_i)]</span> |
| <a name="l00047"></a>00047 <span class="comment"> = \frac{e^{\beta^T x_k} }{ \sum_{j \in R(t_i)} e^{\beta^T x_j}}.</span> |
| <a name="l00048"></a>00048 <span class="comment"> \,.</span> |
| <a name="l00049"></a>00049 <span class="comment">\f]</span> |
| <a name="l00050"></a>00050 <span class="comment"></span> |
| <a name="l00051"></a>00051 <span class="comment">The <b>partial likelihood </b>function can now be generated as the product of </span> |
| <a name="l00052"></a>00052 <span class="comment">conditional probabilities:</span> |
| <a name="l00053"></a>00053 <span class="comment">\f[</span> |
| <a name="l00054"></a>00054 <span class="comment">\mathcal L = \prod_{i = 1}^n </span> |
| <a name="l00055"></a>00055 <span class="comment"> \left(</span> |
| <a name="l00056"></a>00056 <span class="comment"> \frac{e^{\beta^T x_i}}{ \sum_{j \in R(t_i)} e^{\beta^T x_j}}</span> |
| <a name="l00057"></a>00057 <span class="comment"> \right).</span> |
| <a name="l00058"></a>00058 <span class="comment">\f]</span> |
| <a name="l00059"></a>00059 <span class="comment"></span> |
| <a name="l00060"></a>00060 <span class="comment">The log-likelihood form of this equation is</span> |
| <a name="l00061"></a>00061 <span class="comment">\f[</span> |
| <a name="l00062"></a>00062 <span class="comment">L = \sum_{i = 1}^n </span> |
| <a name="l00063"></a>00063 <span class="comment"> \left[ \beta^T x_i</span> |
| <a name="l00064"></a>00064 <span class="comment"> - \log\left(\sum_{j \in R(t_i)} e^{\beta^T x_j }\right)</span> |
| <a name="l00065"></a>00065 <span class="comment"> \right].</span> |
| <a name="l00066"></a>00066 <span class="comment">\f]</span> |
| <a name="l00067"></a>00067 <span class="comment"></span> |
| <a name="l00068"></a>00068 <span class="comment">Using this score function and Hessian matrix, the partial likelihood can be </span> |
| <a name="l00069"></a>00069 <span class="comment">maximized using the <b> Newton-Raphson algorithm </b>.<b> Breslow's method </b> </span> |
| <a name="l00070"></a>00070 <span class="comment">is used to resolved tied times of deaths. The time of death for two records are </span> |
| <a name="l00071"></a>00071 <span class="comment">considered "equal" if they differ by less than 1.0e-6</span> |
| <a name="l00072"></a>00072 <span class="comment"></span> |
| <a name="l00073"></a>00073 <span class="comment">The inverse of the Hessian matrix, evaluated at the estimate of </span> |
| <a name="l00074"></a>00074 <span class="comment">\f$ \boldsymbol \beta \f$, can be used as an <b>approximate variance-covariance </span> |
| <a name="l00075"></a>00075 <span class="comment">matrix </b> for the estimate, and used to produce approximate </span> |
| <a name="l00076"></a>00076 <span class="comment"><b>standard errors</b> for the regression coefficients.</span> |
| <a name="l00077"></a>00077 <span class="comment"></span> |
| <a name="l00078"></a>00078 <span class="comment">\f[</span> |
| <a name="l00079"></a>00079 <span class="comment"> \mathit{se}(c_i) = \left( (H)^{-1} \right)_{ii}</span> |
| <a name="l00080"></a>00080 <span class="comment"> \,.</span> |
| <a name="l00081"></a>00081 <span class="comment">\f]</span> |
| <a name="l00082"></a>00082 <span class="comment">The Wald z-statistic is</span> |
| <a name="l00083"></a>00083 <span class="comment">\f[</span> |
| <a name="l00084"></a>00084 <span class="comment"> z_i = \frac{c_i}{\mathit{se}(c_i)}</span> |
| <a name="l00085"></a>00085 <span class="comment"> \,.</span> |
| <a name="l00086"></a>00086 <span class="comment">\f]</span> |
| <a name="l00087"></a>00087 <span class="comment"></span> |
| <a name="l00088"></a>00088 <span class="comment">The Wald \f$ p \f$-value for coefficient \f$ i \f$ gives the probability (under</span> |
| <a name="l00089"></a>00089 <span class="comment">the assumptions inherent in the Wald test) of seeing a value at least as extreme</span> |
| <a name="l00090"></a>00090 <span class="comment">as the one observed, provided that the null hypothesis (\f$ c_i = 0 \f$) is</span> |
| <a name="l00091"></a>00091 <span class="comment">true. Letting \f$ F \f$ denote the cumulative density function of a standard</span> |
