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http://git-wip-us.apache.org/repos/asf/incubator-madlib-site/blob/c97706e8/docs/latest/group__grp__elasticnet.html
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-<div class="title">Elastic Net Regularization<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></p><ul>
-<li class="level1">
-<a href="#train">Training Function</a> </li>
-<li class="level1">
-<a href="#optimizer">Optimizer Parameters</a> </li>
-<li class="level1">
-<a href="#predict">Prediction Functions</a> </li>
-<li class="level1">
-<a href="#examples">Examples</a> </li>
-<li class="level1">
-<a href="#background">Technical Background</a> </li>
-<li class="level1">
-<a href="#literature">Literature</a> </li>
-<li class="level1">
-<a href="#related">Related Topics</a> </li>
-</ul>
-</div><p>This module implements elastic net regularization for linear and logistic regression problems.</p>
-<p><a class="anchor" id="train"></a></p><dl class="section user"><dt>Training Function</dt><dd>The training function has the following syntax: <pre class="syntax">
-elastic_net_train( tbl_source,
- tbl_result,
- col_dep_var,
- col_ind_var,
- regress_family,
- alpha,
- lambda_value,
- standardize,
- grouping_col,
- optimizer,
- optimizer_params,
- excluded,
- max_iter,
- tolerance
- )
-</pre></dd></dl>
-<p><b>Arguments</b> </p><dl class="arglist">
-<dt>tbl_source </dt>
-<dd><p class="startdd">TEXT. The name of the table containing the training data.</p>
-<p class="enddd"></p>
-</dd>
-<dt>tbl_result </dt>
-<dd><p class="startdd">TEXT. Name of the generated table containing the output model. The output table produced by the <a class="el" href="elastic__net_8sql__in.html#a735038a5090c112505c740a90a203e83" title="Interface for elastic net. ">elastic_net_train()</a> function has the following columns: </p><table class="output">
-<tr>
-<th>regress_family </th><td>The regression type: 'gaussian' or 'binomial'. </td></tr>
-<tr>
-<th>features </th><td>An array of the features (independent variables) passed into the analysis. </td></tr>
-<tr>
-<th>features_selected </th><td>An array of the features selected by the analysis. </td></tr>
-<tr>
-<th>coef_nonzero </th><td>Fitting coefficients for the selected features. </td></tr>
-<tr>
-<th>coef_all </th><td>Coefficients for all selected and unselected features </td></tr>
-<tr>
-<th>intercept </th><td>Fitting intercept for the model. </td></tr>
-<tr>
-<th>log_likelihood </th><td>The negative value of the first equation above (up to a constant depending on the data set). </td></tr>
-<tr>
-<th>standardize </th><td>BOOLEAN. Whether the data was normalized (<em>standardize</em> argument was TRUE). </td></tr>
-<tr>
-<th>iteration_run </th><td>The number of iterations executed. </td></tr>
-</table>
-<p class="enddd"></p>
-</dd>
-<dt>col_dep_var </dt>
-<dd><p class="startdd">TEXT. An expression for the dependent variable.</p>
-<p>Both <em>col_dep_var</em> and <em>col_ind_var</em> can be valid Postgres expressions. For example, <code>col_dep_var = 'log(y+1)'</code>, and <code>col_ind_var = 'array[exp(x[1]), x[2], 1/(1+x[3])]'</code>. In the binomial case, you can use a Boolean expression, for example, <code>col_dep_var = 'y < 0'</code>.</p>
-<p class="enddd"></p>
-</dd>
-<dt>col_ind_var </dt>
-<dd><p class="startdd">TEXT. An expression for the independent variables. Use <code>'*'</code> to specify all columns of <em>tbl_source</em> except those listed in the <em>excluded</em> string. If <em>col_dep_var</em> is a column name, it is automatically excluded from the independent variables. However, if <em>col_dep_var</em> is a valid Postgres expression, any column names used within the expression are only excluded if they are explicitly included in the <em>excluded</em> argument. It is a good idea to add all column names involved in the dependent variable expression to the <em>excluded</em> string.</p>
-<p class="enddd"></p>
-</dd>
-<dt>regress_family </dt>
-<dd><p class="startdd">TEXT. The regression type, either 'gaussian' ('linear') or 'binomial' ('logistic').</p>
-<p class="enddd"></p>
-</dd>
-<dt>alpha </dt>
-<dd><p class="startdd">FLOAT8. Elastic net control parameter, value in [0, 1], 1 for L-1 regularization, 0 for L-2.</p>
-<p class="enddd"></p>
-</dd>
-<dt>lambda_value </dt>
-<dd><p class="startdd">FLOAT8. Regularization parameter, positive.</p>
-<p class="enddd"></p>
-</dd>
-<dt>standardize (optional) </dt>
-<dd><p class="startdd">BOOLEAN, default: TRUE. Whether to normalize the data. Setting this to TRUE usually yields better results and faster convergence.</p>
-<p class="enddd"></p>
-</dd>
-<dt>grouping_col (optional) </dt>
-<dd><p class="startdd">TEXT, default: NULL.</p>
-<dl class="section note"><dt>Note</dt><dd><em>Not currently implemented. Any non-NULL value is ignored. Grouping support will be added in a future release. </em> When implemented, an expression list will be used to group the input dataset into discrete groups, running one regression per group. Similar to the SQL <code>GROUP BY</code> clause. When this value is NULL, no grouping is used and a single result model is generated.</dd></dl>
-</dd>
-<dt>optimizer (optional) </dt>
-<dd><p class="startdd">TEXT, default: 'fista'. Name of optimizer, either 'fista' or 'igd'.</p>
-<p class="enddd"></p>
-</dd>
-<dt>optimizer_params (optional) </dt>
-<dd><p class="startdd">TEXT, default: NULL. Optimizer parameters, delimited with commas. The parameters differ depending on the value of <em>optimizer</em>. See the descriptions below for details.</p>
-<p class="enddd"></p>
-</dd>
-<dt>excluded (optional) </dt>
-<dd><p class="startdd">TEXT, default: NULL. If the <em>col_ind_var</em> input is '*' then <em>excluded</em> can be provided as a comma-delimited list of column names that are to be excluded from the features. For example, <code>'col1, col2'</code>. If the <em>col_ind_var</em> is an array, <em>excluded</em> must be a list of the integer array positions to exclude, for example <code>'1,2'</code>. If this argument is NULL or an empty string <code>''</code>, no columns are excluded.</p>
-<p class="enddd"></p>
-</dd>
-<dt>max_iter (optional) </dt>
-<dd><p class="startdd">INTEGER, default: 1000. The maximum number of iterations that are allowed.</p>
-<p class="enddd"></p>
-</dd>
-<dt>tolerance </dt>
-<dd>FLOAT8, default: default is 1e-6. The criteria to end iterations. Both the 'fista' and 'igd' optimizers compute the difference between the loglikelihood of two consecutive iterations, and when the difference is smaller than <em>tolerance</em> or the iteration number is larger than <em>max_iter</em>, the computation stops. </dd>
