cox_prop_hazards.sql_in

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/* ----------------------------------------------------------------------- *//** 
 *
 * @file cox_prop_hazards.sql_in
 *
 * @brief SQL functions for cox proportional hazards
 * @date July 2012
 *
 * @sa For a brief introduction to cox regression, see the
 *     module description \ref grp_cox_prop_hazards
 *
 *//* ----------------------------------------------------------------------- */

m4_include(`SQLCommon.m4')

/**
@addtogroup grp_cox_prop_hazards

@about
Proportional-Hazard models enable the comparison of various survival models. 
These survival models are functions describing the probability of an one-item 
event (prototypically, this event is death) with respect to time. 
The interval of time before death occurs is the survival time. 
Let T be a random variable representing the survival time, 
with a cumulative probability function P(t). Informally, P(t) is 
the probability that death has happened before time t.

Generally, applications start with a list of \f$ \boldsymbol n \f$ observations, 
each with \f$ \boldsymbol m \f$ covariates and a time of death. From this 
\f$ \boldsymbol n \times m \f$ matrix, we would like to derive the correlation 
between the covariates and the hazard function. This amounts to finding 
the parameters \f$ \boldsymbol \beta \f$ that best fit the model described below.

Let us define:
- \f$ \boldsymbol t \in  \mathbf R^{m} \f$ denote the vector of observed dependent
  variables, with \f$ n \f$ rows.
- \f$ X \in \mathbf R^{m} \f$ denote the design matrix with \f$ m \f$
  columns and \f$ n \f$ rows, containing all observed vectors of independent
  variables \f$ \boldsymbol x_i \f$ as rows.
- \f$ R(t_i) \f$ denote the set of observations still alive at time \f$ t_i \f$

Note that this model <b>does not</b> include a <b>constant term</b>, and the data
cannot contain a column of 1s.

By definition,
\f[
    P[T_k = t_i | \boldsymbol R(t_i)]
    =    \frac{e^{\beta^T x_k} }{ \sum_{j \in R(t_i)} e^{\beta^T x_j}}.
    \,.
\f]

The <b>partial likelihood </b>function can now be generated as the product of 
conditional probabilities:
\f[
\mathcal L = \prod_{i = 1}^n 
                \left(
                \frac{e^{\beta^T x_i}}{ \sum_{j \in R(t_i)} e^{\beta^T x_j}}
                \right).
\f]

The log-likelihood form of this equation is
\f[
L = \sum_{i = 1}^n 
                    \left[  \beta^T x_i
                                    - \log\left(\sum_{j \in R(t_i)} e^{\beta^T x_j }\right)
                    \right].
\f]

Using this score function and Hessian matrix, the partial likelihood can be 
maximized using the <b> Newton-Raphson algorithm </b>.<b> Breslow's method </b> 
is used to resolved tied times of deaths. The time of death for two records are 
considered "equal" if they differ by less than 1.0e-6

The inverse of the Hessian matrix, evaluated at the estimate of 
\f$ \boldsymbol \beta \f$, can be used as an <b>approximate variance-covariance 
matrix </b> for the estimate, and used to produce approximate 
<b>standard errors</b> for the regression coefficients.

\f[
    \mathit{se}(c_i) = \left( (H)^{-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 condition number is computed as \f$ \kappa(H) \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.


@input

The training data is expected to be of the following form:\n
<pre>{TABLE|VIEW} <em>sourceName</em> (
    ...
    <em>dependentVariable</em> FLOAT8,
    <em>independentVariables</em> FLOAT8[],
    ...
)</pre>
Note: Dependent Variables refer to the time of death. There is no need to
pre-sort the data. Additionally, all the data is assumed