| <a name="l00092"></a>00092 <span class="comment">normal distribution, the Wald \f$ p \f$-value for coefficient \f$ i \f$ is</span> |
| <a name="l00093"></a>00093 <span class="comment">therefore</span> |
| <a name="l00094"></a>00094 <span class="comment">\f[</span> |
| <a name="l00095"></a>00095 <span class="comment"> p_i = \Pr(|Z| \geq |z_i|) = 2 \cdot (1 - F( |z_i| ))</span> |
| <a name="l00096"></a>00096 <span class="comment">\f]</span> |
| <a name="l00097"></a>00097 <span class="comment">where \f$ Z \f$ is a standard normally distributed random variable.</span> |
| <a name="l00098"></a>00098 <span class="comment"></span> |
| <a name="l00099"></a>00099 <span class="comment"></span> |
| <a name="l00100"></a>00100 <span class="comment">The condition number is computed as \f$ \kappa(H) \f$ during the iteration</span> |
| <a name="l00101"></a>00101 <span class="comment">immediately <em>preceding</em> convergence (i.e., \f$ A \f$ is computed using</span> |
| <a name="l00102"></a>00102 <span class="comment">the coefficients of the previous iteration). A large condition number (say, more</span> |
| <a name="l00103"></a>00103 <span class="comment">than 1000) indicates the presence of significant multicollinearity.</span> |
| <a name="l00104"></a>00104 <span class="comment"></span> |
| <a name="l00105"></a>00105 <span class="comment"></span> |
| <a name="l00106"></a>00106 <span class="comment">@input</span> |
| <a name="l00107"></a>00107 <span class="comment"></span> |
| <a name="l00108"></a>00108 <span class="comment">The training data is expected to be of the following form:\n</span> |
| <a name="l00109"></a>00109 <span class="comment"><pre>{TABLE|VIEW} <em>sourceName</em> (</span> |
| <a name="l00110"></a>00110 <span class="comment"> ...</span> |
| <a name="l00111"></a>00111 <span class="comment"> <em>dependentVariable</em> FLOAT8,</span> |
| <a name="l00112"></a>00112 <span class="comment"> <em>independentVariables</em> FLOAT8[],</span> |
| <a name="l00113"></a>00113 <span class="comment"> ...</span> |
| <a name="l00114"></a>00114 <span class="comment">)</pre></span> |
| <a name="l00115"></a>00115 <span class="comment">Note: Dependent Variables refer to the time of death. There is no need to</span> |
| <a name="l00116"></a>00116 <span class="comment">pre-sort the data. Additionally, all the data is assumed</span> |
| <a name="l00117"></a>00117 <span class="comment"></span> |
| <a name="l00118"></a>00118 <span class="comment"></span> |
| <a name="l00119"></a>00119 <span class="comment">@usage</span> |
| <a name="l00120"></a>00120 <span class="comment">- Get vector of coefficients \f$ \boldsymbol \beta \f$ and all diagnostic</span> |
| <a name="l00121"></a>00121 <span class="comment"> statistics:\n</span> |
| <a name="l00122"></a>00122 <span class="comment"> <pre>SELECT * FROM \ref cox_prop_hazards(</span> |
| <a name="l00123"></a>00123 <span class="comment"> '<em>sourceName</em>', '<em>dependentVariable</em>', '<em>independentVariables</em>'</span> |
| <a name="l00124"></a>00124 <span class="comment"> [, <em>numberOfIterations</em> [, '<em>optimizer</em>' [, <em>precision</em> ] ] ]</span> |
| <a name="l00125"></a>00125 <span class="comment">);</pre></span> |
| <a name="l00126"></a>00126 <span class="comment"> Output:</span> |
| <a name="l00127"></a>00127 <span class="comment"> Output:</span> |
| <a name="l00128"></a>00128 <span class="comment"> <pre>coef | log_likelihood | std_err | z_stats | p_values | condition_no | num_iterations</span> |
| <a name="l00129"></a>00129 <span class="comment"> ...</span> |
| <a name="l00130"></a>00130 <span class="comment"></pre></span> |
| <a name="l00131"></a>00131 <span class="comment">- Get vector of coefficients \f$ \boldsymbol \beta \f$:\n</span> |
| <a name="l00132"></a>00132 <span class="comment"> <pre>SELECT (\ref cox_prop_hazards('<em>sourceName</em>', '<em>dependentVariable</em>', '<em>independentVariables</em>')).coef;</pre></span> |
| <a name="l00133"></a>00133 <span class="comment">- Get a subset of the output columns, e.g., only the array of coefficients</span> |
| <a name="l00134"></a>00134 <span class="comment"> \f$ \boldsymbol \beta \f$, the log-likelihood of determination:</span> |
| <a name="l00135"></a>00135 <span class="comment"> <pre>SELECT coef, log_likelihood</span> |