-</dl>
-<p><a class="anchor" id="optimizer"></a></p><dl class="section user"><dt>Optimizer Parameters</dt><dd>Optimizer parameters are supplied in a string containing a comma-delimited list of name-value pairs. All of these named parameters are optional, and their order does not matter. You must use the format "<param_name> = <value>" to specify the value of a parameter, otherwise the parameter is ignored.</dd></dl>
-<p>When the <a class="el" href="elastic__net_8sql__in.html#a735038a5090c112505c740a90a203e83">elastic_net_train()</a> <em>optimizer</em> argument value is <b>'fista'</b>, the <em>optimizer_params</em> argument is a string containing name-value pairs with the following format. (Line breaks are inserted for readability.) </p><pre class="syntax">
- 'max_stepsize = <value>,
- eta = <value>,
- warmup = <value>,
- warmup_lambdas = <value>,
- warmup_lambda_no = <value>,
- warmup_tolerance = <value>,
- use_active_set = <value>,
- activeset_tolerance = <value>,
- random_stepsize = <value>'
-</pre><p> <b>Parameters</b> </p><dl class="arglist">
-<dt>max_stepsize </dt>
-<dd>Default: 4.0. Initial backtracking step size. At each iteration, the algorithm first tries <em>stepsize = max_stepsize</em>, and if it does not work out, it then tries a smaller step size, <em>stepsize = stepsize/eta</em>, where <em>eta</em> must be larger than 1. At first glance, this seems to perform repeated iterations for even one step, but using a larger step size actually greatly increases the computation speed and minimizes the total number of iterations. A careful choice of <em>max_stepsize</em> can decrease the computation time by more than 10 times. </dd>
-<dt>eta </dt>
-<dd><p class="startdd">Default: 2. If stepsize does not work <em>stepsize</em> / <em>eta</em> is tried. Must be greater than 1. </p>
-<p class="enddd"></p>
-</dd>
-<dt>warmup </dt>
-<dd><p class="startdd">Default: FALSE. If <em>warmup</em> is TRUE, a series of lambda values, which is strictly descent and ends at the lambda value that the user wants to calculate, is used. The larger lambda gives very sparse solution, and the sparse solution again is used as the initial guess for the next lambda's solution, which speeds up the computation for the next lambda. For larger data sets, this can sometimes accelerate the whole computation and may be faster than computation on only one lambda value.</p>
-<p class="enddd"></p>
-</dd>
-<dt>warmup_lambdas </dt>
-<dd><p class="startdd">Default: NULL. The lambda value series to use when <em>warmup</em> is True. The default is NULL, which means that lambda values will be automatically generated.</p>
-<p class="enddd"></p>
-</dd>
-<dt>warmup_lambda_no </dt>
-<dd><p class="startdd">Default: 15. How many lambdas are used in warm-up. If <em>warmup_lambdas</em> is not NULL, this value is overridden by the number of provided lambda values.</p>
-<p class="enddd"></p>
-</dd>
-<dt>warmup_tolerance </dt>
-<dd><p class="startdd">The value of tolerance used during warmup. The default is the same as the <em>tolerance</em> argument.</p>
-<p class="enddd"></p>
-</dd>
-<dt>use_active_set </dt>
-<dd><p class="startdd">Default: FALSE. If <em>use_active_set</em> is TRUE, an active-set method is used to speed up the computation. Considerable speedup is obtained by organizing the iterations around the active set of features—those with nonzero coefficients. After a complete cycle through all the variables, we iterate on only the active set until convergence. If another complete cycle does not change the active set, we are done, otherwise the process is repeated.</p>
-<p class="enddd"></p>
-</dd>
-<dt>activeset_tolerance </dt>
-<dd><p class="startdd">Default: the value of the tolerance argument. The value of tolerance used during active set calculation. </p>
-<p class="enddd"></p>
-</dd>
-<dt>random_stepsize </dt>
-<dd>Default: FALSE. Whether to add some randomness to the step size. Sometimes, this can speed up the calculation. </dd>
-</dl>
-<p>When the <a class="el" href="elastic__net_8sql__in.html#a735038a5090c112505c740a90a203e83">elastic_net_train()</a> <em>optimizer</em> argument value is <b>'igd'</b>, the <em>optimizer_params</em> argument is a string containing name-value pairs with the following format. (Line breaks are inserted for readability.) </p><pre class="syntax">
- 'stepsize = <value>,
- step_decay = <value>,
- threshold = <value>,
- warmup = <value>,
- warmup_lambdas = <value>,
- warmup_lambda_no = <value>,
- warmup_tolerance = <value>,
- parallel = <value>'
-</pre><p> <b>Parameters</b> </p><dl class="arglist">
-<dt>stepsize </dt>
-<dd>The default is 0.01. </dd>
-<dt>step_decay </dt>
-<dd>The actual setpsize used for current step is (previous stepsize) / exp(setp_decay). The default value is 0, which means that a constant stepsize is used in IGD. </dd>
-<dt>threshold </dt>
-<dd><p class="startdd">Default: 1e-10. When a coefficient is really small, set this coefficient to be 0.</p>
-<p class="enddd">Due to the stochastic nature of SGD, we can only obtain very small values for the fitting coefficients. Therefore, <em>threshold</em> is needed at the end of the computation to screen out tiny values and hard-set them to zeros. This is accomplished as follows: (1) multiply each coefficient with the standard deviation of the corresponding feature; (2) compute the average of absolute values of re-scaled coefficients; (3) divide each rescaled coefficient with the average, and if the resulting absolute value is smaller than <em>threshold</em>, set the original coefficient to zero. </p>
-</dd>
-<dt>warmup </dt>
-<dd>Default: FALSE. If <em>warmup</em> is TRUE, a series of lambda values, which is strictly descent and ends at the lambda value that the user wants to calculate, is used. The larger lambda gives very sparse solution, and the sparse solution again is used as the initial guess for the next lambda's solution, which speeds up the computation for the next lambda. For larger data sets, this can sometimes accelerate the whole computation and may be faster than computation on only one lambda value. </dd>
-<dt>warmup_lambdas </dt>
-<dd>Default: NULL. An array of lambda values to use for warmup. </dd>
-<dt>warmup_lambda_no </dt>
-<dd>The number of lambdas used in warm-up. The default is 15. If <em>warmup_lambdas</em> is not NULL, this argument is overridden by the size of the <em>warmup_lambdas</em> array. </dd>
-<dt>warmup_tolerance </dt>
-<dd>The value of tolerance used during warmup.The default is the same as the <em>tolerance</em> argument. </dd>
-<dt>parallel </dt>
-<dd><p class="startdd">Whether to run the computation on multiple segments. The default is True.</p>
