/* ----------------------------------------------------------------------- *//**
 *
 * @file linear.sql_in
 *
 * @brief SQL functions for linear regression
 * @date January 2011
 *
 * @sa For a brief introduction to linear regression, see the module
 *     description \ref grp_linreg.
 *
 *//* ----------------------------------------------------------------------- */

m4_include(`SQLCommon.m4')

/**
@addtogroup grp_linreg


<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 Ordinary Least Squares Regression, models linear relationship between a dependent variable and one or more independent variables.

Linear regression models a linear relationship of a scalar dependent variable
\f$ y \f$ to one or more explanatory independent variables \f$ x \f$ and builds
a model of coefficients.

@anchor train
@par Training Function

The linear regression training function has the following syntax.
<pre class="syntax">
linregr_train( source_table,
               out_table,
               dependent_varname,
               independent_varname,
               grouping_cols,
               heteroskedasticity_option
             )
</pre>

\b Arguments
<DL class="arglist">
<DT>source_table</DT>
<DD>TEXT. Name of the table containing the training data.</DD>

<DT>out_table</DT>
<DD>TEXT. Name of the generated table containing the output model.

  The output table contains the following columns:
  <table class="output">
  <tr>
  <th>\<...></th>
  <td>Any grouping columns provided during training.
  Present only if the grouping option is used.</td>
  </tr>
  <tr>
  <th>coef</th>
  <td>FLOAT8[]. Vector of the coefficients of the regression.</td>
  </tr>
  <tr>
  <th>r2</th>
  <td>FLOAT8. R-squared coefficient of determination of the model.</td>
  </tr>
  <tr>
  <th>std_err</th>
  <td>FLOAT8[]. Vector of the standard error of the coefficients.</td>
  </tr>
  <tr>
  <th>t_stats</th>
  <td>FLOAT8[]. Vector of the t-statistics of the coefficients.</td>
  </tr>
  <tr>
  <th>p_values</th>
  <td>FLOAT8[]. Vector of the p-values of the coefficients.</td>
  </tr>
  <tr>
  <th>condition_no</th>
  <td>FLOAT8 array. The condition number of the \f$X^{*}X\f$
  matrix. A high condition number is usually an indication that there may be
  some numeric instability in the result yielding a less reliable model. A high
  condition number often results when there is a significant amount of
  colinearity in the underlying design matrix, in which case other regression
  techniques, such as elastic net regression, may be more appropriate.</td>
  </tr>
  <tr>
  <th>bp_stats</th>
  <td>FLOAT8. The Breush-Pagan statistic of heteroskedacity. Present
  only if the heteroskedacity argument was set to True when the model was
  trained.</td>
  </tr>
  <tr>
  <th>bp_p_value</th>
  <td>FLOAT8. The Breush-Pagan calculated p-value. Present only if
  the heteroskedacity parameter was set to True when the model was
  trained.</td>
  </tr>
  <tr>
  <th>num_rows_processed</th>
  <td>BIGINT. The number of rows that are actually used in each group.</td>
  </tr>
  <tr>
  <th>num_missing_rows_skipped</th>
  <td>INTEGER. The number of rows that have NULL values in the dependent and independent variables, and were skipped in the computation for each group.</td></tr>
  <tr>
  <th>variance_covariance</th>
  <td>FLOAT[]. Variance/covariance matrix.</td></tr>
  </table>

  A summary table named \<out_table\>_summary is created together with the output table. It has the following columns:
 <table class="output">
 <tr>
    <th>method</th>
    <td>'linregr' for linear regression.</td>
    </tr>
  <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>num_rows_processed</th>
<td>The total number of rows that were used in the computation.</td></tr>
<tr><th>num_missing_rows_skipped</th>
<td>The total number of rows that were skipped because of NULL values in them.</td>
    </tr>
<tr>
    <th>grouping_cols</th>
    <td>Names of the grouping columns.</td>
    </tr>
</table>
</dd>
<dt></dt>
<dd>@note 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 result may look different, they are
in fact the same.
</dd>

<DT>dependent_varname</DT>
<DD>TEXT. Expression to evaluate for the dependent variable.</DD>

<DT>independent_varname</DT>
<DD>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 <tt>1</tt> term in the independent variable list.</DD>

