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[31/51] [abbrv] [partial] madlib-site git commit: Additional updates
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+<div class="title">Cox-Proportional Hazards Regression<div class="ingroups"><a class="el" href="group__grp__super.html">Supervised Learning</a> » <a class="el" href="group__grp__regml.html">Regression Models</a></div></div> </div>
+</div><!--header-->
+<div class="contents">
+<div class="toc"><b>Contents</b> <ul>
+<li class="level1">
+<a href="#training">Training Function</a> </li>
+<li class="level1">
+<a href="#cox_zph">PHA Test Function</a> </li>
+<li class="level1">
+<a href="#predict">Prediction Function</a> </li>
+<li class="level1">
+<a href="#examples">Examples</a> </li>
+<li class="level1">
+<a href="#background">Technical Background</a> </li>
+<li class="level1">
+<a href="#related">Related Topics</a> </li>
+</ul>
+</div><p>Proportional-Hazard models enable the comparison of various survival models. These survival models are functions describing the probability of a one-item event (prototypically, this event is death) with respect to time. The interval of time before the occurrence of death can be called the survival time. Let T be a random variable representing the survival time, with a cumulative probability function P(t). Informally, P(t) is the probability that death has happened before time t.</p>
+<p><a class="anchor" id="training"></a></p><dl class="section user"><dt>Training Function</dt><dd></dd></dl>
+<p>Following is the syntax for the <a class="el" href="cox__prop__hazards_8sql__in.html#a737450bbfe0f10204b0074a9d45b0cef" title="Compute cox-regression coefficients and diagnostic statistics. ">coxph_train()</a> training function: </p><pre class="syntax">
+coxph_train( source_table,
+ output_table,
+ dependent_variable,
+ independent_variable,
+ right_censoring_status,
+ strata,
+ optimizer_params
+ )
+</pre><p> <b>Arguments</b> </p><dl class="arglist">
+<dt>source_table </dt>
+<dd>TEXT. The name of the table containing input data. </dd>
+<dt>output_table </dt>
+<dd><p class="startdd">TEXT. The name of the table where the output model is saved. The output is saved in the table named by the <em>output_table</em> argument. It has the following columns: </p><table class="output">
+<tr>
+<th>coef </th><td>FLOAT8[]. Vector of the coefficients. </td></tr>
+<tr>
+<th>loglikelihood </th><td>FLOAT8. Log-likelihood value of the MLE estimate. </td></tr>
+<tr>
+<th>std_err </th><td>FLOAT8[]. Vector of the standard error of the coefficients. </td></tr>
+<tr>
+<th>stats </th><td>FLOAT8[]. Vector of the statistics of the coefficients. </td></tr>
+<tr>
+<th>p_values </th><td>FLOAT8[]. Vector of the p-values of the coefficients. </td></tr>
+<tr>
+<th>hessian </th><td>FLOAT8[]. The Hessian matrix computed using the final solution. </td></tr>
+<tr>
+<th>num_iterations </th><td>INTEGER. The number of iterations performed by the optimizer. </td></tr>
+</table>
+<p>Additionally, a summary output table is generated that contains a summary of the parameters used for building the Cox model. It is stored in a table named <output_table>_summary. It has the following columns: </p><table class="output">
+<tr>
+<th>source_table </th><td>The source table name. </td></tr>
+<tr>
+<th>dependent_variable </th><td>The dependent variable name. </td></tr>
+<tr>
+<th>independent_variable </th><td>The independent variable name. </td></tr>
+<tr>
+<th>right_censoring_status </th><td>The right censoring status </td></tr>
+<tr>
+<th>strata </th><td>The stratification columns </td></tr>
+<tr>
+<th>num_processed </th><td>The number of rows that were actually used in the computation. </td></tr>
+<tr>
+<th>num_missing_rows_skipped </th><td>The number of rows that were skipped in the computation due to NULL values in them. </td></tr>
+</table>
+<p class="enddd"></p>
+</dd>
+<dt>dependent_variable </dt>
+<dd>TEXT. A string containing the name of a column that contains an array of numeric values, or a string expression in the format 'ARRAY[1, x1, x2, x3]', where <em>x1</em>, <em>x2</em> and <em>x3</em> are column names. Dependent variables refer to the time of death. There is no need to pre-sort the data. </dd>
+<dt>independent_variable </dt>
+<dd>TEXT. The name of the independent variable. </dd>
+<dt>right_censoring_status (optional) </dt>
+<dd>TEXT, default: TRUE for all observations. A string containing an expression that evaluates to the right-censoring status for the observation—TRUE if the observation is not censored and FALSE if the observation is censored. The string could contain the name of the column containing the right-censoring status, a fixed Boolean expression (i.e., 'true', 'false', '0', '1') that applies to all observations, or a Boolean expression such as 'column_name < 10' that can be evaluated for each observation. </dd>
+<dt>strata (optional) </dt>
+<dd>VARCHAR, default: NULL, which does not do any stratifications. A string of comma-separated column names that are the strata ID variables used to do stratification. </dd>
+<dt>optimizer_params (optional) </dt>
+<dd><p class="startdd">VARCHAR, default: NULL, which uses the default values of optimizer parameters: max_iter=100, optimizer=newton, tolerance=1e-8, array_agg_size=10000000, sample_size=1000000. It should be a string that contains 'key=value' pairs separated by commas. The meanings of these parameters are:</p>
+<ul>
+<li>max_iter — The maximum number of iterations. The computation stops if the number of iterations exceeds this, which usually means that there is no convergence.</li>
+<li>optimizer — The optimization method. Right now, "newton" is the only one supported.</li>
+<li>tolerance — The stopping criteria. When the difference between the log-likelihoods of two consecutive iterations is smaller than this number, the computation has already converged and stops.</li>
+<li>array_agg_size — To speed up the computation, the original data table is cut into multiple pieces, and each pieces of the data is aggregated into one big row. In the process of computation, the whole big row is loaded into memory and thus speed up the computation. This parameter controls approximately how many numbers we want to put into one big row. Larger value of array_agg_size may speed up more, but the size of the big row cannot exceed 1GB due to the restriction of PostgreSQL databases.</li>
+<li>sample_size — To cut the data into approximate equal pieces, we first sample the data, and then find out the break points using this sampled data. A larger sample_size produces more accurate break points. </li>
+</ul>
+</dd>
+</dl>
+<p><a class="anchor" id="cox_zph"></a></p><dl class="section user"><dt>Proportional Hazards Assumption Test Function</dt><dd></dd></dl>
+<p>The <a class="el" href="cox__prop__hazards_8sql__in.html#a682d95d5475ce33e47937067cadc2766" title="Test the proportional hazards assumption for a Cox regression model fit (coxph_train) ...">cox_zph()</a> function tests the proportional hazards assumption (PHA) of a Cox regression.</p>