-<p class="enddd">SGD is a sequential algorithm in nature. When running in a distributed manner, each segment of the data runs its own SGD model and then the models are averaged to get a model for each iteration. This averaging might slow down the convergence speed, although we also acquire the ability to process large datasets on multiple machines. This algorithm, therefore, provides the <em>parallel</em> option to allow you to choose whether to do parallel computation. </p>
-</dd>
-</dl>
-<p><a class="anchor" id="predict"></a></p><dl class="section user"><dt>Prediction Function</dt><dd></dd></dl>
-<h4>Per-Tuple Prediction</h4>
-<p>The prediction function returns a double value for Gaussian family and boolean value for Binomial family.</p>
-<p>The predict function has the following syntax (<a class="el" href="elastic__net_8sql__in.html#a96db4ff4ba3ea363fafbf6c036c19fae" title="Prediction for linear models use learned coefficients for a given example. ">elastic_net_gaussian_predict()</a> and <a class="el" href="elastic__net_8sql__in.html#aa78cde79f1f2caa7c5b38f933001d793" title="Prediction for logistic models use learned coefficients for a given example. ">elastic_net_binomial_predict()</a>): </p><pre class="syntax">
-elastic_net_<family>_predict(
- coefficients,
- intercept,
- ind_var
- )
-</pre><p><b>Arguments</b> </p><dl class="arglist">
-<dt>coefficients </dt>
-<dd>DOUBLE PRECISION[]. Fitting coefficients, usually coef_all or coef_nonzero. </dd>
-<dt>intercept </dt>
-<dd>DOUBLE PRECISION. The intercept for the model. </dd>
-<dt>ind_var </dt>
-<dd>DOUBLE PRECISION[]. Independent variables that correspond to coefficients, use <em>features</em> column in <em>tbl_result</em> for coef_all, and <em>features_selected</em> for coef_nonzero. See also <a href="#additional_example">examples</a>. Note that unexpected results or errors may be returned in the case that this argument is not given properly. </dd>
-</dl>
-<p>For binomial family, there is a function (<a class="el" href="elastic__net_8sql__in.html#a308718fd5234bc1007b971a639aadf71" title="Compute the probability of belonging to the True class for a given observation. ">elastic_net_binomial_prob()</a>) that outputs the probability of the instance being True: </p><pre class="syntax">
-elastic_net_binomial_prob(
- coefficients,
- intercept,
- ind_var
- )
-</pre><h4>Per-Table Prediction</h4>
-<p>Alternatively, you can use another prediction function that stores the prediction result in a table (<a class="el" href="elastic__net_8sql__in.html#a3578608204ac9b2d3442ff42977f632b" title="Prediction and put the result in a table can be used together with General-CV. ">elastic_net_predict()</a>). This is useful if you want to use elastic net together with the general cross validation function. </p><pre class="syntax">
-elastic_net_predict( tbl_model,
- tbl_new_sourcedata,
- col_id,
- tbl_predict
- )
-</pre><p> <b>Arguments</b> </p><dl class="arglist">
-<dt>tbl_model </dt>
-<dd>TEXT. The name of the table containing the output from the training function. </dd>
-<dt>tbl_new_sourcedata </dt>
-<dd>TEXT. The name of the table containing the new source data. </dd>
-<dt>col_id </dt>
-<dd>TEXT. The unique ID associated with each row. </dd>
-<dt>tbl_predict </dt>
-<dd>TEXT. The name of table to store the prediction result. </dd>
-</dl>
-<p>You do not need to specify whether the model is "linear" or "logistic" because this information is already included in the <em>tbl_model</em> table.</p>
-<p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd></dd></dl>
-<ol type="1">
-<li>Display online help for the <a class="el" href="elastic__net_8sql__in.html#a735038a5090c112505c740a90a203e83" title="Interface for elastic net. ">elastic_net_train()</a> function. <pre class="example">
-SELECT madlib.elastic_net_train();
-</pre></li>
-<li>Create an input data set. <pre class="example">
-DROP TABLE IF EXISTS houses;
-CREATE TABLE houses ( id INT,
- tax INT,
- bedroom INT,
- bath FLOAT,
- price INT,
- size INT,
- lot INT
- );
-COPY houses FROM STDIN WITH DELIMITER '|';
- 1 | 590 | 2 | 1 | 50000 | 770 | 22100
- 2 | 1050 | 3 | 2 | 85000 | 1410 | 12000
- 3 | 20 | 3 | 1 | 22500 | 1060 | 3500
- 4 | 870 | 2 | 2 | 90000 | 1300 | 17500
- 5 | 1320 | 3 | 2 | 133000 | 1500 | 30000
- 6 | 1350 | 2 | 1 | 90500 | 820 | 25700
- 7 | 2790 | 3 | 2.5 | 260000 | 2130 | 25000
- 8 | 680 | 2 | 1 | 142500 | 1170 | 22000
- 9 | 1840 | 3 | 2 | 160000 | 1500 | 19000
- 10 | 3680 | 4 | 2 | 240000 | 2790 | 20000
- 11 | 1660 | 3 | 1 | 87000 | 1030 | 17500
- 12 | 1620 | 3 | 2 | 118600 | 1250 | 20000
- 13 | 3100 | 3 | 2 | 140000 | 1760 | 38000
- 14 | 2070 | 2 | 3 | 148000 | 1550 | 14000
- 15 | 650 | 3 | 1.5 | 65000 | 1450 | 12000
-\.
-</pre></li>
-<li>Train the model. <pre class="example">
-DROP TABLE IF EXISTS houses_en;
-SELECT madlib.elastic_net_train( 'houses',
- 'houses_en',
- 'price',
- 'array[tax, bath, size]',
- 'gaussian',
- 0.5,
- 0.1,
- TRUE,
- NULL,
- 'fista',
- '',
- NULL,
- 10000,
- 1e-6
- );
-</pre></li>
-<li>View the resulting model. <pre class="example">
--- Turn on expanded display to make it easier to read results.
-\x on
-SELECT * FROM houses_en;
-</pre> Result: <pre class="result">
--[ RECORD 1 ]-----+--------------------------------------------
-family | gaussian
-features | {tax,bath,size}
-features_selected | {tax,bath,size}
-coef_nonzero | {27.6945611671,11509.0099734,49.0945557639}
-coef_all | {27.6945611671,11509.0099734,49.0945557639}
-intercept | -11145.5061503
-log_likelihood | -490118975.406
-standardize | t
-iteration_run | 322
-</pre></li>
-<li>Use the prediction function to evaluate residuals. <pre class="example">
-\x off
-SELECT id, price, predict, price - predict AS residual
-FROM (
- SELECT
- houses.*,
- madlib.elastic_net_gaussian_predict(
- m.coef_all,
- m.intercept,
- ARRAY[tax,bath,size]
- ) AS predict
- FROM houses, houses_en m) s
-ORDER BY id;
-</pre></li>
-</ol>
-<p><a class="anchor" id="additional_example"></a></p><h4>Additional Example (when coef_nonzero is different from coef_all)</h4>
-<ol type="1">
-<li>Reuse the <a href="#examples">houses</a> table above and train the model with alpha=1 (L-1) and a large lambda (30000). <pre class="example">
-DROP TABLE IF EXISTS houses_en2;
-SELECT madlib.elastic_net_train( 'houses',
- 'houses_en2',
- 'price',
- 'array[tax, bath, size]',
- 'gaussian',
- 1,
- 30000,
- TRUE,
- NULL,
- 'fista',
- '',
- NULL,
- 10000,
- 1e-6
- );
-</pre></li>
-<li>View the resulting model and see coef_nonzero is different from coef_all. <pre class="example">
--- Turn on expanded display to make it easier to read results.