| <a name="l00136"></a>00136 <span class="comment">FROM \ref cox_prop_hazards('<em>sourceName</em>', '<em>dependentVariable</em>', '<em>independentVariables</em>');</pre></span> |
| <a name="l00137"></a>00137 <span class="comment"></span> |
| <a name="l00138"></a>00138 <span class="comment">@examp</span> |
| <a name="l00139"></a>00139 <span class="comment"></span> |
| <a name="l00140"></a>00140 <span class="comment">-# Create the sample data set:</span> |
| <a name="l00141"></a>00141 <span class="comment">@verbatim </span> |
| <a name="l00142"></a>00142 <span class="comment">sql> SELECT * FROM data;</span> |
| <a name="l00143"></a>00143 <span class="comment"> val | time</span> |
| <a name="l00144"></a>00144 <span class="comment">------------|--------------</span> |
| <a name="l00145"></a>00145 <span class="comment"> {0,1.95} | 35</span> |
| <a name="l00146"></a>00146 <span class="comment"> {0,2.20} | 28</span> |
| <a name="l00147"></a>00147 <span class="comment"> {1,1.45} | 32</span> |
| <a name="l00148"></a>00148 <span class="comment"> {1,5.25} | 31</span> |
| <a name="l00149"></a>00149 <span class="comment"> {1,0.38} | 21</span> |
| <a name="l00150"></a>00150 <span class="comment">...</span> |
| <a name="l00151"></a>00151 <span class="comment">@endverbatim</span> |
| <a name="l00152"></a>00152 <span class="comment">-# Run the cox regression function:</span> |
| <a name="l00153"></a>00153 <span class="comment">@verbatim</span> |
| <a name="l00154"></a>00154 <span class="comment">sql> SELECT * FROM cox_prop_hazards('data', 'val', 'time', 100, 'newto', 0.001);</span> |
| <a name="l00155"></a>00155 <span class="comment">---------------|--------------------------------------------------------------</span> |
| <a name="l00156"></a>00156 <span class="comment">coef | {0.881089349817059,-0.0756817768938055}</span> |
| <a name="l00157"></a>00157 <span class="comment">log_likelihood | -4.46535157957394</span> |
| <a name="l00158"></a>00158 <span class="comment">std_err | {1.16954914708414,0.338426252282655}</span> |
| <a name="l00159"></a>00159 <span class="comment">z_stats | {0.753356711368689,-0.223628410729811}</span> |
| <a name="l00160"></a>00160 <span class="comment">p_values | {0.451235588326831,0.823046454908087}</span> |
| <a name="l00161"></a>00161 <span class="comment">condition_no | 12.1135391339082</span> |
| <a name="l00162"></a>00162 <span class="comment">num_iterations | 4</span> |
| <a name="l00163"></a>00163 <span class="comment"></span> |
| <a name="l00164"></a>00164 <span class="comment">@endverbatim</span> |
| <a name="l00165"></a>00165 <span class="comment"></span> |
| <a name="l00166"></a>00166 <span class="comment">@literature</span> |
| <a name="l00167"></a>00167 <span class="comment"></span> |
| <a name="l00168"></a>00168 <span class="comment">A somewhat random selection of nice write-ups, with valuable pointers into</span> |
| <a name="l00169"></a>00169 <span class="comment">further literature:</span> |
| <a name="l00170"></a>00170 <span class="comment"></span> |
| <a name="l00171"></a>00171 <span class="comment">[1] John Fox: Cox Proportional-Hazards Regression for Survival Data,</span> |
| <a name="l00172"></a>00172 <span class="comment"> Appendix to An R and S-PLUS companion to Applied Regression Feb 2012,</span> |
| <a name="l00173"></a>00173 <span class="comment"> http://cran.r-project.org/doc/contrib/Fox-Companion/appendix-cox-regression.pdf</span> |
| <a name="l00174"></a>00174 <span class="comment"></span> |
| <a name="l00175"></a>00175 <span class="comment">[2] Stephen J Walters: What is a Cox model?</span> |
| <a name="l00176"></a>00176 <span class="comment"> http://www.medicine.ox.ac.uk/bandolier/painres/download/whatis/cox_model.pdf</span> |
| <a name="l00177"></a>00177 <span class="comment"></span> |
| <a name="l00178"></a>00178 <span class="comment"></span> |
| <a name="l00179"></a>00179 <span class="comment">@note Source and column names have to be passed as strings (due to limitations </span> |
| <a name="l00180"></a>00180 <span class="comment">of the SQL syntax).</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 cox_prop_hazards.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 cox_prop_hazards </span> |
