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/* ----------------------------------------------------------------------- *//**
*
* @file multilogistic.sql_in
*
* @brief SQL functions for multinomial logistic regression
* @date July 2012
*
* @sa For a brief introduction to multinomial logistic regression, see the
* module description \ref grp_mlogreg.
*
*//* ----------------------------------------------------------------------- */
m4_include(`SQLCommon.m4')
/**
@addtogroup grp_mlogreg
\warning <em> This is an old implementation of multinomial logistic regression. Replacement of this
function is available as the Multinomial regression module \ref grp_multinom</em>
<div class="toc"><b>Contents</b>
<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="#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>
@brief Also called as softmax regression, models the relationship between
one or more independent variables and a categorical dependent variable.
Multinomial logistic regression is a widely used regression analysis tool that
models the outcomes of categorical dependent random variables. The model
assumes that the conditional mean of the dependent categorical variables is
the logistic function of an affine combination of independent variables.
Multinomial logistic regression finds the vector of coefficients that
maximizes the likelihood of the observations.
@anchor train
@par Training Function
The multinomial logistic regression training function has the following syntax:
<pre class="syntax">
mlogregr_train(source_table,
output_table,
dependent_varname,
independent_varname,
ref_category,
optimizer_params
)
</pre>
\b Arguments
<dl class="arglist">
<dt>source_table</dt>
<dd>TEXT. The name of the table containing the input data.
</dd>
<dt>output_table</dt>
<dd>TEXT. The name of the generated table containing the output model.
The output table produced by the multinomial logistic regression training
function contains the following columns:
<table class="output">
<tr>
<th>category</th>
<td>INTEGER. The category. Categories are encoded as integers
with values from {0, 1, 2,..., \e numCategories &ndash; 1}</td>
</tr>
<tr>
<th>ref_category</th>
<td>INTEGER. The reference category. Categories are encoded as integers
with values from {0, 1, 2,..., \e numCategories &ndash; 1}</td>
</tr>
<tr>
<th>coef</th>
<td>FLOAT8[]. An array of coefficients, \f$ \boldsymbol c \f$. </td>
</tr>
<tr>
<th>log_likelihood</th>
<td>FLOAT8. The log-likelihood, \f$ l(\boldsymbol c) \f$.</td>
</tr>
<tr>
<th>std_err</th>
<td>FLOAT8[]. An array of the standard errors.</td>
</tr>
<tr>
<th>z_stats</th>
<td>FLOAT8[]. An array of the Wald z-statistics.</td>
</tr>
<tr>
<th>p_values</th>
<td>FLOAT8[]. An array of the Wald p-values.</td>
</tr>
<tr>
<th>odds_ratios</th>
<td>FLOAT8[]. An array of the odds ratios.</td>
</tr>
<tr>
<th>condition_no</th>
<td>FLOAT8. The condition number of the matrix, computed using the
coefficients of the iteration immediately preceding convergence.</td>
</tr>
<tr>
<th>num_iterations</th>
<td>INTEGER. The number of iterations executed before the
algorithm completed.</td>
</tr>
</table>
A summary table named \<out_table\>_summary is also created at the same time, and it contains the following columns:
<table class="output">
<tr>
<th>source_table</th>
<td>The data source table name.</td>
</tr>
<tr>
<th>out_table</th>
<td>The output table name.</td>
</tr>
<tr>
<th>dependent_varname</th>
<td>The dependent variable.</td>
</tr>
<tr>
<th>independent_varname</th>
<td>The independent variables.</td>
</tr>
<tr>
<th>optimizer_params</th>
<td>The optimizer parameters. It is a copy of the optimizer_params in the training function's arguments.</td>
</tr>
<tr>
<th>ref_category</th>
<td>An integer, the value of reference category used.</td>
</tr>
<tr>
<th>num_rows_processed</th>
<td>INTEGER. The number of rows actually processed, which is equal to the
total number of rows in the source table minus the number of skipped
rows.</td>
</tr>
<tr>
<th>num_missing_rows_skipped</th>
<td>INTEGER. The number of rows skipped during the training. A row
will be skipped if the ind_col is NULL or contains NULL values.</td>
</tr>
</table>
</dd>
<dt>dependent_varname</dt>
<dd>TEXT. The name of the column containing the dependent variable.</dd>
<dt>independent_varname</dt>
<dd>TEXT. Expression list to evaluate for the independent variables.
An intercept variable is not assumed. The number of independent variables
cannot exceed 65535.</dd>
<dt>ref_category (optional)</dt>
<dd>INTEGER, default: 0. The reference category ranges from
[0, \e numCategories &ndash; 1].</dd>
<dt>optimizer_params (optional)</dt>
<dd>VARCHAR, default: NULL, which uses the default values of optimizer parameters.
It should be a string that contains pairs of 'key=value' separated by commas.
Supported parameters with their default values: max_iter=20, optimizer='irls', precision=1e-4.
Currently, only 'irls' and 'newton' are allowed for 'optimizer'.
</dd>
</dl>
@note Table names can be optionally schema qualified and table and column names
should follow the same case-sensitivity and quoting rules as in the database.
@anchor predict
@par Prediction Function
The prediction function is provided to estimate the conditional mean given a new
predictor. It has the following syntax:
<pre class="syntax">
mlogregr_predict(
model_table,
new_data_table,
id_col_name,
output_table,
type)
</pre>
\b Arguments
<DL class="arglist">
<DT>model_table</DT>
<DD>TEXT. Name of the table containing the multilogistic model. This should
be the output table returned from <em>mlogregr_train</em>.</DD>
<DT>new_data_table</DT>
<DD>TEXT. Name of the table containing prediction data. This table is
expected to contain the same features that were used during training. The table
should also contain <em>id_col_name</em> used for identifying each row.</DD>
<DT>id_col_name</DT>
<DD>TEXT. Name of the column containing id information in the source data.