-\x on
-SELECT * FROM houses_en2;
-</pre> Result: <pre class="result">
--[ RECORD 1 ]-----+--------------------------------------------
-features | {tax,bath,size}
-features_selected | {tax,size}
-coef_nonzero | {13.3261747481,22.7347986045}
-coef_all | {13.3261747481,0,22.7347986045}
-intercept | 68877.5045405
-log_likelihood | -1694746275.43
-standardize | t
-iteration_run | 115
-</pre></li>
-<li>We can still use the prediction function with coef_all to evaluate residuals. <pre class="example">
-\x off
-SELECT id, price, predict, price - predict AS residual
-FROM (
- SELECT
- houses.*,
- madlib.elastic_net_gaussian_predict(
- m.coef_all,
- m.intercept,
- ARRAY[tax,bath,size]
- ) AS predict
- FROM houses, houses_en2 m) s
-ORDER BY id;
-</pre></li>
-<li>While we can also speed up the prediction function with coef_nonzero to evaluate residuals. This requires user to examine the feature_selected column in the result table to construct the correct independent variables. <pre class="example">
-\x off
-SELECT id, price, predict, price - predict AS residual
-FROM (
- SELECT
- houses.*,
- madlib.elastic_net_gaussian_predict(
- m.coef_nonzero,
- m.intercept,
- ARRAY[tax,size]
- ) AS predict
- FROM houses, houses_en2 m) s
-ORDER BY id;
-</pre> The two queries are expected to give same residuals: <pre class="result">
- id | price | predict | residual
-----+--------+------------------+-------------------
- 1 | 50000 | 94245.742567344 | -44245.742567344
- 3 | 22500 | 93242.914556232 | -70742.914556232
- 5 | 133000 | 120570.253114742 | 12429.746885258
- 7 | 260000 | 154482.653115284 | 105517.346884716
- 9 | 160000 | 127499.863983754 | 32500.136016246
- 11 | 87000 | 114415.797184981 | -27415.797184981
- 13 | 140000 | 150201.89180353 | -10201.89180353
- 15 | 65000 | 110504.97610329 | -45504.97610329
- 2 | 85000 | 114926.05405835 | -29926.05405835
- 4 | 90000 | 110026.514757197 | -20026.514757197
- 6 | 90500 | 105510.375306125 | -15010.375306125
- 8 | 142500 | 104539.017736473 | 37960.982263527
- 10 | 240000 | 181347.915720063 | 58652.084279937
- 12 | 118600 | 118884.405888047 | -284.405888046997
- 14 | 148000 | 131701.624106042 | 16298.375893958
-(15 rows)
-</pre></li>
-</ol>
-<p><a class="anchor" id="notes"></a></p><dl class="section user"><dt>Note</dt><dd>It is <b>strongly</b> <b>recommended</b> that you run <code><a class="el" href="elastic__net_8sql__in.html#a735038a5090c112505c740a90a203e83" title="Interface for elastic net. ">elastic_net_train()</a></code> on a subset of the data with a limited <em>max_iter</em> before applying it to the full data set with a large <em>max_iter</em>. In the pre-run, you can adjust the parameters to get the best performance and then apply the best set of parameters to the whole data set.</dd></dl>
-<p><a class="anchor" id="background"></a></p><dl class="section user"><dt>Technical Background</dt><dd></dd></dl>
-<p>Elastic net regularization seeks to find a weight vector that, for any given training example set, minimizes: </p><p class="formulaDsp">
-<img class="formulaDsp" alt="\[\min_{w \in R^N} L(w) + \lambda \left(\frac{(1-\alpha)}{2} \|w\|_2^2 + \alpha \|w\|_1 \right)\]" src="form_81.png"/>
-</p>
-<p> where <img class="formulaInl" alt="$L$" src="form_82.png"/> is the metric function that the user wants to minimize. Here <img class="formulaInl" alt="$ \alpha \in [0,1] $" src="form_83.png"/> and <img class="formulaInl" alt="$ lambda \geq 0 $" src="form_84.png"/>. If <img class="formulaInl" alt="$alpha = 0$" src="form_85.png"/>, we have the ridge regularization (known also as Tikhonov regularization), and if <img class="formulaInl" alt="$\alpha = 1$" src="form_86.png"/>, we have the LASSO regularization.</p>
-<p>For the Gaussian response family (or linear model), we have </p><p class="formulaDsp">
-<img class="formulaDsp" alt="\[L(\vec{w}) = \frac{1}{2}\left[\frac{1}{M} \sum_{m=1}^M (w^{t} x_m + w_{0} - y_m)^2 \right] \]" src="form_87.png"/>
-</p>
-<p>For the Binomial response family (or logistic model), we have </p><p class="formulaDsp">
-<img class="formulaDsp" alt="\[ L(\vec{w}) = \sum_{m=1}^M\left[y_m \log\left(1 + e^{-(w_0 + \vec{w}\cdot\vec{x}_m)}\right) + (1-y_m) \log\left(1 + e^{w_0 + \vec{w}\cdot\vec{x}_m}\right)\right]\ , \]" src="form_88.png"/>
-</p>
-<p> where <img class="formulaInl" alt="$y_m \in {0,1}$" src="form_89.png"/>.</p>
-<p>To get better convergence, one can rescale the value of each element of x </p><p class="formulaDsp">
-<img class="formulaDsp" alt="\[ x' \leftarrow \frac{x - \bar{x}}{\sigma_x} \]" src="form_90.png"/>
-</p>
-<p> and for Gaussian case we also let </p><p class="formulaDsp">
-<img class="formulaDsp" alt="\[y' \leftarrow y - \bar{y} \]" src="form_91.png"/>
-</p>
-<p> and then minimize with the regularization terms. At the end of the calculation, the orginal scales will be restored and an intercept term will be obtained at the same time as a by-product.</p>
-<p>Note that fitting after scaling is not equivalent to directly fitting.</p>
-<p><a class="anchor" id="literature"></a></p><dl class="section user"><dt>Literature</dt><dd></dd></dl>
-<p>[1] Elastic net regularization. <a href="http://en.wikipedia.org/wiki/Elastic_net_regularization">http://en.wikipedia.org/wiki/Elastic_net_regularization</a></p>
-<p>[2] Beck, A. and M. Teboulle (2009), A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. on Imaging Sciences 2(1), 183-202.</p>
-<p>[3] Shai Shalev-Shwartz and Ambuj Tewari, Stochastic Methods for l1 Regularized Loss Minimization. Proceedings of the 26th International Conference on Machine Learning, Montreal, Canada, 2009.</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="elastic__net_8sql__in.html" title="SQL functions for elastic net regularization. ">elastic_net.sql_in</a> documenting the SQL functions.</p>
-<p>grp_validation </p>
-</div><!-- contents -->
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- <a href="http://www.doxygen.org/index.html">
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- </ul>
-</div>
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- <div id="projectname">
- <span id="projectnumber">1.9.1</span>
- </div>
- <div id="projectbrief">User Documentation for MADlib</div>
- </td>
- <td> <div id="MSearchBox" class="MSearchBoxInactive">
- <span class="left">
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-</td>
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-<!-- end header part -->
-<!-- Generated by Doxygen 1.8.10 -->
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-<div class="header">
- <div class="headertitle">
-<div class="title">FM (Flajolet-Martin)<div class="ingroups"><a class="el" href="group__grp__early__stage.html">Early Stage Development</a> » <a class="el" href="group__grp__sketches.html">Cardinality Estimators</a></div></div> </div>
-</div><!--header-->
-<div class="contents">
-<div class="toc"><b>Contents</b> </p><ul>
-<li>
-<a href="#syntax">Syntax</a> </li>
-<li>
-<a href="#examples">Examples</a> </li>
-<li>
-<a href="#literature">Literature</a> </li>
-<li>
-<a href="#related">Related Topics</a> </li>
-</ul>