+<p>Proportional-hazard models enable the comparison of various survival models. These PH models, however, assume that the hazard for a given individual is a fixed proportion of the hazard for any other individual, and the ratio of the hazards is constant across time. MADlib does not currently have support for performing any transformation of the time to compute the correlation.</p>
+<p>The <a class="el" href="cox__prop__hazards_8sql__in.html#a682d95d5475ce33e47937067cadc2766" title="Test the proportional hazards assumption for a Cox regression model fit (coxph_train) ...">cox_zph()</a> function is used to test this assumption by computing the correlation of the residual of the <a class="el" href="cox__prop__hazards_8sql__in.html#a737450bbfe0f10204b0074a9d45b0cef" title="Compute cox-regression coefficients and diagnostic statistics. ">coxph_train()</a> model with time.</p>
+<p>Following is the syntax for the <a class="el" href="cox__prop__hazards_8sql__in.html#a682d95d5475ce33e47937067cadc2766" title="Test the proportional hazards assumption for a Cox regression model fit (coxph_train) ...">cox_zph()</a> function: </p><pre class="syntax">
+cox_zph(cox_model_table, output_table)
+</pre><p> <b>Arguments</b> </p><dl class="arglist">
+<dt>cox_model_table </dt>
+<dd><p class="startdd">TEXT. The name of the table containing the Cox Proportional-Hazards model.</p>
+<p class="enddd"></p>
+</dd>
+<dt>output_table </dt>
+<dd>TEXT. The name of the table where the test statistics are saved. The output table is named by the <em>output_table</em> argument and has the following columns: <table class="output">
+<tr>
+<th>covariate </th><td>TEXT. The independent variables. </td></tr>
+<tr>
+<th>rho </th><td>FLOAT8[]. Vector of the correlation coefficients between survival time and the scaled Schoenfeld residuals. </td></tr>
+<tr>
+<th>chi_square </th><td>FLOAT8[]. Chi-square test statistic for the correlation analysis. </td></tr>
+<tr>
+<th>p_value </th><td>FLOAT8[]. Two-side p-value for the chi-square statistic. </td></tr>
+</table>
+</dd>
+</dl>
+<p>Additionally, the residual values are outputted to the table named <em>output_table</em>_residual. The table contains the following columns: </p><table class="output">
+<tr>
+<th><dep_column_name> </th><td>FLOAT8. Time values (dependent variable) present in the original source table. </td></tr>
+<tr>
+<th>residual </th><td>FLOAT8[]. Difference between the original covariate values and the expectation of the covariates obtained from the coxph_train model. </td></tr>
+<tr>
+<th>scaled_residual </th><td>Residual values scaled by the variance of the coefficients. </td></tr>
+</table>
+<p><a class="anchor" id="notes"></a></p><dl class="section user"><dt>Notes</dt><dd></dd></dl>
+<ul>
+<li>Table names can be optionally schema qualified (current_schemas() is used if a schema name is not provided) and table and column names should follow case-sensitivity and quoting rules per the database. For instance, 'mytable' and 'MyTable' both resolve to the same entity—'mytable'. If mixed-case or multi-byte characters are desired for entity names then the string should be double-quoted; in this case the input would be '"MyTable"'.</li>
+<li>The <a class="el" href="cox__prop__hazards_8sql__in.html#a3310cf98478b7c1e400e8fb1b3965d30">cox_prop_hazards_regr()</a> and <a class="el" href="cox__prop__hazards_8sql__in.html#ad778b289eb19ae0bb2b7e02a89bab3bc" title="Cox regression training function. ">cox_prop_hazards()</a> functions have been deprecated; <a class="el" href="cox__prop__hazards_8sql__in.html#a737450bbfe0f10204b0074a9d45b0cef" title="Compute cox-regression coefficients and diagnostic statistics. ">coxph_train()</a> should be used instead.</li>
+</ul>
+<p><a class="anchor" id="predict"></a></p><dl class="section user"><dt>Prediction Function</dt><dd>The prediction function is provided to calculate the linear predictionors, risk or the linear terms for the given prediction data. It has the following syntax: <pre class="syntax">
+coxph_predict(model_table,
+ source_table,
+ id_col_name,
+ output_table,
+ pred_type,
+ reference)
+</pre></dd></dl>
+<p><b>Arguments</b> </p><dl class="arglist">
+<dt>model_table </dt>
+<dd><p class="startdd">TEXT. Name of the table containing the cox model.</p>
+<p class="enddd"></p>
+</dd>
+<dt>source_table </dt>
+<dd><p class="startdd">TEXT. Name of the table containing the prediction data.</p>
+<p class="enddd"></p>
+</dd>
+<dt>id_col_name </dt>
+<dd><p class="startdd">TEXT. Name of the id column in the source table.</p>
+<p class="enddd"></p>
+</dd>
+<dt>output_table </dt>
+<dd><p class="startdd">TEXT. Name of the table to store the prediction results in. The output table is named by the <em>output_table</em> argument and has the following columns: </p><table class="output">
+<tr>
+<th>id </th><td>TEXT. The id column name from the source table. </td></tr>
+<tr>
+<th>predicted_result </th><td>DOUBLE PRECISION. Result of prediction based of the value of the prediction type parameter. </td></tr>
+</table>
+<p class="enddd"></p>
+</dd>
+<dt>pred_type </dt>
+<dd><p class="startdd">TEXT, OPTIONAL. Type of prediction. This can be one of 'linear_predictors', 'risk', or 'terms'. DEFAULT='linear_predictors'.</p><ul>
+<li>'linear_predictors' calculates the dot product of the independent variables and the coefficients.</li>
+<li>'risk' is the exponentiated value of the linear prediction.</li>
+<li>'terms' correspond to the linear terms obtained by multiplying the independent variables with their corresponding coefficients values (without further calculating the sum of these terms) </li>
+</ul>
+<p class="enddd"></p>
+</dd>
+<dt>reference </dt>
+<dd>TEXT, OPTIONAL. Reference level to use for centering predictions. Can be one of 'strata', 'overall'. DEFAULT='strata'. Note that R uses 'sample' instead of 'overall' when referring to the overall mean value of the covariates as being the reference level. </dd>
+</dl>
+<p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl>
+<ol type="1">
+<li>View online help for the proportional hazards training method. <pre class="example">
+SELECT madlib.coxph_train();
+</pre></li>
+<li>Create an input data set. <pre class="example">
+DROP TABLE IF EXISTS sample_data;
+CREATE TABLE sample_data (
+ id INTEGER NOT NULL,
+ grp DOUBLE PRECISION,
+ wbc DOUBLE PRECISION,
+ timedeath INTEGER,
+ status BOOLEAN
+);
+COPY sample_data FROM STDIN WITH DELIMITER '|';
+ 0 | 0 | 1.45 | 35 | t
+ 1 | 0 | 1.47 | 34 | t
+ 3 | 0 | 2.2 | 32 | t
+ 4 | 0 | 1.78 | 25 | t
+ 5 | 0 | 2.57 | 23 | t
+ 6 | 0 | 2.32 | 22 | t
+ 7 | 0 | 2.01 | 20 | t
+ 8 | 0 | 2.05 | 19 | t
+ 9 | 0 | 2.16 | 17 | t
+ 10 | 0 | 3.6 | 16 | t
+ 11 | 1 | 2.3 | 15 | t
+ 12 | 0 | 2.88 | 13 | t
+ 13 | 1 | 1.5 | 12 | t
+ 14 | 0 | 2.6 | 11 | t
+ 15 | 0 | 2.7 | 10 | t
+ 16 | 0 | 2.8 | 9 | t
+ 17 | 1 | 2.32 | 8 | t
+ 18 | 0 | 4.43 | 7 | t
+ 19 | 0 | 2.31 | 6 | t
+ 20 | 1 | 3.49 | 5 | t
+ 21 | 1 | 2.42 | 4 | t
+ 22 | 1 | 4.01 | 3 | t
+ 23 | 1 | 4.91 | 2 | t
+ 24 | 1 | 5 | 1 | t
+\.