<DT>grouping_cols (optional)</DT>
<DD>TEXT, default: NULL. An expression list used to group the input dataset into discrete groups, running one regression per group. Similar to the SQL <tt>GROUP BY</tt> clause. When this value is null, no grouping is used and a
single result model is generated for the whole data set.</DD>

<DT>heteroskedasticity_option (optional)</DT>
<DD>BOOLEAN, default: FALSE. When TRUE, the heteroskedasticity of the model is also calculated and returned with the results.</DD>
</DL>

@anchor warning
@warning The aggregate 'linregr' has been deprecated in favor of the function
'linregr_train'. If the aggregate 'linregr' is used to output the results of
linear regression to a table, it is recommended to follow the general pattern
shown below (replace text within '<...>' with the appropriate variable names).
<pre class="syntax">
CREATE TABLE \<output table\> AS
SELECT (r).*
FROM (
    SELECT linregr(\<dependent variable\>, \<independent variable\>) as r
    FROM \<source table\>
    ) q;
</pre>


@anchor predict
@par Prediction Function
The prediction function is as follows:
<pre class="syntax">
linregr_predict(coef, col_ind)
</pre>
\b Arguments
<dl class="arglist">
<dt>coef</dt>
<dd>FLOAT8[]. Vector of the coefficients of regression from training.</dd>
<dt>col_ind</dt>
<dd>FLOAT8[]. An array containing the independent variable column names,
as was used for the training. </dd>

@anchor examples
@par Examples
-#  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);
INSERT INTO houses VALUES
  (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>
-#  Train a regression model. First, we generate a single regression for all data.
<pre class="example">
DROP TABLE IF EXISTS houses_linregr, houses_linregr_summary;
SELECT madlib.linregr_train( 'houses',
                             'houses_linregr',
                             'price',
                             'ARRAY[1, tax, bath, size]'
                           );
</pre>
(Note that in this example we are dynamically creating the array of independent variables
from column names.  If you have large numbers of independent variables beyond the PostgreSQL
limit of maximum columns per table, you would pre-build the arrays and store them in a
single column.)
-# Next we generate three output models, one for each value of "bedroom".
<pre class="example">
DROP TABLE IF EXISTS houses_linregr_bedroom, houses_linregr_bedroom_summary;
SELECT madlib.linregr_train( 'houses',
                             'houses_linregr_bedroom',
                             'price',
                             'ARRAY[1, tax, bath, size]',
                             'bedroom'
                           );
</pre>
-# Examine the resulting models.
<pre class="example">
-- Set extended display on for easier reading of output
\\x ON
SELECT * FROM houses_linregr;
</pre>
Result:
<pre class="result">
-[ RECORD 1 ]+---------------------------------------------------------------------------
coef                     | {-12849.4168959872,28.9613922651765,10181.6290712648,50.516894915354}
r2                       | 0.768577580597443
std_err                  | {33453.0344331391,15.8992104963997,19437.7710925923,32.928023174087}
t_stats                  | {-0.38410317968819,1.82156166004184,0.523806408809133,1.53416118083605}
p_values                 | {0.708223134615422,0.0958005827189772,0.610804093526536,0.153235085548186}
condition_no             | 9002.50457085737
num_rows_processed       | 15
num_missing_rows_skipped | 0
variance_covariance      | {{1119105512.78479,217782.067878023,-283344228.394562,-616679.69319088}, ...
</pre>
Alternatively you can unnest the results for easier reading of output.
<pre class="example">
\\x OFF
SELECT unnest(ARRAY['intercept','tax','bath','size']) as attribute,
       unnest(coef) as coefficient,
       unnest(std_err) as standard_error,
       unnest(t_stats) as t_stat,
       unnest(p_values) as pvalue
FROM houses_linregr;
</pre>
Result:
<pre class="result">
 attribute |    coefficient    |  standard_error  |      t_stat       |       pvalue
-----------+-------------------+------------------+-------------------+--------------------
 intercept | -12849.4168959872 | 33453.0344331391 | -0.38410317968819 |  0.708223134615422
 tax       |  28.9613922651765 | 15.8992104963997 |  1.82156166004184 | 0.0958005827189772
 bath      |  10181.6290712648 | 19437.7710925923 | 0.523806408809133 |  0.610804093526536
 size      |   50.516894915354 |  32.928023174087 |  1.53416118083605 |  0.153235085548186
(4 rows)
</pre>
-# View the results grouped by bedroom.
<pre class="example">
\\x ON
SELECT * FROM houses_linregr_bedroom ORDER BY bedroom;
</pre>
Result:
<pre class="result">
-[ RECORD 1 ]------------+----------------------------------------------------------------
bedroom                  | 4
coef                     | {0.0112536020318378,41.4132554771633,0.0225072040636757,31.3975496688276}
r2                       | 1
std_err                  | {0,0,0,0}
t_stats                  | {Infinity,Infinity,Infinity,Infinity}
p_values                 |
condition_no             | Infinity
num_rows_processed       | 1
num_missing_rows_skipped | 0
variance_covariance      | {{0,0,0,0},{0,0,0,0},{0,0,0,0},{0,0,0,0}}
-[ RECORD 2 ]------------+----------------------------------------------------------------
bedroom                  | 3
coef                     | {-88155.8292501601,27.1966436294429,41404.0293363612,62.637521075324}
r2                       | 0.841699901311252
std_err                  | {57867.9999702625,17.8272309154689,43643.1321511114,70.8506824863954}
t_stats                  | {-1.52339512849005,1.52556747362508,0.948695185143966,0.884077878676067}
p_values                 | {0.188161432894871,0.187636685729869,0.386340032374927,0.417132778705789}
condition_no             | 11722.6225642147
num_rows_processed       | 9
num_missing_rows_skipped | 0
variance_covariance      | {{3348705420.5583,433697.545104226,-70253017.45773,-2593488.13800193}, ...
-[ RECORD 3 ]------------+----------------------------------------------------------------
bedroom                  | 2
coef                     | {-84242.0345406597,55.4430144648696,-78966.9753675319,225.611910021192}
r2                       | 0.968809546465313
std_err                  | {35018.9991665742,19.5731125320686,23036.8071292552,49.0448678148784}
t_stats                  | {-2.40560942761235,2.83261103077151,-3.42786111480046,4.60011251070697}
p_values                 | {0.250804617665239,0.21605133377602,0.180704400437373,0.136272031474122}
condition_no             | 10086.1048721726
num_rows_processed       | 5
num_missing_rows_skipped | 0
variance_covariance      | {{1226330302.62852,-300921.595596804,551696673.397849,-1544160.63236119}, ...
</pre>