-</div><dl class="section warning"><dt>Warning</dt><dd><em> This MADlib method is still in early stage development. There may be some issues that will be addressed in a future version. Interface and implementation is subject to change. </em></dd></dl>
-<p><a class="el" href="sketch_8sql__in.html#ae27d5aaa5e4b426bcfe55e05a89c8e0b">fmsketch_dcount</a> can be run on a column of any type. It returns an approximation to the number of distinct values (a la <code>COUNT(DISTINCT x)</code>), but faster and approximate. Like any aggregate, it can be combined with a GROUP BY clause to do distinct counts per group.</p>
-<p><a class="anchor" id="syntax"></a></p><dl class="section user"><dt>Syntax</dt><dd></dd></dl>
-<p>Get the number of distinct values in a designated column. </p><pre class="syntax">
-fmsketch_dcount( col_name )
-</pre><p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd><ol type="1">
-<li>Generate some data. <pre class="example">
-CREATE TABLE data(class INT, a1 INT);
-INSERT INTO data SELECT 1,1 FROM generate_series(1,10000);
-INSERT INTO data SELECT 1,2 FROM generate_series(1,15000);
-INSERT INTO data SELECT 1,3 FROM generate_series(1,10000);
-INSERT INTO data SELECT 2,5 FROM generate_series(1,1000);
-INSERT INTO data SELECT 2,6 FROM generate_series(1,1000);
-</pre></li>
-<li>Find the distinct number of values for each class. <pre class="example">
-SELECT class, fmsketch_dcount(a1)
-FROM data
-GROUP BY data.class;
-</pre> Result: <pre class="result">
-class | fmsketch_dcount
- ------+-----------------
- 2 | 2
- 1 | 3
-(2 rows)
-</pre></li>
-</ol>
-</dd></dl>
-<p><a class="anchor" id="literature"></a></p><dl class="section user"><dt>Literature</dt><dd>[1] P. Flajolet and N.G. Martin. Probabilistic counting algorithms for data base applications, Journal of Computer and System Sciences 31(2), pp 182-209, 1985. <a href="http://algo.inria.fr/flajolet/Publications/FlMa85.pdf">http://algo.inria.fr/flajolet/Publications/FlMa85.pdf</a></dd></dl>
-<p><a class="anchor" id="related"></a></p><dl class="section user"><dt>Related Topics</dt><dd>File <a class="el" href="sketch_8sql__in.html" title="SQL functions for sketch-based approximations of descriptive statistics. ">sketch.sql_in</a> documenting the SQL function. </dd></dl>
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-<div class="header">
- <div class="headertitle">
-<div class="title">Generalized Linear Models<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></p><ul>
-<li class="level1">
-<a href="#train">Training Function</a> </li>
-<li class="level1">
-<a href="#predict">Prediction Function</a> </li>
-<li class="level1">
-<a href="#examples">Examples</a> </li>
-<li class="level1">
-<a href="#related">Related Topics</a> </li>
-</ul>
-</div><p>Generalized linear models extends ordinary linear regression by allowing the response variable to follow a more general set of distributions (rather than simply Gaussian distributions), and for a general family of functions of the response variable (the link function) to vary linearly with the predicted values (rather than assuming that the response itself must vary linearly).</p>
-<p>For example, data of counts would typically be modeled with a Poisson distribution and a log link, while binary outcomes would typically be modeled with a Bernoulli distribution (or binomial distribution, depending on exactly how the problem is phrased) and a log-odds (or logit) link function.</p>
-<p>Currently, the implemented distribution families are </p><center> <table class="doxtable">
-<tr>
-<th>Distribution Family </th><th>Link Functions </th></tr>
-<tr>
-<td>Binomial </td><td>logit, probit </td></tr>
-<tr>
-<td>Gamma </td><td>inverse, identity, log </td></tr>
-<tr>
-<td>Gaussian </td><td>identity, inverse, log </td></tr>
-<tr>
-<td>Inverse Gaussian </td><td>inverse of square, inverse, identity, log </td></tr>
-<tr>
-<td>Poisson </td><td>log, identity, square-root<br />
- </td></tr>
-</table>
-</center><p><a class="anchor" id="train"></a></p><dl class="section user"><dt>Training Function</dt><dd>GLM training function has the following format: <pre class="syntax">
-glm(source_table,
- model_table,
- dependent_varname,
- independent_varname,
- family_params,
- grouping_col,
- optim_params,
- verbose
- )
-</pre> <b>Arguments</b> <dl class="arglist">
-<dt>source_table </dt>
-<dd><p class="startdd">TEXT. The name of the table containing the training data.</p>
-<p class="enddd"></p>
-</dd>
-<dt>model_table </dt>
-<dd><p class="startdd">TEXT. Name of the generated table containing the model.</p>
-<p>The model table produced by glm contains the following columns:</p>
-<table class="output">
-<tr>
-<th><...> </th><td><p class="starttd">Text. Grouping columns, if provided in input. This could be multiple columns depending on the <code>grouping_col</code> input. </p>
-<p class="endtd"></p>
-</td></tr>
-<tr>
-<th>coef </th><td><p class="starttd">FLOAT8. Vector of the coefficients in linear predictor. </p>
-<p class="endtd"></p>
-</td></tr>
-<tr>
-<th>log_likelihood </th><td><p class="starttd">FLOAT8. The log-likelihood <img class="formulaInl" alt="$ l(\boldsymbol \beta) $" src="form_92.png"/>. We use the maximum likelihood estimate of dispersion parameter to calculate the log-likelihood while R and Python use deviance estimate and Pearson estimate respectively. </p>
-<p class="endtd"></p>
-</td></tr>
-<tr>
-<th>std_err </th><td><p class="starttd">FLOAT8[]. Vector of the standard error of the coefficients. </p>
-<p class="endtd"></p>
-</td></tr>
-<tr>
-<th>z_stats or t_stats </th><td><p class="starttd">FLOAT8[]. Vector of the z-statistics (in Poisson distribtuion and Binomial distribution) or the t-statistics (in all other distributions) of the coefficients. </p>
-<p class="endtd"></p>
-</td></tr>
-<tr>
-<th>p_values </th><td><p class="starttd">FLOAT8[]. Vector of the p-values of the coefficients. </p>
-<p class="endtd"></p>
-</td></tr>
-<tr>
-<th>dispersion </th><td><p class="starttd">FLOAT8. The dispersion value (Pearson estimate). When family=poisson or family=binomial, the dispersion is always 1. </p>
-<p class="endtd"></p>
-</td></tr>
-<tr>
-<th>num_rows_processed </th><td><p class="starttd">BIGINT. Numbers of rows processed. </p>
-<p class="endtd"></p>
-</td></tr>
-<tr>
-<th>num_rows_skipped </th><td><p class="starttd">BIGINT. Numbers of rows skipped due to missing values or failures. </p>
-<p class="endtd"></p>
-</td></tr>
-<tr>
-<th>num_iterations </th><td>INTEGER. The number of iterations actually completed. This would be different from the <code>nIterations</code> argument if a <code>tolerance</code> parameter is provided and the algorithm converges before all iterations are completed. </td></tr>
-</table>
-<p>A summary table named <model_table>_summary is also created at the same time, which has the following columns: </p><table class="output">
-<tr>
-<th>method </th><td><p class="starttd">'glm' </p>
-<p class="endtd"></p>
-</td></tr>
-<tr>
-<th>source_table </th><td><p class="starttd">The data source table name. </p>
-<p class="endtd"></p>
-</td></tr>
-<tr>