This is a mandatory argument and is used for correlating prediction table rows
with the source. The values of this column are expected to be unique for each tuple.
</DD>
<DT>output_table</DT>
<DD>TEXT. Name of the table to output prediction results to. If this table
already exists then an error is returned.
This output table contains the <em>id_col_name</em> column giving
the 'id' for each prediction.
If <em>type</em> = 'response', then the table has a single additional column with the prediction value of the response.
The type of this column depends on the type of the response variable used during training.
If <em>type</em> = 'prob', then the table has multiple additional columns, one for each possible category. The columns are labeled as
'estimated_prob_<em>category_value</em>', where <em>category_value</em> represents the values of categories (0 to K-1).</DD>
<DT>type</DT>
<DD>TEXT, optional, default: 'response'.
When <em>type</em> = 'prob', the
probabilities of each category (including the reference category) is given.
When <em>type</em> = 'response', a single output is provided which represents
the prediction category for each tuple. This represents the category with the
highest probability.
</DD>
</DL>
@anchor examples
@examp
-# Create the training data table.
<pre class="example">
DROP TABLE IF EXISTS test3;
CREATE TABLE test3 (
feat1 INTEGER,
feat2 INTEGER,
cat INTEGER
);
INSERT INTO test3(feat1, feat2, cat) VALUES
(1,35,1),
(2,33,0),
(3,39,1),
(1,37,1),
(2,31,1),
(3,36,0),
(2,36,1),
(2,31,1),
(2,41,1),
(2,37,1),
(1,44,1),
(3,33,2),
(1,31,1),
(2,44,1),
(1,35,1),
(1,44,0),
(1,46,0),
(2,46,1),
(2,46,2),
(3,49,1),
(2,39,0),
(2,44,1),
(1,47,1),
(1,44,1),
(1,37,2),
(3,38,2),
(1,49,0),
(2,44,0),
(3,61,2),
(1,65,2),
(3,67,1),
(3,65,2),
(1,65,2),
(2,67,2),
(1,65,2),
(1,62,2),
(3,52,2),
(3,63,2),
(2,59,2),
(3,65,2),
(2,59,0),
(3,67,2),
(3,67,2),
(3,60,2),
(3,67,2),
(3,62,2),
(2,54,2),
(3,65,2),
(3,62,2),
(2,59,2),
(3,60,2),
(3,63,2),
(3,65,2),
(2,63,1),
(2,67,2),
(2,65,2),
(2,62,2);
</pre>
-# Run the multilogistic regression function.
<pre class="example">
DROP TABLE IF EXISTS test3_output;
DROP TABLE IF EXISTS test3_output_summary;
SELECT madlib.mlogregr_train('test3',
'test3_output',
'cat',
'ARRAY[1, feat1, feat2]',
0,
'max_iter=20, optimizer=irls, precision=0.0001'
);
</pre>
-# View the result:
<pre class="example">
-- Set extended display on for easier reading of output
\\x on
SELECT * FROM test3_output;
</pre>
Results:
<pre class="result">
-[ RECORD 1 ]--+------------------------------------------------------------
category | 1
ref_category | 0
coef | {1.45474045211601,0.0849956182104023,-0.0172383499601956}
loglikelihood | -39.14759930999
std_err | {2.13085072854143,0.585021661344715,0.0431487356292144}
z_stats | {0.682704063982831,0.145286275409074,-0.39950996729842}
p_values | {0.494793861210936,0.884484850387893,0.689517480964129}
odd_ratios | {4.28337158128448,1.08871229617973,0.982909380301134}
condition_no | 280069.034217586
num_iterations | 5
-[ RECORD 2 ]--+------------------------------------------------------------
category | 2
ref_category | 0
coef | {-7.12908167688326,0.87648787696783,0.127886153027713}
loglikelihood | -39.14759930999
std_err | {2.52104008297868,0.639575886323862,0.0445757462972303}
z_stats | {-2.82783352990566,1.37042045472615,2.86896269049475}
p_values | {0.00468641692252239,0.170555690550421,0.00411820373218956}
odd_ratios | {0.000801455044349486,2.40244718187161,1.13642361694409}
condition_no | 280069.034217586
num_iterations | 5
</pre>
-# View all parameters used during the training
<pre class="example">
\\x on
SELECT * FROM test3_output_summary;
</pre>
Results:
<pre class="result">
-[ RECORD 1 ]------------+--------------------------------------------------
method | mlogregr
source_table | test3
out_table | test3_output
dependent_varname | cat
independent_varname | ARRAY[1, feat1, feat2]
optimizer_params | max_iter=20, optimizer=irls, precision=0.0001
ref_category | 0
num_categories | 3
num_rows_processed | 57
num_missing_rows_skipped | 0
variance_covariance | {{4.54052482732554,3.01080140927409,-0.551901021610841,-0.380754019900586,-0.0784151362989211,-0.0510014701718268},{3.01080140927409,6.35564309998514,-0.351902272617974,-0.766730342510818,-0.051877550252329,-0.0954432017695571},{-0.551901021610841,-0.351902272617974,0.34225034424253,0.231740815080827,-0.00117521831508331,-0.00114043921343171},{-0.380754019900586,-0.766730342510818,0.231740815080827,0.409057314366954,-0.000556498286025567,-0.000404735750986327},{-0.0784151362989211,-0.051877550252329,-0.00117521831508331,-0.000556498286025569,0.00186181338639984,0.00121080293928445},{-0.0510014701718268,-0.0954432017695571,-0.00114043921343171,-0.000404735750986325,0.00121080293928446,0.00198699715795504}}
coef | {{1.45474045211601,0.0849956182104023,-0.0172383499601956},{-7.12908167688326,0.87648787696783,0.127886153027713}}
</pre>
@anchor background
@par Technical Background
Multinomial logistic regression
models the outcomes of categorical dependent random variables (denoted \f$ Y
\in \{ 0,1,2 \ldots k \} \f$). The model assumes that the conditional mean of
the dependent categorical variables is the logistic function of an affine
combination of independent variables (usually denoted \f$ \boldsymbol x \f$).