+</pre></li>
+<li>Run the Cox regression function. <pre class="example">
+SELECT madlib.coxph_train( 'sample_data',
+ 'sample_cox',
+ 'timedeath',
+ 'ARRAY[grp,wbc]',
+ 'status'
+ );
+</pre></li>
+<li>View the results of the regression. <pre class="example">
+\x on
+SELECT * FROM sample_cox;
+</pre> Results: <pre class="result">
+-[ RECORD 1 ]--+----------------------------------------------------------------------------
+coef | {2.54407073265254,1.67172094779487}
+loglikelihood | -37.8532498733
+std_err | {0.677180599294897,0.387195514577534}
+z_stats | {3.7568570855419,4.31751114064138}
+p_values | {0.000172060691513886,1.5779844638453e-05}
+hessian | {{2.78043065745617,-2.25848560642414},{-2.25848560642414,8.50472838284472}}
+num_iterations | 5
+</pre></li>
+<li>Computing predictions using cox model. (This example uses the original data table to perform the prediction. Typically a different test dataset with the same features as the original training dataset would be used.) <pre class="example">
+\x off
+-- Display the linear predictors for the original dataset
+SELECT madlib.coxph_predict('sample_cox',
+ 'sample_data',
+ 'id',
+ 'sample_pred');
+</pre> <pre class="result">
+SELECT * FROM sample_pred;
+ id | predicted_value
+----+--------------------
+ 0 | -2.97110918125034
+ 4 | -2.41944126847803
+ 6 | -1.5167119566688
+ 8 | -1.96807661257341
+ 10 | 0.623090856508638
+ 12 | -0.58054822590367
+ 14 | -1.04863009128623
+ 16 | -0.714285901727259
+ 18 | 2.01061924317838
+ 20 | 2.98327228490375
+ 22 | 3.85256717775708
+ 24 | 5.507570916074
+ 1 | -2.93767476229444
+ 3 | -1.71731847040418
+ 5 | -1.09878171972008
+ 7 | -2.03494545048521
+ 9 | -1.78418730831598
+ 15 | -0.881457996506747
+ 19 | -1.53342916614675
+ 11 | 0.993924357027849
+ 13 | -0.343452401208048
+ 17 | 1.02735877598375
+ 21 | 1.19453087076323
+ 23 | 5.35711603077246
+(24 rows)
+</pre> <pre class="example">
+-- Display the relative risk for the original dataset
+SELECT madlib.coxph_predict('sample_cox',
+ 'sample_data',
+ 'id',
+ 'sample_pred',
+ 'risk');
+</pre> <pre class="result">
+ id | predicted_value
+ ----+--------------------
+ 1 | 0.0529887971503509
+ 3 | 0.179546963459175
+ 5 | 0.33327686110022
+ 7 | 0.130687611255372
+ 9 | 0.167933483703554
+ 15 | 0.414178600294289
+ 19 | 0.215794402223054
+ 11 | 2.70181658768287
+ 13 | 0.709317242984782
+ 17 | 2.79367735395696
+ 21 | 3.30200833843654
+ 23 | 212.112338046551
+ 0 | 0.0512464372091503
+ 4 | 0.0889713146524469
+ 6 | 0.219432204682557
+ 8 | 0.139725343898993
+ 10 | 1.86468261037506
+ 12 | 0.559591499901241
+ 14 | 0.350417460388247
+ 16 | 0.489541567796517
+ 18 | 7.46794038691975
+ 20 | 19.7523463121038
+ 22 | 47.1138577624204
+ 24 | 246.551504798816
+(24 rows)
+</pre></li>
+<li>Run the test for Proportional Hazards assumption to obtain correlation between residuals and time. <pre class="example">
+SELECT madlib.cox_zph( 'sample_cox',
+ 'sample_zph_output'
+ );
+</pre></li>
+<li>View results of the PHA test. <pre class="example">
+SELECT * FROM sample_zph_output;
+</pre> Results: <pre class="result">
+-[ RECORD 1 ]-----------------------------------------
+covariate | ARRAY[grp,wbc]
+rho | {0.00237308357328641,0.0375600568840431}
+chi_square | {0.000100675718191977,0.0232317400546175}
+p_value | {0.991994376850758,0.878855984657948}
+</pre></li>
+</ol>
+<p><a class="anchor" id="background"></a></p><dl class="section user"><dt>Technical Background</dt><dd></dd></dl>
+<p>Generally, proportional-hazard models start with a list of \( \boldsymbol n \) observations, each with \( \boldsymbol m \) covariates and a time of death. From this \( \boldsymbol n \times m \) matrix, we would like to derive the correlation between the covariates and the hazard function. This amounts to finding the parameters \( \boldsymbol \beta \) that best fit the model described below.</p>
+<p>Let us define:</p><ul>
+<li>\( \boldsymbol t \in \mathbf R^{m} \) denote the vector of observed dependent variables, with \( n \) rows.</li>
+<li>\( X \in \mathbf R^{m} \) denote the design matrix with \( m \) columns and \( n \) rows, containing all observed vectors of independent variables \( \boldsymbol x_i \) as rows.</li>
+<li>\( R(t_i) \) denote the set of observations still alive at time \( t_i \)</li>
+</ul>
+<p>Note that this model <b>does not</b> include a <b>constant term</b>, and the data cannot contain a column of 1s.</p>
+<p>By definition, </p><p class="formulaDsp">
+\[ P[T_k = t_i | \boldsymbol R(t_i)] = \frac{e^{\beta^T x_k} }{ \sum_{j \in R(t_i)} e^{\beta^T x_j}}. \,. \]
+</p>
+<p>The <b>partial likelihood </b>function can now be generated as the product of conditional probabilities: </p><p class="formulaDsp">
+\[ \mathcal L = \prod_{i = 1}^n \left( \frac{e^{\beta^T x_i}}{ \sum_{j \in R(t_i)} e^{\beta^T x_j}} \right). \]
+</p>
+<p>The log-likelihood form of this equation is </p><p class="formulaDsp">
+\[ L = \sum_{i = 1}^n \left[ \beta^T x_i - \log\left(\sum_{j \in R(t_i)} e^{\beta^T x_j }\right) \right]. \]
+</p>
+<p>Using this score function and Hessian matrix, the partial likelihood can be maximized using the <b> Newton-Raphson algorithm</b>. <b>Breslow's method</b> is used to resolved tied times of deaths. The time of death for two records are considered "equal" if they differ by less than 1.0e-6</p>
+<p>The inverse of the Hessian matrix, evaluated at the estimate of \( \boldsymbol \beta \), can be used as an <b>approximate variance-covariance matrix </b> for the estimate, and used to produce approximate <b>standard errors</b> for the regression coefficients.</p>
+<p class="formulaDsp">
+\[ \mathit{se}(c_i) = \left( (H)^{-1} \right)_{ii} \,. \]