| <a name="l00187"></a>00187 <span class="comment"> \ref madlib::modules::stats documenting the implementation in C++</span> |
| <a name="l00188"></a>00188 <span class="comment">@endinternal</span> |
| <a name="l00189"></a>00189 <span class="comment"></span> |
| <a name="l00190"></a>00190 <span class="comment">*/</span> |
| <a name="l00191"></a>00191 |
| <a name="l00192"></a>00192 |
| <a name="l00193"></a>00193 DROP TYPE IF EXISTS MADLIB_SCHEMA.intermediate_cox_prop_hazards_result; |
| <a name="l00194"></a>00194 CREATE TYPE MADLIB_SCHEMA.intermediate_cox_prop_hazards_result AS ( |
| <a name="l00195"></a>00195 x DOUBLE PRECISION[], |
| <a name="l00196"></a>00196 exp_coef_x DOUBLE PRECISION, |
| <a name="l00197"></a>00197 x_exp_coef_x DOUBLE PRECISION[], |
| <a name="l00198"></a>00198 x_xTrans_exp_coef_x DOUBLE PRECISION[] |
| <a name="l00199"></a>00199 ); |
| <a name="l00200"></a>00200 |
| <a name="l00201"></a>00201 |
| <a name="l00202"></a>00202 CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.intermediate_cox_prop_hazards( |
| <a name="l00203"></a>00203 /*+ x */ DOUBLE PRECISION[], |
| <a name="l00204"></a>00204 /*+ coef */ DOUBLE PRECISION[]) |
| <a name="l00205"></a>00205 RETURNS MADLIB_SCHEMA.intermediate_cox_prop_hazards_result AS |
| <a name="l00206"></a>00206 'MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00207"></a>00207 <span class="stringliteral">LANGUAGE c IMMUTABLE;</span> |
| <a name="l00208"></a>00208 <span class="stringliteral"></span> |
| <a name="l00209"></a>00209 <span class="stringliteral"></span> |
| <a name="l00210"></a>00210 <span class="stringliteral"></span> |
| <a name="l00211"></a>00211 <span class="stringliteral">DROP TYPE IF EXISTS MADLIB_SCHEMA.cox_prop_hazards_result;</span> |
| <a name="l00212"></a>00212 <span class="stringliteral">CREATE TYPE MADLIB_SCHEMA.cox_prop_hazards_result AS (</span> |
| <a name="l00213"></a>00213 <span class="stringliteral"> coef DOUBLE PRECISION[],</span> |
| <a name="l00214"></a>00214 <span class="stringliteral"> logLikelihood DOUBLE PRECISION,</span> |
| <a name="l00215"></a>00215 <span class="stringliteral"> std_err DOUBLE PRECISION[],</span> |
| <a name="l00216"></a>00216 <span class="stringliteral"> z_stats DOUBLE PRECISION[],</span> |
| <a name="l00217"></a>00217 <span class="stringliteral"> p_values DOUBLE PRECISION[],</span> |
| <a name="l00218"></a>00218 <span class="stringliteral"> condition_no DOUBLE PRECISION,</span> |
| <a name="l00219"></a>00219 <span class="stringliteral"> num_iterations INTEGER</span> |
| <a name="l00220"></a>00220 <span class="stringliteral">);</span> |
| <a name="l00221"></a>00221 <span class="stringliteral"></span> |
| <a name="l00222"></a>00222 <span class="stringliteral"></span> |
| <a name="l00223"></a>00223 <span class="stringliteral"></span> |
| <a name="l00224"></a>00224 <span class="stringliteral"></span> |
| <a name="l00225"></a>00225 <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.cox_prop_hazards_step_final(</span> |
| <a name="l00226"></a>00226 <span class="stringliteral"> state DOUBLE PRECISION[])</span> |
| <a name="l00227"></a>00227 <span class="stringliteral">RETURNS DOUBLE PRECISION[]</span> |
| <a name="l00228"></a>00228 <span class="stringliteral">AS '</span>MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00229"></a>00229 <span class="stringliteral">LANGUAGE C IMMUTABLE STRICT;</span> |
| <a name="l00230"></a>00230 <span class="stringliteral"></span> |
| <a name="l00231"></a>00231 <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.cox_prop_hazards_step_transition(</span> |
| <a name="l00232"></a>00232 <span class="stringliteral"> /*+ state */ DOUBLE PRECISION[],</span> |
| <a name="l00233"></a>00233 <span class="stringliteral"> /*+ x */ DOUBLE PRECISION[],</span> |
| <a name="l00234"></a>00234 <span class="stringliteral"> /*+ y */ DOUBLE PRECISION, </span> |
| <a name="l00235"></a>00235 <span class="stringliteral"> /*+ exp_coef_x */ DOUBLE PRECISION,</span> |
| <a name="l00236"></a>00236 <span class="stringliteral"> /*+ xexp_coef_x */ DOUBLE PRECISION[],</span> |
| <a name="l00237"></a>00237 <span class="stringliteral"> /*+ x_xTrans_exp_coef_x */ DOUBLE PRECISION[],</span> |