-# Compare predicted price with actual.  (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
SELECT houses.*,
       madlib.linregr_predict( m.coef,
                               ARRAY[1,tax,bath,size]
                             ) as predict,
        price -
          madlib.linregr_predict( m.coef,
                                  ARRAY[1,tax,bath,size]
                                ) as residual
FROM houses, houses_linregr m ORDER BY id;
</pre>
Result:
<pre class="result">
 id | tax  | bedroom | bath | price  | size |  lot  |     predict      |     residual
----+------+---------+------+--------+------+-------+------------------+-------------------
  1 |  590 |       2 |    1 |  50000 |  770 | 22100 | 53317.4426965542 | -3317.44269655424
  2 | 1050 |       3 |    2 |  85000 | 1410 | 12000 | 109152.124955627 | -24152.1249556268
  3 |   20 |       3 |    1 |  22500 | 1060 |  3500 | 51459.3486308563 | -28959.3486308563
  4 |  870 |       2 |    2 |  90000 | 1300 | 17500 |  98382.215907206 | -8382.21590720605
  5 | 1320 |       3 |    2 | 133000 | 1500 | 30000 | 121518.221409606 |  11481.7785903937
  6 | 1350 |       2 |    1 |  90500 |  820 | 25700 | 77853.9455638561 |  12646.0544361439
  7 | 2790 |       3 |  2.5 | 260000 | 2130 | 25000 | 201007.926371721 |  58992.0736282788
  8 |  680 |       2 |    1 | 142500 | 1170 | 22000 | 76130.7259665617 |  66369.2740334383
  9 | 1840 |       3 |    2 | 160000 | 1500 | 19000 | 136578.145387498 |  23421.8546125019
 10 | 3680 |       4 |    2 | 240000 | 2790 | 20000 |  255033.90159623 | -15033.9015962295
 11 | 1660 |       3 |    1 |  87000 | 1030 | 17500 | 97440.5250982852 | -10440.5250982852
 12 | 1620 |       3 |    2 | 118600 | 1250 | 20000 | 117577.415360321 |  1022.58463967926
 13 | 3100 |       3 |    2 | 140000 | 1760 | 38000 | 186203.892319613 | -46203.8923196126
 14 | 2070 |       2 |    3 | 148000 | 1550 | 14000 | 155946.739425521 | -7946.73942552117
 15 |  650 |       3 |  1.5 |  65000 | 1450 | 12000 | 94497.4293105379 | -29497.4293105379
(15 rows)
</pre>