@usage
- Get vector of coefficients \f$ \boldsymbol \beta \f$ and all diagnostic
  statistics:\n
  <pre>SELECT * FROM \ref cox_prop_hazards(
    '<em>sourceName</em>', '<em>dependentVariable</em>', '<em>independentVariables</em>'
    [, <em>numberOfIterations</em> [, '<em>optimizer</em>' [, <em>precision</em> ] ] ]
);</pre>
  Output:
  Output:
  <pre>coef | log_likelihood | std_err | z_stats | p_values  | condition_no | num_iterations
                                               ...
</pre>
- Get vector of coefficients \f$ \boldsymbol \beta \f$:\n
  <pre>SELECT (\ref cox_prop_hazards('<em>sourceName</em>', '<em>dependentVariable</em>', '<em>independentVariables</em>')).coef;</pre>
- Get a subset of the output columns, e.g., only the array of coefficients
  \f$ \boldsymbol \beta \f$, the log-likelihood of determination:
  <pre>SELECT coef, log_likelihood
FROM \ref cox_prop_hazards('<em>sourceName</em>', '<em>dependentVariable</em>', '<em>independentVariables</em>');</pre>

@examp

-# Create the sample data set:
@verbatim 
sql> SELECT * FROM data;
      val   | time
------------|--------------
 {0,1.95}   | 35
 {0,2.20}   | 28
 {1,1.45}   | 32
 {1,5.25}   | 31
 {1,0.38}   | 21
...
@endverbatim
-# Run the cox regression function:
@verbatim
sql> SELECT * FROM cox_prop_hazards('data', 'val', 'time', 100, 'newto', 0.001);
---------------|--------------------------------------------------------------
coef           | {0.881089349817059,-0.0756817768938055}
log_likelihood | -4.46535157957394
std_err        | {1.16954914708414,0.338426252282655}
z_stats        | {0.753356711368689,-0.223628410729811}
p_values       | {0.451235588326831,0.823046454908087}
condition_no   | 12.1135391339082
num_iterations | 4

@endverbatim

@literature

A somewhat random selection of nice write-ups, with valuable pointers into
further literature:

[1] John Fox: Cox Proportional-Hazards Regression for Survival Data,
        Appendix to An R and S-PLUS companion to Applied Regression Feb 2012,
    http://cran.r-project.org/doc/contrib/Fox-Companion/appendix-cox-regression.pdf

[2] Stephen J Walters: What is a Cox model?
    http://www.medicine.ox.ac.uk/bandolier/painres/download/whatis/cox_model.pdf


@note Source and column names have to be passed as strings (due to limitations 
of the SQL syntax).


@sa File cox_prop_hazards.sql_in (documenting the SQL functions)

@internal
@sa Namespace cox_prop_hazards 
    \ref madlib::modules::stats documenting the implementation in C++
@endinternal

*/


DROP TYPE IF EXISTS MADLIB_SCHEMA.intermediate_cox_prop_hazards_result;
CREATE TYPE MADLIB_SCHEMA.intermediate_cox_prop_hazards_result AS (
    x DOUBLE PRECISION[],
    exp_coef_x DOUBLE PRECISION,
    x_exp_coef_x DOUBLE PRECISION[],
    x_xTrans_exp_coef_x DOUBLE PRECISION[]
);


CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.intermediate_cox_prop_hazards(
    /*+  x */ DOUBLE PRECISION[],
    /*+  coef */ DOUBLE PRECISION[])
RETURNS MADLIB_SCHEMA.intermediate_cox_prop_hazards_result AS
'MODULE_PATHNAME'
LANGUAGE c IMMUTABLE;



DROP TYPE IF EXISTS MADLIB_SCHEMA.cox_prop_hazards_result;
CREATE TYPE MADLIB_SCHEMA.cox_prop_hazards_result AS (
    coef DOUBLE PRECISION[],
    logLikelihood DOUBLE PRECISION,
    std_err DOUBLE PRECISION[],
    z_stats DOUBLE PRECISION[],
    p_values DOUBLE PRECISION[],
    condition_no DOUBLE PRECISION,
    num_iterations INTEGER
);




CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.cox_prop_hazards_step_final(
    state DOUBLE PRECISION[])
RETURNS DOUBLE PRECISION[]
AS 'MODULE_PATHNAME'
LANGUAGE C IMMUTABLE STRICT;

CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.cox_prop_hazards_step_transition(
    /*+  state */ DOUBLE PRECISION[],
    /*+  x */ DOUBLE PRECISION[],
    /*+  y */ DOUBLE PRECISION,     
    /*+  exp_coef_x */ DOUBLE PRECISION,
    /*+  xexp_coef_x */ DOUBLE PRECISION[],
    /*+  x_xTrans_exp_coef_x */ DOUBLE PRECISION[],
    /*+  previous_state */ DOUBLE PRECISION[])
RETURNS DOUBLE PRECISION[] AS 
'MODULE_PATHNAME'
LANGUAGE C IMMUTABLE;


/**
 * @internal
 * @brief Perform one iteration the Newton-Rhapson method.
 */
CREATE
m4_ifdef(`__GREENPLUM__',m4_ifdef(`__HAS_ORDERED_AGGREGATES__',`ORDERED'))
AGGREGATE MADLIB_SCHEMA.cox_prop_hazards_step(

    /*+  x */ DOUBLE PRECISION[],
    /*+  y */ DOUBLE PRECISION,
    /*+  exp_coef_x */ DOUBLE PRECISION,
    /*+  xexp_coef_x */ DOUBLE PRECISION[],
    /*+  x_xTrans_exp_coef_x */ DOUBLE PRECISION[],
    /*+ previous_state */ DOUBLE PRECISION[]) (    
    STYPE=DOUBLE PRECISION[],
    SFUNC=MADLIB_SCHEMA.cox_prop_hazards_step_transition,
    FINALFUNC=MADLIB_SCHEMA.cox_prop_hazards_step_final,
    INITCOND='{0,0,0,0,0,0,0}'
);



CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.internal_cox_prop_hazards_step_distance(
    /*+ state1 */ DOUBLE PRECISION[],
    /*+ state2 */ DOUBLE PRECISION[])
RETURNS DOUBLE PRECISION AS
'MODULE_PATHNAME'
LANGUAGE c IMMUTABLE STRICT;

CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.internal_cox_prop_hazards_result(
    /*+ state */ DOUBLE PRECISION[])
RETURNS MADLIB_SCHEMA.cox_prop_hazards_result AS
'MODULE_PATHNAME'
LANGUAGE c IMMUTABLE STRICT;


-- We only need to document the last one (unfortunately, in Greenplum we have to
-- use function overloading instead of default arguments).
CREATE FUNCTION MADLIB_SCHEMA.compute_cox_prop_hazards(
    "source" VARCHAR,
    "indepColumn" VARCHAR,
    "depColumn" VARCHAR,
    "maxNumIterations" INTEGER,
    "optimizer" VARCHAR,
    "precision" DOUBLE PRECISION)
RETURNS INTEGER
AS $$PythonFunction(stats, cox_prop_hazards, compute_cox_prop_hazards)$$
LANGUAGE plpythonu VOLATILE;