| <a name="l00238"></a>00238 <span class="stringliteral"> /*+ previous_state */ DOUBLE PRECISION[])</span> |
| <a name="l00239"></a>00239 <span class="stringliteral">RETURNS DOUBLE PRECISION[] AS </span> |
| <a name="l00240"></a>00240 <span class="stringliteral">'</span>MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00241"></a>00241 <span class="stringliteral">LANGUAGE C IMMUTABLE;</span> |
| <a name="l00242"></a>00242 <span class="stringliteral"></span> |
| <a name="l00243"></a>00243 <span class="stringliteral"></span><span class="comment"></span> |
| <a name="l00244"></a>00244 <span class="comment">/**</span> |
| <a name="l00245"></a>00245 <span class="comment"> * @internal</span> |
| <a name="l00246"></a>00246 <span class="comment"> * @brief Perform one iteration the Newton-Rhapson method.</span> |
| <a name="l00247"></a>00247 <span class="comment"> */</span> |
| <a name="l00248"></a>00248 CREATE |
| <a name="l00249"></a>00249 m4_ifdef(`__GREENPLUM__',m4_ifdef(`__HAS_ORDERED_AGGREGATES__<span class="stringliteral">',`ORDERED'</span>)) |
| <a name="l00250"></a>00250 AGGREGATE MADLIB_SCHEMA.cox_prop_hazards_step( |
| <a name="l00251"></a>00251 |
| <a name="l00252"></a>00252 <span class="comment">/*+ x */</span> DOUBLE PRECISION[], |
| <a name="l00253"></a>00253 <span class="comment">/*+ y */</span> DOUBLE PRECISION, |
| <a name="l00254"></a>00254 <span class="comment">/*+ exp_coef_x */</span> DOUBLE PRECISION, |
| <a name="l00255"></a>00255 <span class="comment">/*+ xexp_coef_x */</span> DOUBLE PRECISION[], |
| <a name="l00256"></a>00256 <span class="comment">/*+ x_xTrans_exp_coef_x */</span> DOUBLE PRECISION[], |
| <a name="l00257"></a>00257 <span class="comment">/*+ previous_state */</span> DOUBLE PRECISION[]) ( |
| <a name="l00258"></a>00258 STYPE=DOUBLE PRECISION[], |
| <a name="l00259"></a>00259 SFUNC=MADLIB_SCHEMA.cox_prop_hazards_step_transition, |
| <a name="l00260"></a>00260 FINALFUNC=MADLIB_SCHEMA.cox_prop_hazards_step_final, |
| <a name="l00261"></a>00261 INITCOND='{0,0,0,0,0,0,0}<span class="stringliteral">'</span> |
| <a name="l00262"></a>00262 <span class="stringliteral">);</span> |
| <a name="l00263"></a>00263 <span class="stringliteral"></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">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.internal_cox_prop_hazards_step_distance(</span> |
| <a name="l00267"></a>00267 <span class="stringliteral"> /*+ state1 */ DOUBLE PRECISION[],</span> |
| <a name="l00268"></a>00268 <span class="stringliteral"> /*+ state2 */ DOUBLE PRECISION[])</span> |
| <a name="l00269"></a>00269 <span class="stringliteral">RETURNS DOUBLE PRECISION AS</span> |
| <a name="l00270"></a>00270 <span class="stringliteral">'</span>MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00271"></a>00271 <span class="stringliteral">LANGUAGE c IMMUTABLE STRICT;</span> |
| <a name="l00272"></a>00272 <span class="stringliteral"></span> |
| <a name="l00273"></a>00273 <span class="stringliteral">CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.internal_cox_prop_hazards_result(</span> |
| <a name="l00274"></a>00274 <span class="stringliteral"> /*+ state */ DOUBLE PRECISION[])</span> |
| <a name="l00275"></a>00275 <span class="stringliteral">RETURNS MADLIB_SCHEMA.cox_prop_hazards_result AS</span> |
| <a name="l00276"></a>00276 <span class="stringliteral">'</span>MODULE_PATHNAME<span class="stringliteral">'</span> |
| <a name="l00277"></a>00277 <span class="stringliteral">LANGUAGE c IMMUTABLE STRICT;</span> |
| <a name="l00278"></a>00278 <span class="stringliteral"></span> |
| <a name="l00279"></a>00279 <span class="stringliteral"></span> |
| <a name="l00280"></a>00280 <span class="stringliteral">-- We only need to document the last one (unfortunately, in Greenplum we have to</span> |
| <a name="l00281"></a>00281 <span class="stringliteral">-- use function overloading instead of default arguments).</span> |
| <a name="l00282"></a>00282 <span class="stringliteral">CREATE FUNCTION MADLIB_SCHEMA.compute_cox_prop_hazards(</span> |
| <a name="l00283"></a>00283 <span class="stringliteral"> "source" VARCHAR,</span> |
| <a name="l00284"></a>00284 <span class="stringliteral"> "indepColumn" VARCHAR,</span> |
| <a name="l00285"></a>00285 <span class="stringliteral"> "depColumn" VARCHAR,</span> |