-<th>model_table </th><td><p class="starttd">The model table name. </p>
-<p class="endtd"></p>
-</td></tr>
-<tr>
-<th>dependent_varname </th><td><p class="starttd">The dependent variable. </p>
-<p class="endtd"></p>
-</td></tr>
-<tr>
-<th>independent_varname </th><td><p class="starttd">The independent variables </p>
-<p class="endtd"></p>
-</td></tr>
-<tr>
-<th>family_params </th><td><p class="starttd">A string that contains family parameters, and has the form of 'family=..., link=...' </p>
-<p class="endtd"></p>
-</td></tr>
-<tr>
-<th>grouping_col </th><td><p class="starttd">Name of grouping columns. </p>
-<p class="endtd"></p>
-</td></tr>
-<tr>
-<th>optimizer_params </th><td><p class="starttd">A string that contains optimizer parameters, and has the form of 'optimizer=..., max_iter=..., tolerance=...' </p>
-<p class="endtd"></p>
-</td></tr>
-<tr>
-<th>num_all_groups </th><td><p class="starttd">Number of groups in glm training. </p>
-<p class="endtd"></p>
-</td></tr>
-<tr>
-<th>num_failed_groups </th><td><p class="starttd">Number of failed groups in glm training. </p>
-<p class="endtd"></p>
-</td></tr>
-<tr>
-<th>total_rows_processed </th><td><p class="starttd">BIGINT. Total numbers of rows processed in all groups. </p>
-<p class="endtd"></p>
-</td></tr>
-<tr>
-<th>total_rows_skipped </th><td><p class="starttd">BIGINT. Total numbers of rows skipped in all groups due to missing values or failures. </p>
-<p class="endtd"></p>
-</td></tr>
-</table>
-<p class="enddd"></p>
-</dd>
-<dt>dependent_varname </dt>
-<dd><p class="startdd">TEXT. Name of the dependent variable column.</p>
-<p class="enddd"></p>
-</dd>
-<dt>independent_varname </dt>
-<dd><p class="startdd">TEXT. Expression list to evaluate for the independent variables. An intercept variable is not assumed. It is common to provide an explicit intercept term by including a single constant <code>1</code> term in the independent variable list.</p>
-<p class="enddd"></p>
-</dd>
-<dt>family_params (optional) </dt>
-<dd><p class="startdd">TEXT, Parameters for distribution family. Currently, we support</p>
-<p>(1) family=poisson and link=[log or identity or sqrt].</p>
-<p>(2) family=gaussian and link=[identity or log or inverse]. And when family=gaussian and link=identity, the GLM model is exactly the same as the linear regression.</p>
-<p>(3) family=gamma and link=[inverse or identity or log].</p>
-<p>(4) family=inverse_gaussian and link=[sqr_inverse or log or identity or inverse].</p>
-<p>(5) family=binomial and link=[probit or logit]. </p>
-<p class="enddd"></p>
-</dd>
-<dt>grouping_col (optional) </dt>
-<dd><p class="startdd">TEXT, default: NULL. An expression list used to group the input dataset into discrete groups, running one regression per group. Similar to the SQL "GROUP BY" clause. When this value is NULL, no grouping is used and a single model is generated.</p>
-<p class="enddd"></p>
-</dd>
-<dt>optim_params (optional) </dt>
-<dd><p class="startdd">TEXT, default: 'max_iter=100,optimizer=irls,tolerance=1e-6'. Parameters for optimizer. Currently, we support tolerance=[tolerance for relative error between log-likelihoods], max_iter=[maximum iterations to run], optimizer=irls.</p>
-<p class="enddd"></p>
-</dd>
-<dt>verbose (optional) </dt>
-<dd>BOOLEAN, default: FALSE. Provides verbose output of the results of training. </dd>
-</dl>
-</dd></dl>
-<dl class="section note"><dt>Note</dt><dd>For p-values, we just return the computation result directly. Other statistical packages, like 'R', produce the same result, but on printing the result to screen, another format function is used and any p-value that is smaller than the machine epsilon (the smallest positive floating-point number 'x' such that '1 + x != 1') will be printed on screen as "< xxx" (xxx is the value of the machine epsilon). Although the results may look different, they are in fact the same. </dd></dl>
-<p><a class="anchor" id="predict"></a></p><dl class="section user"><dt>Prediction Function</dt><dd>The prediction function is provided to estimate the conditional mean given a new predictor. It has the following syntax: <pre class="syntax">
-glm_predict(coef,
- col_ind_var
- link)
-</pre></dd></dl>
-<p><b>Arguments</b> </p><dl class="arglist">
-<dt>coef </dt>
-<dd><p class="startdd">DOUBLE PRECISION[]. Model coefficients obtained from <a class="el" href="glm_8sql__in.html#a3f8eb219013e05675626acb8cf4612cc">glm()</a>.</p>
-<p class="enddd"></p>
-</dd>
-<dt>col_ind_var </dt>
-<dd><p class="startdd">New predictor, as a DOUBLE array. This should be the same length as the array obtained by evaluation of the 'independent_varname' argument in <a class="el" href="glm_8sql__in.html#a3f8eb219013e05675626acb8cf4612cc">glm()</a>.</p>
-<p class="enddd"></p>
-</dd>
-<dt>link </dt>
-<dd>link function, as a string. This should match the link function the user inputted in <a class="el" href="glm_8sql__in.html#a3f8eb219013e05675626acb8cf4612cc">glm()</a>. </dd>
-</dl>
-<p><a class="anchor" id="examples"></a></p><dl class="section user"><dt>Examples</dt><dd><ol type="1">
-<li>Create the training data table. <pre class="example">
-CREATE TABLE warpbreaks(
- id serial,
- breaks integer,
- wool char(1),
- tension char(1)
-);
-INSERT INTO warpbreaks(breaks, wool, tension) VALUES
-(26, 'A', 'L'),
-(30, 'A', 'L'),
-(54, 'A', 'L'),
-(25, 'A', 'L'),
-(70, 'A', 'L'),
-(52, 'A', 'L'),
-(51, 'A', 'L'),
-(26, 'A', 'L'),
-(67, 'A', 'L'),
-(18, 'A', 'M'),
-(21, 'A', 'M'),
-(29, 'A', 'M'),
-(17, 'A', 'M'),
-(12, 'A', 'M'),
-(18, 'A', 'M'),
-(35, 'A', 'M'),
-(30, 'A', 'M'),
-(36, 'A', 'M'),
-(36, 'A', 'H'),
-(21, 'A', 'H'),
-(24, 'A', 'H'),
-(18, 'A', 'H'),
-(10, 'A', 'H'),
-(43, 'A', 'H'),
-(28, 'A', 'H'),
-(15, 'A', 'H'),
-(26, 'A', 'H'),
-(27, 'B', 'L'),
-(14, 'B', 'L'),
-(29, 'B', 'L'),
-(19, 'B', 'L'),
-(29, 'B', 'L'),
-(31, 'B', 'L'),
-(41, 'B', 'L'),
-(20, 'B', 'L'),
-(44, 'B', 'L'),
-(42, 'B', 'M'),
-(26, 'B', 'M'),
-(19, 'B', 'M'),
-(16, 'B', 'M'),
-(39, 'B', 'M'),
-(28, 'B', 'M'),
-(21, 'B', 'M'),
-(39, 'B', 'M'),
-(29, 'B', 'M'),
-(20, 'B', 'H'),
-(21, 'B', 'H'),
-(24, 'B', 'H'),
-(17, 'B', 'H'),
-(13, 'B', 'H'),
-(15, 'B', 'H'),
-(15, 'B', 'H'),
-(16, 'B', 'H'),
-(28, 'B', 'H');
-SELECT create_indicator_variables('warpbreaks', 'warpbreaks_dummy', 'wool,tension');
-</pre></li>
-<li>Train a GLM model. <pre class="example">
-SELECT glm('warpbreaks_dummy',
- 'glm_model',
- 'breaks',
- 'ARRAY[1.0,"wool_B","tension_M", "tension_H"]',
- 'family=poisson, link=log');
-</pre></li>
-<li>View the regression results. <pre class="example">
--- Set extended display on for easier reading of output
-\x on
-SELECT * FROM glm_model;
-</pre> Result: <pre class="result">
-coef | {3.69196314494079,-0.205988442638621,-0.321320431600611,-0.51848849651156}
-log_likelihood | -242.527983208979
-std_err | {0.04541079434248,0.0515712427835191,0.0602659166951256,0.0639595193956924}
-z_stats | {81.3014438174473,-3.99425011926316,-5.3317106786264,-8.10651020224019}
-p_values | {0,6.48993254938271e-05,9.72918600322907e-08,5.20943463005751e-16}
-num_rows_processed | 54
-num_rows_skipped | 0