-# Compare predicted price with actual with grouping.
It means a different model is used depending on the number of bedrooms.
<pre class="example">
\\x OFF
SELECT houses.*,
       madlib.linregr_predict( m.coef,
                              ARRAY[1,tax,bath,size]
                             ) as predict,
        price -
          madlib.linregr_predict( m.coef,
                                 ARRAY[1,tax,bath,size]
                                ) as residual
FROM houses, houses_linregr_bedroom m
WHERE houses.bedroom = m.bedroom
ORDER BY id;
</pre>
Result:
<pre class="result">
 id | tax  | bedroom | bath | price  | size |  lot  |     predict      |     residual
----+------+---------+------+--------+------+-------+------------------+-------------------
  1 |  590 |       2 |    1 |  50000 |  770 | 22100 | 43223.5393423978 |  6776.46065760222
  2 | 1050 |       3 |    2 |  85000 | 1410 | 12000 | 111527.609949684 |  -26527.609949684
  3 |   20 |       3 |    1 |  22500 | 1060 |  3500 | 20187.9052986341 |  2312.09470136587
  4 |  870 |       2 |    2 |  90000 | 1300 | 17500 | 99354.9203362612 | -9354.92033626116
  5 | 1320 |       3 |    2 | 133000 | 1500 | 30000 | 124508.080626412 |  8491.91937358756
  6 | 1350 |       2 |    1 |  90500 |  820 | 25700 | 96640.8258367579 |  -6140.8258367579
  7 | 2790 |       3 |  2.5 | 260000 | 2130 | 25000 | 224650.799707327 |  35349.2002926733
  8 |  680 |       2 |    1 | 142500 | 1170 | 22000 | 138458.174652714 |  4041.82534728572
  9 | 1840 |       3 |    2 | 160000 | 1500 | 19000 | 138650.335313722 |  21349.6646862777
 10 | 3680 |       4 |    2 | 240000 | 2790 | 20000 |           240000 |                 0
 11 | 1660 |       3 |    1 |  87000 | 1030 | 17500 | 62911.2752186594 |  24088.7247813406
 12 | 1620 |       3 |    2 | 118600 | 1250 | 20000 | 117007.693446414 |  1592.30655358579
 13 | 3100 |       3 |    2 | 140000 | 1760 | 38000 | 189203.861766403 | -49203.8617664034
 14 | 2070 |       2 |    3 | 148000 | 1550 | 14000 | 143322.539831869 |  4677.46016813093
 15 |  650 |       3 |  1.5 |  65000 | 1450 | 12000 | 82452.4386727394 | -17452.4386727394
(15 rows)
</pre>


@anchor notes
@par Note
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"').


@anchor background
@par Technical Background

Ordinary least-squares (OLS) linear regression refers to a stochastic model in
which the conditional mean of the dependent variable (usually denoted \f$ Y \f$)
is an affine function of the vector of independent variables (usually denoted
\f$ \boldsymbol x \f$). That is,
\f[
    E[Y \mid \boldsymbol x] = \boldsymbol c^T \boldsymbol x
\f]
for some unknown vector of coefficients \f$ \boldsymbol c \f$. The assumption is
that the residuals are i.i.d. distributed Gaussians. That is, the (conditional)
probability density of \f$ Y \f$ is given by
\f[
    f(y \mid \boldsymbol x)
    =   \frac{1}{\sqrt{2 \pi \sigma^2}}
        \cdot \exp\left(-\frac{1}{2 \sigma^2}
            \cdot (y - \boldsymbol x^T \boldsymbol c)^2 \right)
    \,.
\f]
OLS linear regression finds the vector of coefficients \f$ \boldsymbol c \f$
that maximizes the likelihood of the observations.