+</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 condition number is computed as \( \kappa(H) \) 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>
+<p><a class="anchor" id="Literature"></a></p><dl class="section user"><dt>Literature</dt><dd></dd></dl>
+<p>A somewhat random selection of nice write-ups, with valuable pointers into further literature:</p>
+<p>[1] John Fox: Cox Proportional-Hazards Regression for Survival Data, Appendix to An R and S-PLUS companion to Applied Regression Feb 2012, <a href="http://cran.r-project.org/doc/contrib/Fox-Companion/appendix-cox-regression.pdf">http://cran.r-project.org/doc/contrib/Fox-Companion/appendix-cox-regression.pdf</a></p>
+<p>[2] Stephen J Walters: What is a Cox model? <a href="http://www.medicine.ox.ac.uk/bandolier/painres/download/whatis/cox_model.pdf">http://www.medicine.ox.ac.uk/bandolier/painres/download/whatis/cox_model.pdf</a></p>
+<p><a class="anchor" id="notes"></a></p><dl class="section user"><dt>Notes</dt><dd></dd></dl>
+<p>If number of ties in the source table is very large, a memory allocation error may be raised. The limitation is about \((10^8 / m)\), where \(m\) is number of featrues. For instance, if there are 100 featrues, the number of ties should be fewer than 1 million.</p>
+<p><a class="anchor" id="related"></a></p><dl class="section user"><dt>Related Topics</dt><dd></dd></dl>
+<p>File <a class="el" href="cox__prop__hazards_8sql__in.html" title="SQL functions for cox proportional hazards. ">cox_prop_hazards.sql_in</a> documenting the functions</p>
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+ <div class="headertitle">
+<div class="title">Conditional Random Field<div class="ingroups"><a class="el" href="group__grp__super.html">Supervised Learning</a></div></div> </div>
+</div><!--header-->
+<div class="contents">
+<div class="toc"><b>Contents</b> <ul>
+<li>
+<a href="#train_feature">Training Feature Generation</a> </li>
+<li>
+<a href="#train">CRF Training Function</a> </li>
+<li>
+<a href="#test_feature">Testing Feature Generation</a> </li>
+<li>
+<a href="#inference">Inference using Viterbi</a> </li>
+<li>
+<a href="#usage">Using CRF</a> </li>
+<li>
+<a href="#examples">Examples</a> </li>
+<li>
+<a href="#background">Technical Background</a> </li>
+<li>
+<a href="#literature">Literature</a> </li>
+<li>
+<a href="#related">Related Topics</a> </li>
+</ul>
+</div><p>A conditional random field (CRF) is a type of discriminative, undirected probabilistic graphical model. A linear-chain CRF is a special type of CRF that assumes the current state depends only on the previous state.</p>
+<p>Feature extraction modules are provided for text-analysis tasks such as part-of-speech (POS) tagging and named-entity resolution (NER). Currently, six feature types are implemented:</p>
+<ul>
+<li>Edge Feature: transition feature that encodes the transition feature weight from current label to next label.</li>
+<li>Start Feature: fired when the current token is the first token in a sequence.</li>
+<li>End Feature: fired when the current token is the last token in a sequence.</li>
+<li>Word Feature: fired when the current token is observed in the trained dictionary.</li>
+<li>Unknown Feature: fired when the current token is not observed in the trained dictionary for at least a certain number of times (default 1).</li>
+<li>Regex Feature: fired when the current token can be matched by a regular expression.</li>
+</ul>
+<p>A Viterbi implementation is also provided to get the best label sequence and the conditional probability \( \Pr( \text{best label sequence} \mid \text{sequence}) \).</p>
+<p>Following steps are required for CRF Learning and Inference:</p><ol type="1">
+<li><a href="#train_feature">Training Feature Generation</a></li>
+<li><a href="#train">CRF Training</a></li>
+<li><a href="#test_feature">Testing Feature Generation</a></li>
+<li><a href="#inference">Inference using Viterbi</a></li>
+</ol>
+<p><a class="anchor" id="train_feature"></a></p><dl class="section user"><dt>Training Feature Generation</dt><dd>The function takes <code>train_segment_tbl</code> and <code>regex_tbl</code> as input and does feature generation generating three tables <code>dictionary_tbl</code>, <code>train_feature_tbl</code> and <code>train_featureset_tbl</code>, that are required as an input for CRF training. <pre class="syntax">
+crf_train_fgen(train_segment_tbl,
+ regex_tbl,
+ label_tbl,
+ dictionary_tbl,
+ train_feature_tbl,
+ train_featureset_tbl)
+</pre> <b>Arguments</b> <dl class="arglist">
+<dt>train_segment_tbl </dt>
+<dd>TEXT. Name of the training segment table. The table is expected to have the following columns: <table class="output">
+<tr>
+<th>doc_id </th><td>INTEGER. Document id column </td></tr>
+<tr>
+<th>start_pos </th><td>INTEGER. Index of a particular term in the respective document </td></tr>
+<tr>
+<th>seg_text </th><td>TEXT. Term at the respective <code>start_pos</code> in the document </td></tr>
+<tr>
+<th>label </th><td>INTEGER. Label id for the term corresponding to the actual label from <code>label_tbl</code> </td></tr>
+</table>
+</dd>
+<dt>regex_tbl </dt>
+<dd>TEXT. Name of the regular expression table. The table is expected to have the following columns: <table class="output">
+<tr>
+<th>pattern </th><td>TEXT. Regular Expression </td></tr>
+<tr>
+<th>name </th><td>TEXT. Regular Expression name </td></tr>
+</table>
+</dd>
+<dt>label_tbl </dt>
+<dd>TEXT. Name of the table containing unique labels and their id's. The table is expected to have the following columns: <table class="output">
+<tr>
+<th>id </th><td>INTEGER. Unique label id. NOTE: Must range from 0 to total number of labels in the table - 1. </td></tr>
+<tr>
+<th>label </th><td>TEXT. Label name </td></tr>