-iteration | 5
-</pre> Alternatively, unnest the arrays in the results for easier reading of output: <pre class="example">
-\x off
-SELECT unnest(coef) as coefficient,
- unnest(std_err) as standard_error,
- unnest(z_stats) as z_stat,
- unnest(p_values) as pvalue
-FROM glm_model;
-</pre></li>
-<li>Predicting dependent variable using GLM 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 for prediction.) <pre class="example">
-\x off
--- Display predicted mean value on the original dataset
-SELECT
- w.id,
- madlib.glm_predict(
- coef,
- ARRAY[1, "wool_B", "tension_M", "tension_H"]::float8[],
- 'log') AS mu
-FROM warpbreaks_dummy w, glm_model m
-ORDER BY w.id;
-</pre> <pre class="example">
--- Display predicted counts (which are predicted mean values rounded to the nearest integral value) on the original dataset
-SELECT
- w.id,
- madlib.glm_predict_poisson(
- coef,
- ARRAY[1, "wool_B", "tension_M", "tension_H"]::float8[],
- 'log') AS poisson_count
-FROM warpbreaks_dummy w, glm_model m
-ORDER BY w.id;
-</pre></li>
-</ol>
-</dd></dl>
-<p><b>Example for Gaussian family:</b></p>
-<ol type="1">
-<li>Create a testing data table <pre class="example">
-CREATE TABLE abalone (
- id integer,
- sex text,
- length double precision,
- diameter double precision,
- height double precision,
- whole double precision,
- shucked double precision,
- viscera double precision,
- shell double precision,
- rings integer
-);
-INSERT INTO abalone VALUES
-(3151, 'F', 0.655000000000000027, 0.505000000000000004, 0.165000000000000008, 1.36699999999999999, 0.583500000000000019, 0.351499999999999979, 0.396000000000000019, 10),
-(2026, 'F', 0.550000000000000044, 0.469999999999999973, 0.149999999999999994, 0.920499999999999985, 0.381000000000000005, 0.243499999999999994, 0.267500000000000016, 10),
-(3751, 'I', 0.434999999999999998, 0.375, 0.110000000000000001, 0.41549999999999998, 0.170000000000000012, 0.0759999999999999981, 0.14499999999999999, 8),
-(720, 'I', 0.149999999999999994, 0.100000000000000006, 0.0250000000000000014, 0.0149999999999999994, 0.00449999999999999966, 0.00400000000000000008, 0.0050000000000000001, 2),
-(1635, 'F', 0.574999999999999956, 0.469999999999999973, 0.154999999999999999, 1.1160000000000001, 0.509000000000000008, 0.237999999999999989, 0.340000000000000024, 10),
-(2648, 'I', 0.5, 0.390000000000000013, 0.125, 0.582999999999999963, 0.293999999999999984, 0.132000000000000006, 0.160500000000000004, 8),
-(1796, 'F', 0.57999999999999996, 0.429999999999999993, 0.170000000000000012, 1.47999999999999998, 0.65349999999999997, 0.32400000000000001, 0.41549999999999998, 10),
-(209, 'F', 0.525000000000000022, 0.41499999999999998, 0.170000000000000012, 0.832500000000000018, 0.275500000000000023, 0.168500000000000011, 0.309999999999999998, 13),
-(1451, 'I', 0.455000000000000016, 0.33500000000000002, 0.135000000000000009, 0.501000000000000001, 0.274000000000000021, 0.0995000000000000051, 0.106499999999999997, 7),
-(1108, 'I', 0.510000000000000009, 0.380000000000000004, 0.115000000000000005, 0.515499999999999958, 0.214999999999999997, 0.113500000000000004, 0.166000000000000009, 8),
-(3675, 'F', 0.594999999999999973, 0.450000000000000011, 0.165000000000000008, 1.08099999999999996, 0.489999999999999991, 0.252500000000000002, 0.279000000000000026, 12),
-(2108, 'F', 0.675000000000000044, 0.550000000000000044, 0.179999999999999993, 1.68849999999999989, 0.562000000000000055, 0.370499999999999996, 0.599999999999999978, 15),
-(3312, 'F', 0.479999999999999982, 0.380000000000000004, 0.135000000000000009, 0.507000000000000006, 0.191500000000000004, 0.13650000000000001, 0.154999999999999999, 12),
-(882, 'M', 0.655000000000000027, 0.520000000000000018, 0.165000000000000008, 1.40949999999999998, 0.585999999999999965, 0.290999999999999981, 0.405000000000000027, 9),
-(3402, 'M', 0.479999999999999982, 0.395000000000000018, 0.149999999999999994, 0.681499999999999995, 0.214499999999999996, 0.140500000000000014, 0.2495, 18),
-(829, 'I', 0.409999999999999976, 0.325000000000000011, 0.100000000000000006, 0.394000000000000017, 0.20799999999999999, 0.0655000000000000027, 0.105999999999999997, 6),
-(1305, 'M', 0.535000000000000031, 0.434999999999999998, 0.149999999999999994, 0.716999999999999971, 0.347499999999999976, 0.14449999999999999, 0.194000000000000006, 9),
-(3613, 'M', 0.599999999999999978, 0.46000000000000002, 0.179999999999999993, 1.1399999999999999, 0.422999999999999987, 0.257500000000000007, 0.364999999999999991, 10),
-(1068, 'I', 0.340000000000000024, 0.265000000000000013, 0.0800000000000000017, 0.201500000000000012, 0.0899999999999999967, 0.0475000000000000006, 0.0550000000000000003, 5),
-(2446, 'M', 0.5, 0.380000000000000004, 0.135000000000000009, 0.583500000000000019, 0.22950000000000001, 0.126500000000000001, 0.179999999999999993, 12),
-(1393, 'M', 0.635000000000000009, 0.474999999999999978, 0.170000000000000012, 1.19350000000000001, 0.520499999999999963, 0.269500000000000017, 0.366499999999999992, 10),
-(359, 'M', 0.744999999999999996, 0.584999999999999964, 0.214999999999999997, 2.49900000000000011, 0.92649999999999999, 0.471999999999999975, 0.699999999999999956, 17),
-(549, 'F', 0.564999999999999947, 0.450000000000000011, 0.160000000000000003, 0.79500000000000004, 0.360499999999999987, 0.155499999999999999, 0.23000000000000001, 12),
-(1154, 'F', 0.599999999999999978, 0.474999999999999978, 0.160000000000000003, 1.02649999999999997, 0.484999999999999987, 0.2495, 0.256500000000000006, 9),
-(1790, 'F', 0.54500000000000004, 0.385000000000000009, 0.149999999999999994, 1.11850000000000005, 0.542499999999999982, 0.244499999999999995, 0.284499999999999975, 9),
-(3703, 'F', 0.665000000000000036, 0.540000000000000036, 0.195000000000000007, 1.76400000000000001, 0.850500000000000034, 0.361499999999999988, 0.469999999999999973, 11),
-(1962, 'F', 0.655000000000000027, 0.515000000000000013, 0.179999999999999993, 1.41199999999999992, 0.619500000000000051, 0.248499999999999999, 0.496999999999999997, 11),
-(1665, 'I', 0.604999999999999982, 0.469999999999999973, 0.14499999999999999, 0.802499999999999991, 0.379000000000000004, 0.226500000000000007, 0.220000000000000001, 9),
-(635, 'M', 0.359999999999999987, 0.294999999999999984, 0.100000000000000006, 0.210499999999999993, 0.0660000000000000031, 0.0524999999999999981, 0.0749999999999999972, 9),
-(3901, 'M', 0.445000000000000007, 0.344999999999999973, 0.140000000000000013, 0.475999999999999979, 0.205499999999999988, 0.101500000000000007, 0.108499999999999999, 15),
-(2734, 'I', 0.41499999999999998, 0.33500000000000002, 0.100000000000000006, 0.357999999999999985, 0.169000000000000011, 0.067000000000000004, 0.104999999999999996, 7),
-(3856, 'M', 0.409999999999999976, 0.33500000000000002, 0.115000000000000005, 0.440500000000000003, 0.190000000000000002, 0.0850000000000000061, 0.135000000000000009, 8),
-(827, 'I', 0.395000000000000018, 0.28999999999999998, 0.0950000000000000011, 0.303999999999999992, 0.127000000000000002, 0.0840000000000000052, 0.076999999999999999, 6),