That is,
\f[
E[Y \mid \boldsymbol x] = \sigma(\boldsymbol c^T \boldsymbol x)
\f]
for some unknown vector of coefficients \f$ \boldsymbol c \f$ and where \f$
\sigma(x) = \frac{1}{1 + \exp(-x)} \f$ is the logistic function. Multinomial
logistic regression finds the vector of coefficients \f$ \boldsymbol c \f$ that
maximizes the likelihood of the observations.
Let
- \f$ \boldsymbol y \in \{ 0,1 \}^{n \times k} \f$ denote the vector of observed
dependent variables, with \f$ n \f$ rows and \f$ k \f$ columns, containing the
observed values of the dependent variable,
- \f$ X \in \mathbf R^{n \times k} \f$ denote the design matrix with \f$ k \f$
columns and \f$ n \f$ rows, containing all observed vectors of independent
variables \f$ \boldsymbol x_i \f$ as rows.
By definition,
\f[
P[Y = y_i | \boldsymbol x_i]
= \sigma((-1)^{y_i} \cdot \boldsymbol c^T \boldsymbol x_i)
\,.
\f]
Maximizing the likelihood
\f$ \prod_{i=1}^n \Pr(Y = y_i \mid \boldsymbol x_i) \f$
is equivalent to maximizing the log-likelihood
\f$ \sum_{i=1}^n \log \Pr(Y = y_i \mid \boldsymbol x_i) \f$, which simplifies to
\f[
l(\boldsymbol c) =
-\sum_{i=1}^n \log(1 + \exp((-1)^{y_i}
\cdot \boldsymbol c^T \boldsymbol x_i))
\,.
\f]
The Hessian of this objective is \f$ H = -X^T A X \f$ where
\f$ A = \text{diag}(a_1, \dots, a_n) \f$ is the diagonal matrix with
\f$
a_i = \sigma(\boldsymbol c^T \boldsymbol x)
\cdot
\sigma(-\boldsymbol c^T \boldsymbol x)
\,.
\f$
Since \f$ H \f$ is non-positive definite, \f$ l(\boldsymbol c) \f$ is convex.
There are many techniques for solving convex optimization problems. Currently,
logistic regression in MADlib can use:
- Iteratively Reweighted Least Squares
We estimate the standard error for coefficient \f$ i \f$ as
\f[
\mathit{se}(c_i) = \left( (X^T A X)^{-1} \right)_{ii}
\,.
\f]
The Wald z-statistic is
\f[
z_i = \frac{c_i}{\mathit{se}(c_i)}
\,.
\f]
The Wald \f$ p \f$-value for coefficient \f$ i \f$ 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 (\f$ c_i = 0 \f$) is
true. Letting \f$ F \f$ denote the cumulative density function of a standard
normal distribution, the Wald \f$ p \f$-value for coefficient \f$ i \f$ is
therefore
\f[
p_i = \Pr(|Z| \geq |z_i|) = 2 \cdot (1 - F( |z_i| ))
\f]
where \f$ Z \f$ is a standard normally distributed random variable.
The odds ratio for coefficient \f$ i \f$ is estimated as \f$ \exp(c_i) \f$.
The condition number is computed as \f$ \kappa(X^T A X) \f$ during the iteration
immediately <em>preceding</em> convergence (i.e., \f$ A \f$ is computed using
the coefficients of the previous iteration). A large condition number (say, more
than 1000) indicates the presence of significant multicollinearity.
The multinomial logistic regression uses a default reference category of zero,
and the regression coefficients in the output are in the order described below.
For a problem with
\f$ K \f$ dependent variables \f$ (1, ..., K) \f$ and \f$ J \f$ categories \f$ (0, ..., J-1)
\f$, let \f$ {m_{k,j}} \f$ denote the coefficient for dependent variable \f$ k
\f$ and category \f$ j \f$. The output is \f$ {m_{k_1, j_0}, m_{k_1, j_1}
\ldots m_{k_1, j_{J-1}}, m_{k_2, j_0}, m_{k_2, j_1}, \ldots m_{k_2, j_{J-1}} \ldots m_{k_K, j_{J-1}}} \f$.
The order is NOT CONSISTENT with the multinomial regression marginal effect
calculation with function <em>marginal_mlogregr</em>. This is deliberate
because the interfaces of all multinomial regressions (robust, clustered, ...)
will be moved to match that used in marginal.