+</table>
+</dd>
+<dt>dictionary_tbl </dt>
+<dd>TEXT. Name of the dictionary table to be created containing unique terms along with their counts. The table will have the following columns: <table class="output">
+<tr>
+<th>token </th><td>TEXT. Contains all the unique terms found in <code>train_segment_tbl</code> </td></tr>
+<tr>
+<th>total </th><td>INTEGER. Respective counts for the terms </td></tr>
+</table>
+</dd>
+<dt>train_feature_tbl</dt>
+<dd></dd>
+<dt></dt>
+<dd><p class="startdd">TEXT. Name of the training feature table to be created. The table will have the following columns: </p><table class="output">
+<tr>
+<th>doc_id </th><td>INTEGER. Document id </td></tr>
+<tr>
+<th>f_size </th><td>INTEGER. Feature set size. This value will be same for all the tuples in the table </td></tr>
+<tr>
+<th>sparse_r </th><td>DOUBLE PRECISION[]. Array union of individual single state features (previous label, label, feature index, start position, training existance indicator), ordered by their start position. </td></tr>
+<tr>
+<th>dense_m </th><td>DOUBLE PRECISION[]. Array union of (previous label, label, feature index, start position, training existance indicator) of edge features ordered by start position. </td></tr>
+<tr>
+<th>sparse_m </th><td>DOUBLE PRECISION[]. Array union of (feature index, previous label, label) of edge features ordered by feature index. </td></tr>
+</table>
+<p class="enddd"></p>
+</dd>
+<dt>train_featureset_tbl </dt>
+<dd>TEXT. Name of the table to be created containing distinct featuresets generated from training feature extraction. The table will have the following columns: <table class="output">
+<tr>
+<th>f_index </th><td>INTEGER. Column containing distinct featureset ids </td></tr>
+<tr>
+<th>f_name </th><td>TEXT. Feature name </td></tr>
+<tr>
+<th>feature </th><td>ARRAY. Feature value. The value is of the form [L1, L2] <br />
+ - If L1 = -1: represents single state feature with L2 being the current label id. <br />
+ - If L1 != -1: represents transition feature with L1 be the previous label and L2 be the current label. </td></tr>
+</table>
+</dd>
+</dl>
+</dd></dl>
+<p><a class="anchor" id="train"></a></p><dl class="section user"><dt>Linear Chain CRF Training Function</dt><dd>The function takes <code>train_feature_tbl</code> and <code>train_featureset_tbl</code> tables generated in the training feature generation steps as input along with other required parameters and produces two output tables <code>crf_stats_tbl</code> and <code>crf_weights_tbl</code>.</dd></dl>
+<pre class="syntax">
+lincrf_train(train_feature_tbl,
+ train_featureset_tbl,
+ label_tbl,
+ crf_stats_tbl,
+ crf_weights_tbl
+ max_iterations
+ )
+</pre><p> <b>Arguments</b> </p><dl class="arglist">
+<dt>train_feature_tbl </dt>
+<dd><p class="startdd">TEXT. Name of the feature table generated during training feature generation</p>
+<p class="enddd"></p>
+</dd>
+<dt>train_featureset_tbl </dt>
+<dd><p class="startdd">TEXT. Name of the featureset table generated during training feature generation</p>
+<p class="enddd"></p>
+</dd>
+<dt>label_tbl </dt>
+<dd><p class="startdd">TEXT. Name of the label table used</p>
+<p class="enddd"></p>
+</dd>
+<dt>crf_stats_table </dt>
+<dd>TEXT. Name of the table to be created containing statistics for CRF training. The table has the following columns: <table class="output">
+<tr>
+<th>coef </th><td>DOUBLE PRECISION[]. Array of coefficients </td></tr>
+<tr>
+<th>log_likelihood </th><td>DOUBLE. Log-likelihood </td></tr>
+<tr>
+<th>num_iterations </th><td>INTEGER. The number of iterations at which the algorithm terminated </td></tr>
+</table>
+</dd>
+<dt>crf_weights_table </dt>
+<dd><p class="startdd">TEXT. Name of the table to be created creating learned feature weights. The table has the following columns: </p><table class="output">
+<tr>
+<th>id </th><td>INTEGER. Feature set id </td></tr>
+<tr>
+<th>name </th><td>TEXT. Feature name </td></tr>
+<tr>
+<th>prev_label_id </th><td>INTEGER. Label for the previous token encountered </td></tr>
+<tr>
+<th>label_id </th><td>INTEGER. Label of the token with the respective feature </td></tr>
+<tr>
+<th>weight </th><td>DOUBLE PRECISION. Weight for the respective feature set </td></tr>
+</table>
+<p class="enddd"></p>
+</dd>
+<dt>max_iterations </dt>
+<dd>INTEGER. The maximum number of iterations </dd>
+</dl>
+<p><a class="anchor" id="test_feature"></a></p><dl class="section user"><dt>Testing Feature Generation</dt><dd></dd></dl>
+<pre class="syntax">
+crf_test_fgen(test_segment_tbl,
+ dictionary_tbl,
+ label_tbl,
+ regex_tbl,
+ crf_weights_tbl,
+ viterbi_mtbl,
+ viterbi_rtbl
+ )
+</pre><p> <b>Arguments</b> </p><dl class="arglist">
+<dt>test_segment_tbl </dt>
+<dd><p class="startdd">TEXT. Name of the testing segment table. The table is expected to have the following columns: </p><table class="output">
+<tr>
+<th>doc_id </th><td>INTEGER. Document id column </td></tr>
+<tr>
+<th>start_pos </th><td>INTEGER. Index of a particular term in the respective document </td></tr>
+<tr>
+<th>seg_text </th><td>TEXT. Term at the respective <code>start_pos</code> in the document </td></tr>
+</table>
+<p class="enddd"></p>
+</dd>
+<dt>dictionary_tbl </dt>
+<dd><p class="startdd">TEXT. Name of the dictionary table created during training feature generation (<code>crf_train_fgen</code>)</p>
+<p class="enddd"></p>
+</dd>
+<dt>label_tbl </dt>
+<dd><p class="startdd">TEXT. Name of the label table</p>
+<p class="enddd"></p>
+</dd>
+<dt>regex_tbl </dt>
+<dd><p class="startdd">TEXT. Name of the regular expression table</p>
+<p class="enddd"></p>
+</dd>
+<dt>crf_weights_tbl </dt>
+<dd><p class="startdd">TEXT. Name of the weights table generated during CRF training (<code>lincrf_train</code>)</p>