| <a name="l00286"></a>00286 <span class="stringliteral"> "maxNumIterations" INTEGER,</span> |
| <a name="l00287"></a>00287 <span class="stringliteral"> "optimizer" VARCHAR,</span> |
| <a name="l00288"></a>00288 <span class="stringliteral"> "precision" DOUBLE PRECISION)</span> |
| <a name="l00289"></a>00289 <span class="stringliteral">RETURNS INTEGER</span> |
| <a name="l00290"></a>00290 <span class="stringliteral">AS $$PythonFunction(stats, cox_prop_hazards, compute_cox_prop_hazards)$$</span> |
| <a name="l00291"></a>00291 <span class="stringliteral">LANGUAGE plpythonu VOLATILE;</span> |
| <a name="l00292"></a>00292 <span class="stringliteral"></span><span class="comment"></span> |
| <a name="l00293"></a>00293 <span class="comment">/**</span> |
| <a name="l00294"></a>00294 <span class="comment"> * @brief Compute cox-regression coefficients and diagnostic statistics</span> |
| <a name="l00295"></a>00295 <span class="comment"> *</span> |
| <a name="l00296"></a>00296 <span class="comment"> * To include an intercept in the model, set one coordinate in the</span> |
| <a name="l00297"></a>00297 <span class="comment"> * <tt>independentVariables</tt> array to 1.</span> |
| <a name="l00298"></a>00298 <span class="comment"> * </span> |
| <a name="l00299"></a>00299 <span class="comment"> * @param source Name of the source relation containing the training data</span> |
| <a name="l00300"></a>00300 <span class="comment"> * @param indepColumn Name of the independent column</span> |
| <a name="l00301"></a>00301 <span class="comment"> * @param depColumn Name of the dependant column measuring time of death</span> |
| <a name="l00302"></a>00302 <span class="comment"> * @param maxNumIterations The maximum number of iterations</span> |
| <a name="l00303"></a>00303 <span class="comment"> * @param optimizer The optimizer to use (either</span> |
| <a name="l00304"></a>00304 <span class="comment"> * <tt>'newton'</tt>/<tt>'newton'</tt> for the newton method</span> |
| <a name="l00305"></a>00305 <span class="comment"> * @param precision The difference between log-likelihood values in successive</span> |
| <a name="l00306"></a>00306 <span class="comment"> * iterations that should indicate convergence. Note that a non-positive</span> |
| <a name="l00307"></a>00307 <span class="comment"> * value here disables the convergence criterion, and execution will only</span> |
| <a name="l00308"></a>00308 <span class="comment"> * stop after \c maxNumIterations iterations.</span> |
| <a name="l00309"></a>00309 <span class="comment"> *</span> |
| <a name="l00310"></a>00310 <span class="comment"> * @return A composite value:</span> |
| <a name="l00311"></a>00311 <span class="comment"> * - <tt>coef FLOAT8[]</tt> - Array of coefficients, \f$ \boldsymbol \beta \f$</span> |
| <a name="l00312"></a>00312 <span class="comment"> * - <tt>log_likelihood FLOAT8</tt> - Log-likelihood \f$l(\boldsymbol \beta)\f$</span> |
| <a name="l00313"></a>00313 <span class="comment"> * - <tt>std_err FLOAT8[]</tt> - Array of standard errors,</span> |
| <a name="l00314"></a>00314 <span class="comment"> * \f$ \mathit{se}(c_1), \dots, \mathit{se}(c_k) \f$</span> |
| <a name="l00315"></a>00315 <span class="comment"> * - <tt>z_stats FLOAT8[]</tt> - Array of Wald z-statistics, \f$ \boldsymbol z \f$</span> |
| <a name="l00316"></a>00316 <span class="comment"> * - <tt>p_values FLOAT8[]</tt> - Array of Wald p-values, \f$ \boldsymbol p \f$</span> |
| <a name="l00317"></a>00317 <span class="comment"> * - <tt>condition_no FLOAT8</tt> - The condition number of matrix</span> |
| <a name="l00318"></a>00318 <span class="comment"> * \f$ H \f$ during the iteration immediately <em>preceding</em></span> |
| <a name="l00319"></a>00319 <span class="comment"> * convergence (i.e., \f$ H \f$ is computed using the coefficients of the</span> |
| <a name="l00320"></a>00320 <span class="comment"> * previous iteration)</span> |
| <a name="l00321"></a>00321 <span class="comment"> * - <tt>num_iterations INTEGER</tt> - The number of iterations before the</span> |
| <a name="l00322"></a>00322 <span class="comment"> * algorithm terminated</span> |
| <a name="l00323"></a>00323 <span class="comment"> *</span> |