@anchor literature
@literature
A collection of nice write-ups, with valuable pointers into
further literature:
[1] Annette J. Dobson: An Introduction to Generalized Linear Models, Second Edition. Nov 2001
[2] Cosma Shalizi: Statistics 36-350: Data Mining, Lecture Notes, 18 November
2009, http://www.stat.cmu.edu/~cshalizi/350/lectures/26/lecture-26.pdf
[3] Scott A. Czepiel: Maximum Likelihood Estimation
of Logistic Regression Models: Theory and Implementation,
Retrieved Jul 12 2012, http://czep.net/stat/mlelr.pdf
@anchor related
@par Related Topics
File multilogistic.sql_in documenting the multinomial logistic regression functions
\ref grp_logreg
@internal
@sa Namespace multilogistic (documenting the driver/outer loop implemented in
Python), Namespace
\ref madlib::modules::regress documenting the implementation in C++
@endinternal
*/
DROP TYPE IF EXISTS MADLIB_SCHEMA.mlogregr_result CASCADE;
CREATE TYPE MADLIB_SCHEMA.mlogregr_result AS
(
ref_category INTEGER,
coef DOUBLE PRECISION[],
log_likelihood DOUBLE PRECISION,
std_err DOUBLE PRECISION[],
z_stats DOUBLE PRECISION[],
p_values DOUBLE PRECISION[],
odds_ratios DOUBLE PRECISION[],
condition_no DOUBLE PRECISION,
num_iterations INTEGER,
num_processed BIGINT,
variance_covariance DOUBLE PRECISION[]
);
DROP TYPE IF EXISTS MADLIB_SCHEMA.mlogregr_summary_result CASCADE;
CREATE TYPE MADLIB_SCHEMA.mlogregr_summary_result AS
(
coef DOUBLE PRECISION[],
variance_covariance DOUBLE PRECISION[]
);
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.__mlogregr_irls_step_transition
(
state DOUBLE PRECISION[],
y INTEGER,
num_categories INTEGER,
ref_category INTEGER,
x DOUBLE PRECISION[],
prev_state DOUBLE PRECISION[]
)
RETURNS DOUBLE PRECISION[]
AS 'MODULE_PATHNAME'
LANGUAGE C IMMUTABLE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `NO SQL', `');
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.__mlogregr_irls_step_merge_states
(
state1 DOUBLE PRECISION[],
state2 DOUBLE PRECISION[]
)
RETURNS DOUBLE PRECISION[]
AS 'MODULE_PATHNAME'
LANGUAGE C IMMUTABLE STRICT
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `NO SQL', `');
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.__mlogregr_irls_step_final
(
state DOUBLE PRECISION[]
)
RETURNS DOUBLE PRECISION[]
AS 'MODULE_PATHNAME'
LANGUAGE C IMMUTABLE STRICT
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `NO SQL', `');
/**
* @internal
* @brief Perform one iteration of the iteratively-reweighted-least-squares
* method for computing linear regression
*/
DROP AGGREGATE IF EXISTS MADLIB_SCHEMA.__mlogregr_irls_step(
INTEGER, INTEGER, INTEGER, DOUBLE PRECISION [], DOUBLE PRECISION []);
CREATE AGGREGATE MADLIB_SCHEMA.__mlogregr_irls_step(
/*+ y */ INTEGER,
/*+ numCategories */ INTEGER,
/*+ ref_category */ INTEGER,
/*+ x */ DOUBLE PRECISION[],
/*+ previous_state */ DOUBLE PRECISION[]) (
STYPE=DOUBLE PRECISION[],
SFUNC=MADLIB_SCHEMA.__mlogregr_irls_step_transition,
m4_ifdef(`__POSTGRESQL__', `', `prefunc=MADLIB_SCHEMA.__mlogregr_irls_step_merge_states,')
FINALFUNC=MADLIB_SCHEMA.__mlogregr_irls_step_final,
INITCOND='{0,0,0,0,0,0}'
);
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.__internal_mlogregr_irls_step_distance(
/*+ state1 */ DOUBLE PRECISION[],
/*+ state2 */ DOUBLE PRECISION[])
RETURNS DOUBLE PRECISION AS
'MODULE_PATHNAME'
LANGUAGE c IMMUTABLE STRICT
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `NO SQL', `');
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.__internal_mlogregr_irls_result(
/*+ state */ DOUBLE PRECISION[])
RETURNS MADLIB_SCHEMA.mlogregr_result AS
'MODULE_PATHNAME'
LANGUAGE c IMMUTABLE STRICT
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `NO SQL', `');
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.__internal_mlogregr_summary_results(
/*+ state */ DOUBLE PRECISION[])
RETURNS MADLIB_SCHEMA.mlogregr_summary_result AS
'MODULE_PATHNAME'
LANGUAGE c IMMUTABLE STRICT
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `NO SQL', `');
/**
* @brief Compute multinomial logistic regression coefficients
*
* To include an intercept in the model, set one coordinate in the
* <tt>independentVariables</tt> array to 1.