+<p class="enddd"></p>
+</dd>
+<dt>viterbi_mtbl </dt>
+<dd><p class="startdd">TEXT. Name of the Viterbi M table to be created</p>
+<p class="enddd"></p>
+</dd>
+<dt>viterbi_rtbl </dt>
+<dd>TEXT. Name of the Viterbi R table to be created </dd>
+</dl>
+<p><a class="anchor" id="inference"></a></p><dl class="section user"><dt>Inference using Viterbi</dt><dd><pre class="syntax">
+vcrf_label(test_segment_tbl,
+ viterbi_mtbl,
+ viterbi_rtbl,
+ label_tbl,
+ result_tbl)
+</pre> <b>Arguments</b> <dl class="arglist">
+<dt>test_segment_tbl </dt>
+<dd>TEXT. Name of the testing segment table. For required table schema, please refer to arguments in previous section </dd>
+<dt>viterbi_mtbl </dt>
+<dd>TEXT. Name of the table <code>viterbi_mtbl</code> generated from testing feature generation <code>crf_test_fgen</code>. </dd>
+<dt>viterbi_rtbl </dt>
+<dd>TEXT. Name of the table <code>viterbi_rtbl</code> generated from testing feature generation <code>crf_test_fgen</code>. </dd>
+<dt>label_tbl </dt>
+<dd>TEXT. Name of the label table. </dd>
+<dt>result_tbl </dt>
+<dd>TEXT. Name of the result table to be created containing extracted best label sequences. </dd>
+</dl>
+</dd></dl>
+<p><a class="anchor" id="usage"></a></p><dl class="section user"><dt>Using CRF</dt><dd></dd></dl>
+<p>Generate text features, calculate their weights, and output the best label sequence for test data:<br />
+</p><ol type="1">
+<li>Perform feature generation on training data i.e. <code>train_segment_tbl</code> generating <code>train_feature_tbl</code> and <code>train_featureset_tbl</code>. <pre>SELECT madlib.crf_train_fgen(
+ '<em>train_segment_tbl</em>',
+ '<em>regex_tbl</em>',
+ '<em>label_tbl</em>',
+ '<em>dictionary_tbl</em>',
+ '<em>train_feature_tbl</em>',
+ '<em>train_featureset_tbl</em>');</pre></li>
+<li>Use linear-chain CRF for training providing <code>train_feature_tbl</code> and <code>train_featureset_tbl</code> generated from previous step as an input. <pre>SELECT madlib.lincrf_train(
+ '<em>train_feature_tbl</em>',
+ '<em>train_featureset_tbl</em>',
+ '<em>label_tbl</em>',
+ '<em>crf_stats_tbl</em>',
+ '<em>crf_weights_tbl</em>',
+ <em>max_iterations</em>);</pre></li>
+<li>Perform feature generation on testing data <code>test_segment_tbl</code> generating <code>viterbi_mtbl</code> and <code>viterbi_rtbl</code> required for inferencing. <pre>SELECT madlib.crf_test_fgen(
+ '<em>test_segment_tbl</em>',
+ '<em>dictionary_tbl</em>',
+ '<em>label_tbl</em>',
+ '<em>regex_tbl</em>',
+ '<em>crf_weights_tbl</em>',
+ '<em>viterbi_mtbl</em>',
+ '<em>viterbi_rtbl</em>');</pre></li>
+<li>Run the Viterbi function to get the best label sequence and the conditional probability \( \Pr( \text{best label sequence} \mid \text{sequence}) \). <pre>SELECT madlib.vcrf_label(
+ '<em>test_segment_tbl</em>',
+ '<em>viterbi_mtbl</em>',
+ '<em>viterbi_rtbl</em>',
+ '<em>label_tbl</em>',
+ '<em>result_tbl</em>');</pre></li>
+</ol>
+<p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd>This example uses a trivial training and test data set.</dd></dl>
+<ol type="1">
+<li>Load the label table, the regular expressions table, and the training segment table: <pre class="example">
+SELECT * FROM crf_label ORDER BY id;
+</pre> Result: <pre class="result">
+ id | label
+ ---+-------
+ 0 | #
+ 1 | $
+ 2 | ''
+...
+ 8 | CC
+ 9 | CD
+ 10 | DT
+ 11 | EX
+ 12 | FW
+ 13 | IN
+ 14 | JJ
+...
+</pre> The regular expressions table: <pre class="example">
+SELECT * from crf_regex;
+</pre> <pre class="result">
+ pattern | name
+ --------------+----------------------
+ ^.+ing$ | endsWithIng
+ ^[A-Z][a-z]+$ | InitCapital
+ ^[A-Z]+$ | isAllCapital
+ ^.*[0-9]+.*$ | containsDigit
+...
+</pre> The training segment table: <pre class="example">
+SELECT * from train_segmenttbl ORDER BY doc_id, start_pos;
+</pre> <pre class="result">
+ doc_id | start_pos | seg_text | label
+ -------+-----------+------------+-------
+ 0 | 0 | Confidence | 18
+ 0 | 1 | in | 13
+ 0 | 2 | the | 10
+ 0 | 3 | pound | 18
+ 0 | 4 | is | 38
+ 0 | 5 | widely | 26
+...
+ 1 | 0 | Chancellor | 19
+ 1 | 1 | of | 13
+ 1 | 2 | the | 10
+ 1 | 3 | Exchequer | 19
+ 1 | 4 | Nigel | 19
+...
+</pre></li>
+<li>Generate the training features: <pre class="example">
+SELECT crf_train_fgen( 'train_segmenttbl',
+ 'crf_regex',
+ 'crf_label',
+ 'crf_dictionary',
+ 'train_featuretbl',
+ 'train_featureset'
+ );
+SELECT * from crf_dictionary;
+</pre> Result: <pre class="result">
+ token | total
+ ----------------+-------
+ Hawthorne | 1
+ Mercedes-Benzes | 1
+ Wolf | 3
+ best-known | 1
+ hairline | 1
+ accepting | 2
+ purchases | 14
+ trash | 5
+ co-venture | 1
+ restaurants | 7
+...
+</pre> <pre class="example">
+SELECT * from train_featuretbl;
+</pre> Result: <pre class="result">
+ doc_id | f_size | sparse_r | dense_m | sparse_m
+ -------+--------+-------------------------------+---------------------------------+-----------------------
+ 2 | 87 | {-1,13,12,0,1,-1,13,9,0,1,..} | {13,31,79,1,1,31,29,70,2,1,...} | {51,26,2,69,29,17,...}
+ 1 | 87 | {-1,13,0,0,1,-1,13,9,0,1,...} | {13,0,62,1,1,0,13,54,2,1,13,..} | {51,26,2,69,29,17,...}
+</pre> <pre class="example">
+SELECT * from train_featureset;
+</pre> <pre class="result">
+ f_index | f_name | feature
+ --------+---------------+---------
+ 1 | R_endsWithED | {-1,29}
+ 13 | W_outweigh | {-1,26}
+ 29 | U | {-1,5}
+ 31 | U | {-1,29}
+ 33 | U | {-1,12}
+ 35 | W_a | {-1,2}
+ 37 | W_possible | {-1,6}
+ 15 | W_signaled | {-1,29}
+ 17 | End. | {-1,43}
+ 49 | W_'s | {-1,16}
+ 63 | W_acquire | {-1,26}
+ 51 | E. | {26,2}
+ 69 | E. | {29,17}
+ 71 | E. | {2,11}
+ 83 | W_the | {-1,2}
+ 85 | E. | {16,11}
+ 4 | W_return | {-1,11}
+...