-(3381, 'I', 0.190000000000000002, 0.130000000000000004, 0.0449999999999999983, 0.0264999999999999993, 0.00899999999999999932, 0.0050000000000000001, 0.00899999999999999932, 5),
-(3972, 'I', 0.400000000000000022, 0.294999999999999984, 0.0950000000000000011, 0.252000000000000002, 0.110500000000000001, 0.0575000000000000025, 0.0660000000000000031, 6),
-(1155, 'M', 0.599999999999999978, 0.455000000000000016, 0.170000000000000012, 1.1915, 0.695999999999999952, 0.239499999999999991, 0.239999999999999991, 8),
-(3467, 'M', 0.640000000000000013, 0.5, 0.170000000000000012, 1.4544999999999999, 0.642000000000000015, 0.357499999999999984, 0.353999999999999981, 9),
-(2433, 'F', 0.609999999999999987, 0.484999999999999987, 0.165000000000000008, 1.08699999999999997, 0.425499999999999989, 0.232000000000000012, 0.380000000000000004, 11),
-(552, 'I', 0.614999999999999991, 0.489999999999999991, 0.154999999999999999, 0.988500000000000045, 0.41449999999999998, 0.195000000000000007, 0.344999999999999973, 13),
-(1425, 'F', 0.729999999999999982, 0.57999999999999996, 0.190000000000000002, 1.73750000000000004, 0.678499999999999992, 0.434499999999999997, 0.520000000000000018, 11),
-(2402, 'F', 0.584999999999999964, 0.41499999999999998, 0.154999999999999999, 0.69850000000000001, 0.299999999999999989, 0.145999999999999991, 0.195000000000000007, 12),
-(1748, 'F', 0.699999999999999956, 0.535000000000000031, 0.174999999999999989, 1.77299999999999991, 0.680499999999999994, 0.479999999999999982, 0.512000000000000011, 15),
-(3983, 'I', 0.57999999999999996, 0.434999999999999998, 0.149999999999999994, 0.891499999999999959, 0.362999999999999989, 0.192500000000000004, 0.251500000000000001, 6),
-(335, 'F', 0.739999999999999991, 0.599999999999999978, 0.195000000000000007, 1.97399999999999998, 0.597999999999999976, 0.408499999999999974, 0.709999999999999964, 16),
-(1587, 'I', 0.515000000000000013, 0.349999999999999978, 0.104999999999999996, 0.474499999999999977, 0.212999999999999995, 0.122999999999999998, 0.127500000000000002, 10),
-(2448, 'I', 0.275000000000000022, 0.204999999999999988, 0.0800000000000000017, 0.096000000000000002, 0.0359999999999999973, 0.0184999999999999991, 0.0299999999999999989, 6),
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-</pre></li>
-<li>Train a model with family=gaussian and link=identity <pre class="example">
-SELECT madlib.glm(
- 'abalone',
- 'abalone_out',
- 'rings',
- 'ARRAY[1, length, diameter, height, whole, shucked, viscera, shell]',
- 'family=gaussian, link=identity');
-</pre></li>
-</ol>
-<p><b>Example for Gamma family:</b> (reuse the dataset in Gaussian case)</p>
-<ol type="1">
-<li>Reuse the test data set in Gaussian</li>
-<li>Train a model with family=gamma and link=inverse <pre class="example">
-SELECT madlib.glm(
- 'abalone',
- 'abalone_out',
- 'rings',
- 'ARRAY[1, length, diameter, height, whole, shucked, viscera, shell]',
- 'family=gamma, link=inverse');
-</pre></li>
-</ol>
-<p><b>Example for Inverse Gaussian family:</b> (reuse the dataset in Gaussian case)</p>
-<ol type="1">
-<li>Reuse the test data set in Gaussian</li>
-<li>Train a model with family=inverse_gaussian and link=sqr_inverse <pre class="example">
-SELECT madlib.glm(
- 'abalone',
- 'abalone_out',
- 'rings',
- 'ARRAY[1, length, diameter, height, whole, shucked, viscera, shell]',
- 'family=inverse_gaussian, link=sqr_inverse');
-</pre></li>
-</ol>
-<p><b>Example for Binomial family:</b> (reuse the dataset in Gaussian case)</p>
-<ol type="1">
-<li>Reuse the test data set in Gaussian</li>
-<li>Train a model with family=binomial and link=probit <pre class="example">
-SELECT madlib.glm(
- 'abalone',
- 'abalone_out',
- 'rings < 10',
- 'ARRAY[1, length, diameter, height, whole, shucked, viscera, shell]',
- 'family=binomial, link=probit');
-</pre></li>
-<li>Predict output probabilities <pre class="example">
-SELECT madlib.glm_predict(
- coef,
- ARRAY[1, length, diameter, height, whole, shucked, viscera, shell]::float8[],
- 'probit')
-FROM abalone_out, abalone;
-</pre></li>
-<li>Predict output categories <pre class="example">
-SELECT madlib.glm_predict(
-SELECT madlib.glm_predict_binomial(
- coef,
- ARRAY[1, length, diameter, height, whole, shucked, viscera, shell]::float8[],
- 'probit')
-FROM abalone_out, abalone;
-</pre></li>
-</ol>
-<p><a class="anchor" id="notes"></a></p><dl class="section user"><dt>Notes</dt><dd>All table names can be optionally schema qualified (current_schemas() would be searched if a schema name is not provided) and all table and column names should follow case-sensitivity and quoting rules per the database. (For instance, 'mytable' and 'MyTable' both resolve to the same entity, i.e. 'mytable'. If mixed-case or multi-byte characters are desired for entity names then the string should be double-quoted; in this case the input would be '"MyTable"').</dd></dl>
-<p>Currently implementation uses Newton's method and, according to performance tests, when number of features are over 1000, this GLM function could be running slowly.</p>
-<p>Functions in <a class="el" href="group__grp__linreg.html">Linear Regression</a> is prefered to GLM with family=gaussian,link=identity, as the former require only a single pass over the training data. In addition, if user expects to use robust variance, clustered variance, or marginal effects on top of the trained model, functions in <a class="el" href="group__grp__linreg.html">Linear Regression</a> and <a class="el" href="group__grp__logreg.html">Logistic Regression</a> should be used.</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="glm_8sql__in.html" title="SQL functions for GLM (Poisson) ">glm.sql_in</a> documenting the training function</p>
-<p><a class="el" href="group__grp__linreg.html">Linear Regression</a></p>
-<p><a class="el" href="group__grp__logreg.html">Logistic Regression</a></p>
-<p><a class="el" href="group__grp__mlogreg.html">Multinomial Logistic Regression</a></p>
-<p><a class="el" href="group__grp__robust.html">Robust Variance</a></p>
-<p><a class="el" href="group__grp__clustered__errors.html">Clustered Variance</a></p>
-<p><a class="el" href="group__grp__validation.html">Cross Validation</a></p>
-<p><a class="el" href="group__grp__marginal.html">Marginal Effects</a></p>
-</div><!-- contents -->
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- </ul>
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-</html>
http://git-wip-us.apache.org/repos/asf/incubator-madlib-site/blob/c97706e8/docs/latest/group__grp__inf__stats.html
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- <span class="left">
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- <div id="nav-tree">
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-</div><!--header-->
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-</table>
-</div><!-- contents -->
-</div><!-- doc-content -->
-<!-- start footer part -->
-<div id="nav-path" class="navpath"><!-- id is needed for treeview function! -->
- <ul>
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- <a href="http://www.doxygen.org/index.html">
- <img class="footer" src="doxygen.png" alt="doxygen"/></a> 1.8.10 </li>
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