/**
 * @brief Compute cox-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 indepColumn Name of the independent column
 * @param depColumn Name of the dependant column measuring time of death
 * @param maxNumIterations The maximum number of iterations
 * @param optimizer The optimizer to use (either
 *        <tt>'newton'</tt>/<tt>'newton'</tt> for the newton method
 * @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 \c maxNumIterations iterations.
 *
 * @return A composite value:
 *  - <tt>coef FLOAT8[]</tt> - Array of coefficients, \f$ \boldsymbol \beta \f$
 *  - <tt>log_likelihood FLOAT8</tt> - Log-likelihood \f$l(\boldsymbol \beta)\f$
 *  - <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>condition_no FLOAT8</tt> - The condition number of matrix
 *    \f$ H \f$ during the iteration immediately <em>preceding</em>
 *    convergence (i.e., \f$ H \f$ is computed using the coefficients of the
 *    previous iteration)
 *  - <tt>num_iterations INTEGER</tt> - The number of iterations before the
 *    algorithm terminated
 *
 * - Get vector of coefficients \f$ \boldsymbol \beta \f$ and all diagnostic
 *  statistics:\n
 *  <pre>SELECT * FROM \ref cox_prop_hazards(
 *    '<em>sourceName</em>', '<em>dependentVariable</em>', 
 *      '<em>independentVariables</em>'
 *    [, <em>numberOfIterations</em> [, '<em>optimizer</em>' [, <em>precision</em> ] ] ]
 * );</pre>
 * - Get vector of coefficients \f$ \boldsymbol \beta \f$:\n
 *  <pre>SELECT (\ref cox_prop_hazards('<em>sourceName</em>', 
 * '<em>dependentVariable</em>', '<em>independentVariables</em>')).coef;</pre>
 * - Get a subset of the output columns, e.g., only the array of coefficients
 *  \f$ \boldsymbol \beta \f$, the log-likelihood of determination:
 *  <pre>SELECT coef, log_likelihood
 * FROM \ref cox_prop_hazards('<em>sourceName</em>', '<em>dependentVariable</em>',
 * '<em>independentVariables</em>');</pre>
 */
CREATE FUNCTION MADLIB_SCHEMA.cox_prop_hazards(
    "source" VARCHAR,
    "indepColumn" VARCHAR,
    "depColumn" VARCHAR,
    "maxNumIterations" INTEGER /*+ DEFAULT 20 */,
    "optimizer" VARCHAR /*+ DEFAULT 'newton' */,
    "precision" DOUBLE PRECISION /*+ DEFAULT 0.0001 */)
RETURNS MADLIB_SCHEMA.cox_prop_hazards_result AS $$
DECLARE
    theIteration INTEGER;
    fnName VARCHAR;
    theResult MADLIB_SCHEMA.cox_prop_hazards_result;
BEGIN
    theIteration := (
        SELECT MADLIB_SCHEMA.compute_cox_prop_hazards($1, $2, $3, $4, $5, $6)
    );
    IF optimizer = 'newton' THEN
        fnName := 'internal_cox_prop_hazards_result';
    ELSE
        RAISE EXCEPTION 'Unknown optimizer (''%'')', optimizer;
    END IF;
    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;


CREATE FUNCTION MADLIB_SCHEMA.cox_prop_hazards(
    "source" VARCHAR,
    "indepColumn" VARCHAR,
    "depColumn" VARCHAR)
RETURNS MADLIB_SCHEMA.cox_prop_hazards_result AS
$$SELECT MADLIB_SCHEMA.cox_prop_hazards($1, $2, $3, 20, 'newton', 0.0001);$$
LANGUAGE sql VOLATILE;

CREATE FUNCTION MADLIB_SCHEMA.cox_prop_hazards(
    "source" VARCHAR,
    "indepColumn" VARCHAR,
    "depColumn" VARCHAR,
    "maxNumIterations" INTEGER)
RETURNS MADLIB_SCHEMA.cox_prop_hazards_result AS
$$SELECT MADLIB_SCHEMA.cox_prop_hazards($1, $2, $3, $4, 'newton', 0.0001);$$
LANGUAGE sql VOLATILE;

CREATE FUNCTION MADLIB_SCHEMA.cox_prop_hazards(
    "source" VARCHAR,
    "indepColumn" VARCHAR,
    "depColumn" VARCHAR,
    "maxNumIterations" INTEGER,
    "optimizer" VARCHAR)
RETURNS MADLIB_SCHEMA.cox_prop_hazards_result AS
$$SELECT MADLIB_SCHEMA.cox_prop_hazards($1, $2, $3, $4, $5, 0.0001);$$
LANGUAGE sql VOLATILE;