*
* @param source_table Name of the source relation containing the training data
* @param output_table Name of the output relation to contain the resulting model
* @param dependent_varname Name of the dependent column (of type INTEGER)
* @param independent_varname Name of the independent column (or an array expression)
* @param ref_category The reference category specified by the user
* @param optimizer_params Comma-separated list of parameters for the optimizer function
*
* @return
* An output table (named 'output_table' above) containing following columns:
* - <tt>ref_category INTEGER</tt> - Reference category
* - <tt>coef FLOAT8[]</tt> - Array of coefficients, \f$ \boldsymbol c \f$
* - <tt>log_likelihood FLOAT8</tt> - Log-likelihood \f$ l(\boldsymbol c) \f$
* - <tt>std_err FLOAT8[]</tt> - Array of standard errors,
* \f$ \mathit{se}(c_1), \dots, \mathit{se}(c_k) \f$
* - <tt>z_stats FLOAT8[]</tt> - Array of Wald z-statistics, \f$ \boldsymbol z \f$
* - <tt>p_values FLOAT8[]</tt> - Array of Wald p-values, \f$ \boldsymbol p \f$
* - <tt>odds_ratios FLOAT8[]</tt>: Array of odds ratios,
* \f$ \mathit{odds}(c_1), \dots, \mathit{odds}(c_k) \f$
* - <tt>condition_no FLOAT8</tt> - The condition number of matrix
* \f$ X^T A X \f$ during the iteration immediately <em>preceding</em>
* convergence (i.e., \f$ A \f$ is computed using the coefficients of the
* previous iteration)
* An output table (named 'output_table'_summary) containing following columns:
* - <tt>regression_type VARCHAR</tt> - The regression type run (in this case it will be 'mlogit')
* - <tt>source_table VARCHAR</tt> - Source table containing the training data
* - <tt>output_table VARCHAR</tt> - Output table containing the trained model
* - <tt>dependent_varname VARCHAR</tt> - Name of the dependent column used for training
* - <tt>independent_varname VARCHAR</tt> - Name of the independent column used for training (or the ARRAY expression used for training)
* - <tt>ref_category INTEGER</tt> - The reference category specified by the user
* - <tt>num_iterations INTEGER</tt> - The number of iterations before the algorithm terminated
* - <tt>num_rows_processed INTEGER</tt> - The number of rows from training data used for training
* - <tt>num_missing_rows_skipped INTEGER</tt> - The number of rows skipped during training
*
* @usage
* - Get vector of coefficients \f$ \boldsymbol c \f$ and all diagnostic
* statistics:\n
* <pre>SELECT mlogregr_train('<em>sourceName</em>', '<em>outputName</em>',
'<em>dependentVariable</em>', '<em>independentVariables</em>');
SELECT * from <em>outputName</em>;
</pre>
* - Get vector of coefficients \f$ \boldsymbol c \f$:\n
* <pre>SELECT coef from <em>outputName</em>;</pre>
* - Get a subset of the output columns, e.g., only the array of coefficients
* \f$ \boldsymbol c \f$, the log-likelihood of determination
* \f$ l(\boldsymbol c) \f$, and the array of p-values \f$ \boldsymbol p \f$:
* <pre>SELECT coef, log_likelihood, p_values
* FROM <em>outputName</em>;</pre>
*
*
* @internal
* @sa This function is a wrapper for multilogistic::__compute_mlogregr(), which
* sets the default values.
*/
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlogregr_train(
source_table VARCHAR -- Name of the source table containing training data
, output_table VARCHAR -- Name of the output table to hold the trained model
, dependent_varname VARCHAR -- Name of the column containing the category values (the values must be integers from 0 to num_categories-1)
, independent_varname VARCHAR -- Name of the column containing the independent variables. Can also be an ARRAY expression.
, ref_category INTEGER -- The value of reference category (the value must be between 0 and num_categories-1)
, optimizer_params VARCHAR -- a comma-separated string with optimizer parameters
) RETURNS VOID AS $$
PythonFunction(regress, multilogistic, mlogregr_train)
$$ LANGUAGE plpythonu VOLATILE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA', `');
-- Overloaded functions --------------------------------------------------------
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlogregr_train(
source_table VARCHAR -- Name of the source table containing training data
, output_table VARCHAR -- Name of the output table to hold the trained model
, dependent_varname VARCHAR -- Name of the column containing the category values (the values must be integers from 0 to num_categories-1)
, independent_varname VARCHAR -- Name of the column containing the independent variables. Can also be an ARRAY expression.
, ref_category INTEGER -- The value of reference category (the value must be between 0 and num_categories-1)
) RETURNS VOID AS $$
SELECT MADLIB_SCHEMA.mlogregr_train($1, $2, $3, $4, $5, NULL::VARCHAR);
$$ LANGUAGE sql VOLATILE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA', `');
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlogregr_train(
source_table VARCHAR -- Name of the source table containing training data
, output_table VARCHAR -- Name of the output table to hold the trained model
, dependent_varname VARCHAR -- Name of the column containing the category values (the values must be integers from 0 to num_categories-1)
, independent_varname VARCHAR -- Name of the column containing the independent variables. Can also be an ARRAY expression.
) RETURNS VOID AS $$
SELECT MADLIB_SCHEMA.mlogregr_train($1, $2, $3, $4, 0, NULL::VARCHAR);
$$ LANGUAGE sql VOLATILE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA', `');
-- Help messages -----------------------------------------------------------
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlogregr_train(
message VARCHAR -- usage string
)
RETURNS VARCHAR AS $$
# PythonFunctionBodyOnly(`regress', `multilogistic')
# return multilogistic.mlogregr_help_message(message)
PythonFunction(regress, multilogistic, mlogregr_help_message)
$$ LANGUAGE plpythonu IMMUTABLE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `CONTAINS SQL', `');
----------------------------------------------------------------------
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlogregr_train()
RETURNS VARCHAR AS $$
SELECT MADLIB_SCHEMA.mlogregr_train('')
$$ LANGUAGE SQL IMMUTABLE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `CONTAINS SQL', `');
-- End of overloaded functions --------------------------------------------------------
-------------------------------------------------------------------------------
-- DEPRECATION NOTICE ----------------------------------------------------------
-- The function 'mlogregr' has been deprecated and replaced by 'mlogregr_train'
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.__compute_mlogregr
(
source_table VARCHAR,
dependent_varname VARCHAR,
independent_varname VARCHAR,
num_categories INTEGER,
max_iter INTEGER,
optimizer VARCHAR,
"precision" DOUBLE PRECISION,
ref_category INTEGER
)
RETURNS INTEGER
AS $$
PythonFunction(regress, multilogistic, compute_mlogregr)
$$ LANGUAGE plpythonu VOLATILE STRICT
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA', `');
/**
* @brief Compute logistic-regression coefficients and diagnostic statistics
*
* To include an intercept in the model, set one coordinate in the
* <tt>independentVariables</tt> array to 1.