+</pre></li>
+<li>Train using linear CRF: <pre class="example">
+SELECT lincrf_train( 'train_featuretbl',
+ 'train_featureset',
+ 'crf_label',
+ 'crf_stats_tbl',
+ 'crf_weights_tbl',
+ 20
+ );
+</pre> <pre class="result">
+ lincrf_train
+ -----------------------------------------------------------------------------------
+ CRF Train successful. Results stored in the specified CRF stats and weights table
+ lincrf
+</pre> View the feature weight table. <pre class="example">
+SELECT * from crf_weights_tbl;
+</pre> Result: <pre class="result">
+ id | name | prev_label_id | label_id | weight
+ ---+---------------+---------------+----------+-------------------
+ 1 | R_endsWithED | -1 | 29 | 1.54128249293937
+ 13 | W_outweigh | -1 | 26 | 1.70691232223653
+ 29 | U | -1 | 5 | 1.40708515869008
+ 31 | U | -1 | 29 | 0.830356200936407
+ 33 | U | -1 | 12 | 0.769587378281239
+ 35 | W_a | -1 | 2 | 2.68470625883726
+ 37 | W_possible | -1 | 6 | 3.41773107604468
+ 15 | W_signaled | -1 | 29 | 1.68187039165771
+ 17 | End. | -1 | 43 | 3.07687845517082
+ 49 | W_'s | -1 | 16 | 2.61430312229883
+ 63 | W_acquire | -1 | 26 | 1.67247047385797
+ 51 | E. | 26 | 2 | 3.0114240119435
+ 69 | E. | 29 | 17 | 2.82385531733866
+ 71 | E. | 2 | 11 | 3.00970493772732
+ 83 | W_the | -1 | 2 | 2.58742315259326
+...
+</pre></li>
+<li>To find the best labels for a test set using the trained linear CRF model, repeat steps #1-2 and generate the test features, except instead of creating a new dictionary, use the dictionary generated from the training set. <pre class="example">
+SELECT * from test_segmenttbl ORDER BY doc_id, start_pos;
+</pre> Result: <pre class="result">
+ doc_id | start_pos | seg_text
+ -------+-----------+---------------
+ 0 | 0 | Rockwell
+ 0 | 1 | International
+ 0 | 2 | Corp.
+ 0 | 3 | 's
+ 0 | 4 | Tulsa
+ 0 | 5 | unit
+ 0 | 6 | said
+...
+ 1 | 0 | Rockwell
+ 1 | 1 | said
+ 1 | 2 | the
+ 1 | 3 | agreement
+ 1 | 4 | calls
+...
+</pre> <pre class="example">
+SELECT crf_test_fgen( 'test_segmenttbl',
+ 'crf_dictionary',
+ 'crf_label',
+ 'crf_regex',
+ 'crf_weights_tbl',
+ 'viterbi_mtbl',
+ 'viterbi_rtbl'
+ );
+</pre></li>
+<li>Calculate the best label sequence and save in the table <code>extracted_best_labels</code>. <pre class="example">
+SELECT vcrf_label( 'test_segmenttbl',
+ 'viterbi_mtbl',
+ 'viterbi_rtbl',
+ 'crf_label',
+ 'extracted_best_labels'
+ );
+</pre> View the best labels. <pre class="example">
+SELECT * FROM extracted_best_labels;
+</pre> Result: <pre class="result">
+ doc_id | start_pos | seg_text | label | id | max_pos | prob
+ -------+-----------+---------------+-------+----+---------+----------
+ 0 | 0 | Rockwell | NNP | 19 | 27 | 0.000269
+ 0 | 1 | International | NNP | 19 | 27 | 0.000269
+ 0 | 2 | Corp. | NNP | 19 | 27 | 0.000269
+ 0 | 3 | 's | NNP | 19 | 27 | 0.000269
+...
+ 1 | 0 | Rockwell | NNP | 19 | 16 | 0.000168
+ 1 | 1 | said | NNP | 19 | 16 | 0.000168
+ 1 | 2 | the | DT | 10 | 16 | 0.000168
+ 1 | 3 | agreement | JJ | 14 | 16 | 0.000168
+ 1 | 4 | calls | NNS | 21 | 16 | 0.000168
+...
+</pre></li>
+</ol>
+<p><a class="anchor" id="background"></a></p><dl class="section user"><dt>Technical Background</dt><dd></dd></dl>
+<p>Specifically, a linear-chain CRF is a distribution defined by </p><p class="formulaDsp">
+\[ p_\lambda(\boldsymbol y | \boldsymbol x) = \frac{\exp{\sum_{m=1}^M \lambda_m F_m(\boldsymbol x, \boldsymbol y)}}{Z_\lambda(\boldsymbol x)} \,. \]
+</p>
+<p>where</p><ul>
+<li>\( F_m(\boldsymbol x, \boldsymbol y) = \sum_{i=1}^n f_m(y_i,y_{i-1},x_i) \) is a global feature function that is a sum along a sequence \( \boldsymbol x \) of length \( n \)</li>
+<li>\( f_m(y_i,y_{i-1},x_i) \) is a local feature function dependent on the current token label \( y_i \), the previous token label \( y_{i-1} \), and the observation \( x_i \)</li>
+<li>\( \lambda_m \) is the corresponding feature weight</li>
+<li>\( Z_\lambda(\boldsymbol x) \) is an instance-specific normalizer <p class="formulaDsp">
+\[ Z_\lambda(\boldsymbol x) = \sum_{\boldsymbol y'} \exp{\sum_{m=1}^M \lambda_m F_m(\boldsymbol x, \boldsymbol y')} \]
+</p>
+</li>
+</ul>
+<p>A linear-chain CRF estimates the weights \( \lambda_m \) by maximizing the log-likelihood of a given training set \( T=\{(x_k,y_k)\}_{k=1}^N \).</p>
+<p>The log-likelihood is defined as </p><p class="formulaDsp">
+\[ \ell_{\lambda}=\sum_k \log p_\lambda(y_k|x_k) =\sum_k[\sum_{m=1}^M \lambda_m F_m(x_k,y_k) - \log Z_\lambda(x_k)] \]
+</p>
+<p>and the zero of its gradient </p><p class="formulaDsp">
+\[ \nabla \ell_{\lambda}=\sum_k[F(x_k,y_k)-E_{p_\lambda(Y|x_k)}[F(x_k,Y)]] \]
+</p>
+<p>is found since the maximum likelihood is reached when the empirical average of the global feature vector equals its model expectation. The MADlib implementation uses limited-memory BFGS (L-BFGS), a limited-memory variation of the Broyden–Fletcher–Goldfarb–Shanno (BFGS) update, a quasi-Newton method for unconstrained optimization.</p>
+<p>\(E_{p_\lambda(Y|x)}[F(x,Y)]\) is found by using a variant of the forward-backward algorithm: </p><p class="formulaDsp">
+\[ E_{p_\lambda(Y|x)}[F(x,Y)] = \sum_y p_\lambda(y|x)F(x,y) = \sum_i\frac{\alpha_{i-1}(f_i*M_i)\beta_i^T}{Z_\lambda(x)} \]
+</p>
+ <p class="formulaDsp">
+\[ Z_\lambda(x) = \alpha_n.1^T \]
+</p>
+<p> where \(\alpha_i\) and \( \beta_i\) are the forward and backward state cost vectors defined by </p><p class="formulaDsp">
+\[ \alpha_i = \begin{cases} \alpha_{i-1}M_i, & 0<i<=n\\ 1, & i=0 \end{cases}\\ \]
+</p>
+ <p class="formulaDsp">
+\[ \beta_i^T = \begin{cases} M_{i+1}\beta_{i+1}^T, & 1<=i<n\\ 1, & i=n \end{cases} \]
+</p>