| <a name="l00324"></a>00324 <span class="comment"> * - Get vector of coefficients \f$ \boldsymbol \beta \f$ and all diagnostic</span> |
| <a name="l00325"></a>00325 <span class="comment"> * statistics:\n</span> |
| <a name="l00326"></a>00326 <span class="comment"> * <pre>SELECT * FROM \ref cox_prop_hazards(</span> |
| <a name="l00327"></a>00327 <span class="comment"> * '<em>sourceName</em>', '<em>dependentVariable</em>', </span> |
| <a name="l00328"></a>00328 <span class="comment"> * '<em>independentVariables</em>'</span> |
| <a name="l00329"></a>00329 <span class="comment"> * [, <em>numberOfIterations</em> [, '<em>optimizer</em>' [, <em>precision</em> ] ] ]</span> |
| <a name="l00330"></a>00330 <span class="comment"> * );</pre></span> |
| <a name="l00331"></a>00331 <span class="comment"> * - Get vector of coefficients \f$ \boldsymbol \beta \f$:\n</span> |
| <a name="l00332"></a>00332 <span class="comment"> * <pre>SELECT (\ref cox_prop_hazards('<em>sourceName</em>', </span> |
| <a name="l00333"></a>00333 <span class="comment"> * '<em>dependentVariable</em>', '<em>independentVariables</em>')).coef;</pre></span> |
| <a name="l00334"></a>00334 <span class="comment"> * - Get a subset of the output columns, e.g., only the array of coefficients</span> |
| <a name="l00335"></a>00335 <span class="comment"> * \f$ \boldsymbol \beta \f$, the log-likelihood of determination:</span> |
| <a name="l00336"></a>00336 <span class="comment"> * <pre>SELECT coef, log_likelihood</span> |
| <a name="l00337"></a>00337 <span class="comment"> * FROM \ref cox_prop_hazards('<em>sourceName</em>', '<em>dependentVariable</em>',</span> |
| <a name="l00338"></a>00338 <span class="comment"> * '<em>independentVariables</em>');</pre></span> |
| <a name="l00339"></a>00339 <span class="comment"> */</span> |
| <a name="l00340"></a>00340 CREATE FUNCTION MADLIB_SCHEMA.cox_prop_hazards( |
| <a name="l00341"></a>00341 "source" VARCHAR, |
| <a name="l00342"></a>00342 "indepColumn" VARCHAR, |
| <a name="l00343"></a>00343 "depColumn" VARCHAR, |
| <a name="l00344"></a>00344 "maxNumIterations" INTEGER /*+ DEFAULT 20 */, |
| <a name="l00345"></a>00345 "optimizer" VARCHAR /*+ DEFAULT 'newton<span class="stringliteral">' */,</span> |
| <a name="l00346"></a>00346 <span class="stringliteral"> "precision" DOUBLE PRECISION /*+ DEFAULT 0.0001 */)</span> |
| <a name="l00347"></a>00347 <span class="stringliteral">RETURNS MADLIB_SCHEMA.cox_prop_hazards_result AS $$</span> |
| <a name="l00348"></a>00348 <span class="stringliteral">DECLARE</span> |
| <a name="l00349"></a>00349 <span class="stringliteral"> theIteration INTEGER;</span> |
| <a name="l00350"></a>00350 <span class="stringliteral"> fnName VARCHAR;</span> |
| <a name="l00351"></a>00351 <span class="stringliteral"> theResult MADLIB_SCHEMA.cox_prop_hazards_result;</span> |
| <a name="l00352"></a>00352 <span class="stringliteral">BEGIN</span> |
| <a name="l00353"></a>00353 <span class="stringliteral"> theIteration := (</span> |
| <a name="l00354"></a>00354 <span class="stringliteral"> SELECT MADLIB_SCHEMA.compute_cox_prop_hazards($1, $2, $3, $4, $5, $6)</span> |
| <a name="l00355"></a>00355 <span class="stringliteral"> );</span> |
| <a name="l00356"></a>00356 <span class="stringliteral"> IF optimizer = '</span>newton<span class="stringliteral">' THEN</span> |
| <a name="l00357"></a>00357 <span class="stringliteral"> fnName := '</span>internal_cox_prop_hazards_result<span class="stringliteral">';</span> |
| <a name="l00358"></a>00358 <span class="stringliteral"> ELSE</span> |
| <a name="l00359"></a>00359 <span class="stringliteral"> RAISE EXCEPTION '</span>Unknown optimizer (<span class="stringliteral">''</span>%<span class="stringliteral">''</span>)<span class="stringliteral">', optimizer;</span> |
| <a name="l00360"></a>00360 <span class="stringliteral"> END IF;</span> |
| <a name="l00361"></a>00361 <span class="stringliteral"> EXECUTE</span> |
| <a name="l00362"></a><a class="code" href="cox__prop__hazards_8sql__in.html#a9c04e1fd1353bb3cfb942b6251851179">00362</a> <span class="stringliteral"> $sql$</span> |
| <a name="l00363"></a>00363 <span class="stringliteral"> SELECT (result).*</span> |
| <a name="l00364"></a>00364 <span class="stringliteral"> FROM (</span> |