*
* @param source Name of the source relation containing the training data
* @param depvar Name of the dependent column (of type INTEGER < numcategories)
* @param indepvar Name of the independent column (of type DOUBLE PRECISION[])
* @param max_num_iterations The maximum number of iterations
* @param optimizer The optimizer to use (
* <tt>'irls'</tt>/<tt>'newton'</tt> for iteratively reweighted least
* squares)
* @param precision The difference between log-likelihood values in successive
* iterations that should indicate convergence. Note that a non-positive
* value here disables the convergence criterion, and execution will only
* stop after \ max_num_iterations iterations.
* @param ref_category The reference category specified by the user
*
* @return A composite value:
* - <tt>ref_category INTEGER</tt> - Reference category
* - <tt>coef FLOAT8[]</tt> - Array of coefficients, \f$ \boldsymbol c \f$
* - <tt>log_likelihood FLOAT8</tt> - Log-likelihood \f$ l(\boldsymbol c) \f$
* - <tt>std_err FLOAT8[]</tt> - Array of standard errors,
* \f$ \mathit{se}(c_1), \dots, \mathit{se}(c_k) \f$
* - <tt>z_stats FLOAT8[]</tt> - Array of Wald z-statistics, \f$ \boldsymbol z \f$
* - <tt>p_values FLOAT8[]</tt> - Array of Wald p-values, \f$ \boldsymbol p \f$
* - <tt>odds_ratios FLOAT8[]</tt>: Array of odds ratios,
* \f$ \mathit{odds}(c_1), \dots, \mathit{odds}(c_k) \f$
* - <tt>condition_no FLOAT8</tt> - The condition number of matrix
* \f$ X^T A X \f$ during the iteration immediately <em>preceding</em>
* convergence (i.e., \f$ A \f$ is computed using the coefficients of the
* previous iteration)
* - <tt>num_iterations INTEGER</tt> - The number of iterations before the
* algorithm terminated
*
* @usage
* - Get vector of coefficients \f$ \boldsymbol c \f$ and all diagnostic
* statistics:\n
* <pre>SELECT * FROM mlogregr('<em>sourceName</em>', '<em>dependentVariable</em>',
* '<em>numCategories</em>', '<em>independentVariables</em>');</pre>
* - Get vector of coefficients \f$ \boldsymbol c \f$:\n
* <pre>SELECT (mlogregr('<em>sourceName</em>', '<em>dependentVariable</em>',
* '<em>numCategories</em>', '<em>independentVariables</em>')).coef;</pre>
* - Get a subset of the output columns, e.g., only the array of coefficients
* \f$ \boldsymbol c \f$, the log-likelihood of determination
* \f$ l(\boldsymbol c) \f$, and the array of p-values \f$ \boldsymbol p \f$:
* <pre>SELECT coef, log_likelihood, p_values
* FROM mlogregr('<em>sourceName</em>', '<em>dependentVariable</em>',
* '<em>numCategories</em>', '<em>independentVariables</em>');</pre>
*
* @note This function starts an iterative algorithm. It is not an aggregate
* function. Source and column names have to be passed as strings (due to
* limitations of the SQL syntax).
*
* @internal
* @sa This function is a wrapper for multilogistic::__compute_mlogregr(), which
* sets the default values.
*/
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlogregr
(
source VARCHAR,
depvar VARCHAR,
indepvar VARCHAR,
max_num_iterations INTEGER /*+ DEFAULT 20 */,
optimizer VARCHAR /*+ DEFAULT 'irls' */,
"precision" DOUBLE PRECISION /*+ DEFAULT 0.0001 */,
ref_category INTEGER
)
RETURNS MADLIB_SCHEMA.mlogregr_result AS $$
DECLARE
observed_count INTEGER;
theIteration INTEGER;
fnName VARCHAR;
theResult MADLIB_SCHEMA.mlogregr_result;
numcategories INTEGER;
min_category INTEGER;
max_category INTEGER;
BEGIN
RAISE WARNING $sql$This function has been deprecated. Please use 'mlogregr_train' instead.$sql$;
IF (source IS NULL OR trim(source) = '') THEN
RAISE EXCEPTION 'Invalid source table given';
END IF;
IF (depvar IS NULL OR trim(depvar) = '') THEN
RAISE EXCEPTION 'Invalid depvar given';
END IF;
IF (indepvar IS NULL OR trim(indepvar) = '') THEN
RAISE EXCEPTION 'Invalid indepvar given';
END IF;
IF (max_num_iterations IS NULL OR max_num_iterations < 1) THEN
RAISE EXCEPTION 'Number of max iterations must be positive';
END IF;
IF (optimizer IS NULL OR trim(optimizer) = '') THEN
RAISE EXCEPTION 'Invalid optimizer given';
END IF;
IF (precision IS NULL) THEN
RAISE EXCEPTION 'Invalid precision given.';
END IF;
IF (ref_category IS NULL OR ref_category < 0) THEN
RAISE EXCEPTION 'Invalid ref_category given.';
END IF;
IF (SELECT atttypid::regtype <> 'INTEGER'::regtype
FROM pg_attribute
WHERE attrelid = source::regclass AND attname = depvar) THEN
RAISE EXCEPTION 'The dependent variable column should be of type INTEGER';
END IF;
EXECUTE $sql$ SELECT count(DISTINCT $sql$ || depvar || $sql$ )
FROM $sql$ || textin(regclassout(source))
INTO observed_count;
numcategories := observed_count;
EXECUTE $sql$ SELECT max($sql$ || depvar || $sql$ )
FROM $sql$ || textin(regclassout(source))
INTO max_category;
EXECUTE $sql$ SELECT min($sql$ || depvar || $sql$ )
FROM $sql$ || textin(regclassout(source))
INTO min_category;
IF max_category != numcategories - 1 OR min_category != 0 THEN
RAISE EXCEPTION 'The value of the dependent variable should be in the
range of [0, %)', numcategories;
END IF;
IF ref_category > numcategories -1 OR ref_category < 0 THEN
RAISE EXCEPTION 'The value of the reference category should be in the
range of [0, "%")', numcategories;
END IF;
IF optimizer = 'irls' OR optimizer = 'newton' THEN
fnName := '__internal_mlogregr_irls_result';
ELSE
RAISE EXCEPTION 'Unknown optimizer (''%''). Must be "newton" or "irls"', optimizer;
END IF;
theIteration := (
SELECT MADLIB_SCHEMA.__compute_mlogregr(
$1, $2, $3, numcategories, $4, $5, $6, $7)
);
-- Because of Greenplum bug MPP-10050, we have to use dynamic SQL (using
-- EXECUTE) in the following
-- Because of Greenplum bug MPP-6731, we have to hide the tuple-returning
-- function in a subquery
EXECUTE
$sql$
SELECT (result).*
FROM (
SELECT
MADLIB_SCHEMA.$sql$ || fnName || $sql$(_madlib_state) AS result
FROM _madlib_iterative_alg
WHERE _madlib_iteration = $sql$ || theIteration || $sql$
) subq
$sql$
INTO theResult;
-- The number of iterations are not updated in the C++ code. We do it here.