+<p>To avoid overfitting, we penalize the likelihood with a spherical Gaussian weight prior: </p><p class="formulaDsp">
+\[ \ell_{\lambda}^\prime=\sum_k[\sum_{m=1}^M \lambda_m F_m(x_k,y_k) - \log Z_\lambda(x_k)] - \frac{\lVert \lambda \rVert^2}{2\sigma ^2} \]
+</p>
+<p class="formulaDsp">
+\[ \nabla \ell_{\lambda}^\prime=\sum_k[F(x_k,y_k) - E_{p_\lambda(Y|x_k)}[F(x_k,Y)]] - \frac{\lambda}{\sigma ^2} \]
+</p>
+<dl class="section user"><dt>Literature</dt><dd>[1] F. Sha, F. Pereira. Shallow Parsing with Conditional Random Fields, <a href="http://www-bcf.usc.edu/~feisha/pubs/shallow03.pdf">http://www-bcf.usc.edu/~feisha/pubs/shallow03.pdf</a></dd></dl>
+<p>[2] Wikipedia, Conditional Random Field, <a href="http://en.wikipedia.org/wiki/Conditional_random_field">http://en.wikipedia.org/wiki/Conditional_random_field</a></p>
+<p>[3] A. Jaiswal, S.Tawari, I. Mansuri, K. Mittal, C. Tiwari (2012), CRF, <a href="http://crf.sourceforge.net/">http://crf.sourceforge.net/</a></p>
+<p>[4] D. Wang, ViterbiCRF, <a href="http://www.cs.berkeley.edu/~daisyw/ViterbiCRF.html">http://www.cs.berkeley.edu/~daisyw/ViterbiCRF.html</a></p>
+<p>[5] Wikipedia, Viterbi Algorithm, <a href="http://en.wikipedia.org/wiki/Viterbi_algorithm">http://en.wikipedia.org/wiki/Viterbi_algorithm</a></p>
+<p>[6] J. Nocedal. Updating Quasi-Newton Matrices with Limited Storage (1980), Mathematics of Computation 35, pp. 773-782</p>
+<p>[7] J. Nocedal, Software for Large-scale Unconstrained Optimization, <a href="http://users.eecs.northwestern.edu/~nocedal/lbfgs.html">http://users.eecs.northwestern.edu/~nocedal/lbfgs.html</a></p>
+<p><a class="anchor" id="related"></a></p><dl class="section user"><dt>Related Topics</dt><dd></dd></dl>
+<p>File <a class="el" href="crf_8sql__in.html" title="SQL functions for conditional random field. ">crf.sql_in</a> <a class="el" href="crf__feature__gen_8sql__in.html" title="SQL function for POS/NER feature extraction. ">crf_feature_gen.sql_in</a> <a class="el" href="viterbi_8sql__in.html" title="concatenate a set of input values into arrays to feed into viterbi c function and create a human read...">viterbi.sql_in</a> (documenting the SQL functions) </p>
+</div><!-- contents -->
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+<!-- start footer part -->
+<div id="nav-path" class="navpath"><!-- id is needed for treeview function! -->
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+ <a href="http://www.doxygen.org/index.html">
+ <img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.13 </li>
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+ <td style="padding-left: 0.5em;">
+ <div id="projectname">
+ <span id="projectnumber">1.13</span>
+ </div>
+ <div id="projectbrief">User Documentation for MADlib</div>
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+<a href="#groups">Modules</a> </div>
+ <div class="headertitle">
+<div class="title">Data Types and Transformations</div> </div>
+</div><!--header-->
+<div class="contents">
+<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
+<p>Data types and transformation operations </p>
+<table class="memberdecls">
+<tr class="heading"><td colspan="2"><h2 class="groupheader"><a name="groups"></a>
+Modules</h2></td></tr>
+<tr class="memitem:group__grp__arraysmatrix"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__arraysmatrix.html">Arrays and Matrices</a></td></tr>
+<tr class="memdesc:group__grp__arraysmatrix"><td class="mdescLeft"> </td><td class="mdescRight">Mathematical operations for arrays and matrices. <br /></td></tr>
+<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
+<tr class="memitem:group__grp__pca"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__pca.html">Dimensionality Reduction</a></td></tr>
+<tr class="memdesc:group__grp__pca"><td class="mdescLeft"> </td><td class="mdescRight">A collection of methods for dimensionality reduction. <br /></td></tr>
+<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
+<tr class="memitem:group__grp__encode__categorical"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__encode__categorical.html">Encoding Categorical Variables</a></td></tr>
+<tr class="memdesc:group__grp__encode__categorical"><td class="mdescLeft"> </td><td class="mdescRight">Provides functions to encode categorical variables. <br /></td></tr>
+<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
+<tr class="memitem:group__grp__pivot"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__pivot.html">Pivot</a></td></tr>
+<tr class="memdesc:group__grp__pivot"><td class="mdescLeft"> </td><td class="mdescRight">Provides pivoting functions helpful for data preparation before modeling. <br /></td></tr>
+<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
+<tr class="memitem:group__grp__stemmer"><td class="memItemLeft" align="right" valign="top"> </td><td class="memItemRight" valign="bottom"><a class="el" href="group__grp__stemmer.html">Stemming</a></td></tr>
+<tr class="memdesc:group__grp__stemmer"><td class="mdescLeft"> </td><td class="mdescRight">Provides porter stemmer operations supporting other MADlib modules. <br /></td></tr>
+<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
+</table>
+</div><!-- contents -->
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+var group__grp__datatrans =
+[
+ [ "Arrays and Matrices", "group__grp__arraysmatrix.html", "group__grp__arraysmatrix" ],
+ [ "Dimensionality Reduction", "group__grp__pca.html", "group__grp__pca" ],
+ [ "Encoding Categorical Variables", "group__grp__encode__categorical.html", null ],
+ [ "Pivot", "group__grp__pivot.html", null ],
+ [ "Stemming", "group__grp__stemmer.html", null ]
+];
\ No newline at end of file