| <a name="l00365"></a>00365 <span class="stringliteral"> SELECT</span> |
| <a name="l00366"></a>00366 <span class="stringliteral"> MADLIB_SCHEMA.$sql$ || fnName || $sql$(_madlib_state) AS result</span> |
| <a name="l00367"></a>00367 <span class="stringliteral"> FROM _madlib_iterative_alg</span> |
| <a name="l00368"></a>00368 <span class="stringliteral"> WHERE _madlib_iteration = $sql$ || theIteration || $sql$</span> |
| <a name="l00369"></a>00369 <span class="stringliteral"> ) subq</span> |
| <a name="l00370"></a>00370 <span class="stringliteral"> $sql$</span> |
| <a name="l00371"></a>00371 <span class="stringliteral"> INTO theResult;</span> |
| <a name="l00372"></a>00372 <span class="stringliteral"> </span> |
| <a name="l00373"></a>00373 <span class="stringliteral"> -- The number of iterations are not updated in the C++ code. We do it here.</span> |
| <a name="l00374"></a>00374 <span class="stringliteral"> IF NOT (theResult IS NULL) THEN</span> |
| <a name="l00375"></a>00375 <span class="stringliteral"> theResult.num_iterations = theIteration;</span> |
| <a name="l00376"></a>00376 <span class="stringliteral"> END IF;</span> |
| <a name="l00377"></a>00377 <span class="stringliteral"> RETURN theResult;</span> |
| <a name="l00378"></a>00378 <span class="stringliteral">END;</span> |
| <a name="l00379"></a>00379 <span class="stringliteral">$$ LANGUAGE plpgsql VOLATILE;</span> |
| <a name="l00380"></a>00380 <span class="stringliteral"></span> |
| <a name="l00381"></a>00381 <span class="stringliteral"></span> |
| <a name="l00382"></a>00382 <span class="stringliteral">CREATE FUNCTION MADLIB_SCHEMA.cox_prop_hazards(</span> |
| <a name="l00383"></a>00383 <span class="stringliteral"> "source" VARCHAR,</span> |
| <a name="l00384"></a>00384 <span class="stringliteral"> "indepColumn" VARCHAR,</span> |
| <a name="l00385"></a>00385 <span class="stringliteral"> "depColumn" VARCHAR)</span> |
| <a name="l00386"></a>00386 <span class="stringliteral">RETURNS MADLIB_SCHEMA.cox_prop_hazards_result AS</span> |
| <a name="l00387"></a>00387 <span class="stringliteral">$$SELECT MADLIB_SCHEMA.cox_prop_hazards($1, $2, $3, 20, '</span>newton<span class="stringliteral">', 0.0001);$$</span> |
| <a name="l00388"></a>00388 <span class="stringliteral">LANGUAGE sql VOLATILE;</span> |
| <a name="l00389"></a>00389 <span class="stringliteral"></span> |
| <a name="l00390"></a>00390 <span class="stringliteral">CREATE FUNCTION MADLIB_SCHEMA.cox_prop_hazards(</span> |
| <a name="l00391"></a>00391 <span class="stringliteral"> "source" VARCHAR,</span> |
| <a name="l00392"></a>00392 <span class="stringliteral"> "indepColumn" VARCHAR,</span> |
| <a name="l00393"></a>00393 <span class="stringliteral"> "depColumn" VARCHAR,</span> |
| <a name="l00394"></a>00394 <span class="stringliteral"> "maxNumIterations" INTEGER)</span> |
| <a name="l00395"></a>00395 <span class="stringliteral">RETURNS MADLIB_SCHEMA.cox_prop_hazards_result AS</span> |
| <a name="l00396"></a>00396 <span class="stringliteral">$$SELECT MADLIB_SCHEMA.cox_prop_hazards($1, $2, $3, $4, '</span>newton<span class="stringliteral">', 0.0001);$$</span> |
| <a name="l00397"></a>00397 <span class="stringliteral">LANGUAGE sql VOLATILE;</span> |
| <a name="l00398"></a>00398 <span class="stringliteral"></span> |
| <a name="l00399"></a>00399 <span class="stringliteral">CREATE FUNCTION MADLIB_SCHEMA.cox_prop_hazards(</span> |
| <a name="l00400"></a>00400 <span class="stringliteral"> "source" VARCHAR,</span> |
| <a name="l00401"></a>00401 <span class="stringliteral"> "indepColumn" VARCHAR,</span> |
| <a name="l00402"></a>00402 <span class="stringliteral"> "depColumn" VARCHAR,</span> |
| <a name="l00403"></a>00403 <span class="stringliteral"> "maxNumIterations" INTEGER,</span> |
| <a name="l00404"></a>00404 <span class="stringliteral"> "optimizer" VARCHAR)</span> |
| <a name="l00405"></a>00405 <span class="stringliteral">RETURNS MADLIB_SCHEMA.cox_prop_hazards_result AS</span> |
| <a name="l00406"></a>00406 <span class="stringliteral">$$SELECT MADLIB_SCHEMA.cox_prop_hazards($1, $2, $3, $4, $5, 0.0001);$$</span> |
| <a name="l00407"></a>00407 <span class="stringliteral">LANGUAGE sql VOLATILE;</span> |
| </pre></div></div> |
| </div> |
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