IF NOT (theResult IS NULL) THEN
theResult.num_iterations = theIteration;
END IF;
RETURN theResult;
END;
$$ LANGUAGE plpgsql VOLATILE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA', `');
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlogregr
(
source VARCHAR,
depvar VARCHAR,
indepvar VARCHAR
)
RETURNS MADLIB_SCHEMA.mlogregr_result AS
$$
SELECT MADLIB_SCHEMA.mlogregr($1, $2, $3, 20, 'irls', 0.0001, 0);
$$ LANGUAGE sql VOLATILE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA', `');
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlogregr(
source VARCHAR,
depvar VARCHAR,
indepvar VARCHAR,
max_num_iterations INTEGER
)
RETURNS MADLIB_SCHEMA.mlogregr_result AS
$$
SELECT MADLIB_SCHEMA.mlogregr($1, $2, $3, $4, 'irls', 0.0001, 0);
$$ LANGUAGE sql VOLATILE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA', `');
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlogregr(
source VARCHAR,
depvar VARCHAR,
indepvar VARCHAR,
max_num_iterations INTEGER,
optimizer VARCHAR
)
RETURNS MADLIB_SCHEMA.mlogregr_result AS
$$
SELECT MADLIB_SCHEMA.mlogregr($1, $2, $3, $4, $5, 0.0001, 0);
$$ LANGUAGE sql VOLATILE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA', `');
-- -----------------------------------------------------------------------
-- New interface: The 'mlogregr' call is to be deprecated
-- -----------------------------------------------------------------------
/**
* @brief Compute multinomial logistic regression coefficients
*
* To include an intercept in the model, set one coordinate in the
* <tt>independentVariables</tt> array to 1.
*
* @param source_table Name of the source relation containing the training data
* @param output_table Name of the output relation to contain the resulting model
* @param dependent_varname Name of the dependent column (of type INTEGER)
* @param independent_varname Name of the independent column (or an array expression)
* @param ref_category The reference category specified by the user
* @param optimizer_params Comma-separated list of parameters for the optimizer function
*
* @return
* An output table (named 'output_table' above) containing following columns:
* - <tt>ref_category INTEGER</tt> - Reference category
* - <tt>coef FLOAT8[]</tt> - Array of coefficients, \f$ \boldsymbol c \f$
* - <tt>log_likelihood FLOAT8</tt> - Log-likelihood \f$ l(\boldsymbol c) \f$
* - <tt>std_err FLOAT8[]</tt> - Array of standard errors,
* \f$ \mathit{se}(c_1), \dots, \mathit{se}(c_k) \f$
* - <tt>z_stats FLOAT8[]</tt> - Array of Wald z-statistics, \f$ \boldsymbol z \f$
* - <tt>p_values FLOAT8[]</tt> - Array of Wald p-values, \f$ \boldsymbol p \f$
* - <tt>odds_ratios FLOAT8[]</tt>: Array of odds ratios,
* \f$ \mathit{odds}(c_1), \dots, \mathit{odds}(c_k) \f$
* - <tt>condition_no FLOAT8</tt> - The condition number of matrix
* \f$ X^T A X \f$ during the iteration immediately <em>preceding</em>
* convergence (i.e., \f$ A \f$ is computed using the coefficients of the
* previous iteration)
* An output table (named 'output_table'_summary) containing following columns:
* - <tt>regression_type VARCHAR</tt> - The regression type run (in this case it will be 'mlogit')
* - <tt>source_table VARCHAR</tt> - Source table containing the training data
* - <tt>output_table VARCHAR</tt> - Output table containing the trained model
* - <tt>dependent_varname VARCHAR</tt> - Name of the dependent column used for training
* - <tt>independent_varname VARCHAR</tt> - Name of the independent column used for training (or the ARRAY expression used for training)
* - <tt>ref_category INTEGER</tt> - The reference category specified by the user
* - <tt>num_iterations INTEGER</tt> - The number of iterations before the algorithm terminated
* - <tt>num_rows_processed INTEGER</tt> - The number of rows from training data used for training
* - <tt>num_missing_rows_skipped INTEGER</tt> - The number of rows skipped during training
*
* @usage
* - Get vector of coefficients \f$ \boldsymbol c \f$ and all diagnostic
* statistics:\n
* <pre>SELECT mlogregr_train('<em>sourceName</em>', '<em>outputName</em>',
'<em>dependentVariable</em>', '<em>independentVariables</em>');
SELECT * from <em>outputName</em>;
</pre>
* - Get vector of coefficients \f$ \boldsymbol c \f$:\n
* <pre>SELECT coef from <em>outputName</em>;</pre>
* - Get a subset of the output columns, e.g., only the array of coefficients
* \f$ \boldsymbol c \f$, the log-likelihood of determination
* \f$ l(\boldsymbol c) \f$, and the array of p-values \f$ \boldsymbol p \f$:
* <pre>SELECT coef, log_likelihood, p_values
* FROM <em>outputName</em>;</pre>
*
*
* @internal
* @sa This function is a wrapper for multilogistic::__compute_mlogregr(), which
* sets the default values.
*/
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlogregr_train (
source_table VARCHAR -- Name of the source table containing training data
, output_table VARCHAR -- Name of the output table to hold the trained model
, dependent_varname VARCHAR -- Name of the column containing the category values (the values must be integers from 0 to num_categories-1)
, independent_varname VARCHAR -- Name of the column containing the independent variables. Can also be an ARRAY expression.
, ref_category INTEGER -- The value of reference category (the value must be between 0 and num_categories-1)
, optimizer_params VARCHAR -- a comma-separated string with optimizer parameters
) RETURNS VOID AS $$
PythonFunction(regress, multilogistic, mlogregr_train)
$$ LANGUAGE plpythonu
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA', `');
-- Overloaded functions --------------------------------------------------------
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlogregr_train (
source_table VARCHAR -- Name of the source table containing training data
, output_table VARCHAR -- Name of the output table to hold the trained model
, dependent_varname VARCHAR -- Name of the column containing the category values (the values must be integers from 0 to num_categories-1)
, independent_varname VARCHAR -- Name of the column containing the independent variables. Can also be an ARRAY expression.
, ref_category INTEGER -- The value of reference category (the value must be between 0 and num_categories-1)
) RETURNS VOID AS $$
SELECT MADLIB_SCHEMA.mlogregr_train($1, $2, $3, $4, $5, NULL::VARCHAR);
$$ LANGUAGE sql VOLATILE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA', `');
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlogregr_train (
source_table VARCHAR -- Name of the source table containing training data
, output_table VARCHAR -- Name of the output table to hold the trained model
, dependent_varname VARCHAR -- Name of the column containing the category values (the values must be integers from 0 to num_categories-1)
, independent_varname VARCHAR -- Name of the column containing the independent variables. Can also be an ARRAY expression.
) RETURNS VOID AS $$
SELECT MADLIB_SCHEMA.mlogregr_train($1, $2, $3, $4, 0, NULL::VARCHAR);
$$ LANGUAGE sql VOLATILE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA', `');
-- Help messages -----------------------------------------------------------
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlogregr_train()
RETURNS VARCHAR AS $$
BEGIN
RETURN MADLIB_SCHEMA.mlogregr_train('');
END;
$$ LANGUAGE plpgsql IMMUTABLE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `NO SQL', `');
----------------------------------------------------------------------
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlogregr_train(
message VARCHAR -- usage string
)
RETURNS VARCHAR AS $$
PythonFunction(regress, multilogistic, mlogregr_help_message)
$$ LANGUAGE plpythonu IMMUTABLE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `NO SQL', `');
----------------------------------------------------------------------
DROP TYPE IF EXISTS MADLIB_SCHEMA.__mlogregr_cat_coef CASCADE;
CREATE TYPE MADLIB_SCHEMA.__mlogregr_cat_coef AS (
category INTEGER,
coef DOUBLE PRECISION[]
);
--------------------------------------
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.__mlogregr_format(
coef DOUBLE PRECISION[],
num_feature INTEGER,
num_category INTEGER,
ref_category INTEGER
) RETURNS SETOF MADLIB_SCHEMA.__mlogregr_cat_coef
AS 'MODULE_PATHNAME'
LANGUAGE C IMMUTABLE STRICT
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `NO SQL', `');
------------------------------------------------------------
-- returns the probabilities of each categories
-- The user has to get the coef matrix from the summary table
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.__mlogregr_predict_prob(
coef DOUBLE PRECISION[],
ref_category INTEGER,
col_ind_var DOUBLE PRECISION[]
) RETURNS DOUBLE PRECISION[] AS
'MODULE_PATHNAME', 'mlogregr_predict_prob'
LANGUAGE C STRICT IMMUTABLE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `NO SQL', `');
------------------------------------------------------------
-- returns the index of the category that has the largest probability
-- The user has to get the coef matrix from the summary table
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.__mlogregr_predict_response(
coef DOUBLE PRECISION[],
ref_category INTEGER,
col_ind_var DOUBLE PRECISION[]
) RETURNS INTEGER AS
'MODULE_PATHNAME', 'mlogregr_predict_response'
LANGUAGE C STRICT IMMUTABLE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `NO SQL', `');
-------------------------------------------------------------------------
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlogregr_predict(
model TEXT,
source TEXT,
id_col_name TEXT,
output TEXT,
pred_type TEXT
) RETURNS VOID AS $$
PythonFunction(regress, multilogistic, mlogregr_predict)
$$ LANGUAGE plpythonu VOLATILE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA', `');
-------------------------------------------------------------------------
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlogregr_predict(
model TEXT,
source TEXT,
id_col_name TEXT,
output TEXT
) RETURNS VOID AS $$
SELECT MADLIB_SCHEMA.mlogregr_predict($1, $2, $3, $4, 'response');
$$ LANGUAGE SQL VOLATILE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA', `');
-------------------------------------------------------------------------
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.mlogregr_predict(
message TEXT
) RETURNS TEXT AS $$
PythonFunction(regress, multilogistic, mlogregr_predict_help_message)
$$ LANGUAGE plpythonu VOLATILE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `CONTAINS SQL', `');