Let
- \f$ \boldsymbol y \in \mathbf R^n \f$ denote the vector of observed dependent
    variables, with \f$ n \f$ rows, 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,
- \f$ X^T \f$ denote the transpose of \f$ X \f$,
- \f$ X^+ \f$ denote the pseudo-inverse of \f$ X \f$.

Maximizing the likelihood is equivalent to maximizing the log-likelihood
\f$ \sum_{i=1}^n \log f(y_i \mid \boldsymbol x_i) \f$, which simplifies to
minimizing the <b>residual sum of squares</b> \f$ RSS \f$ (also called sum of
squared residuals or sum of squared errors of prediction),
\f[
    RSS = \sum_{i=1}^n ( y_i - \boldsymbol c^T \boldsymbol x_i )^2
        = (\boldsymbol y - X \boldsymbol c)^T (\boldsymbol y - X \boldsymbol c)
    \,.
\f]
The first-order conditions yield that the \f$ RSS \f$ is minimized at
\f[
    \boldsymbol c = (X^T X)^+ X^T \boldsymbol y
    \,.
\f]

Computing the <b>total sum of squares</b> \f$ TSS \f$, the <b>explained
sum of squares</b> \f$ ESS \f$ (also called the regression sum of
squares), and the <b>coefficient of determination</b> \f$ R^2 \f$ is
done according to the following formulas:
\f{align*}{
    ESS & = \boldsymbol y^T X \boldsymbol c
        -   \frac{ \| y \|_1^2 }{n} \\
    TSS & = \sum_{i=1}^n y_i^2
        -   \frac{ \| y \|_1^2 }{n} \\
    R^2 & = \frac{ESS}{TSS}
\f}
Note: The last equality follows from the definition
\f$ R^2 = 1 - \frac{RSS}{TSS} \f$ and the fact that for linear regression
\f$ TSS = RSS + ESS \f$. A proof of the latter can be found, e.g., at:
http://en.wikipedia.org/wiki/Sum_of_squares

We estimate the variance
\f$ Var[Y - \boldsymbol c^T \boldsymbol x \mid \boldsymbol x] \f$ as
\f[
    \sigma^2 = \frac{RSS}{n - k}
\f]
and compute the t-statistic for coefficient \f$ i \f$ as
\f[
    t_i = \frac{c_i}{\sqrt{\sigma^2 \cdot \left( (X^T X)^{-1} \right)_{ii} }}
    \,.
\f]

The \f$ p \f$-value for coefficient \f$ i \f$ gives the probability 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_\nu \f$ denote the
cumulative density function of student-t with \f$ \nu \f$ degrees of freedom,
the \f$ p \f$-value for coefficient \f$ i \f$
is therefore
\f[
    p_i = \Pr(|T| \geq |t_i|) = 2 \cdot (1 - F_{n - k}( |t_i| ))
\f]
where \f$ T \f$ is a student-t distributed random variable with mean 0.

The condition number [2] \f$ \kappa(X) = \|X\|_2\cdot\|X^{-1}\|_2\f$ is computed
as the product of two spectral norms [3]. The spectral norm of a matrix \f$X\f$
is the largest singular value of \f$X\f$ i.e. the square root of the largest
eigenvalue of the positive-semidefinite matrix \f$X^{*}X\f$:

\f[
    \|X\|_2 = \sqrt{\lambda_{\max}\left(X^{*}X\right)}\ ,
\f]
where \f$X^{*}\f$ is the conjugate transpose of \f$X\f$.
The condition number of a linear regression problem
is a worst-case measure of how sensitive the
result is to small perturbations of the input. A large condition number (say,
more than 1000) indicates the presence of significant multicollinearity.

@anchor literature
@literature

[1] Cosma Shalizi: Statistics 36-350: Data Mining, Lecture Notes, 21 October
    2009, http://www.stat.cmu.edu/~cshalizi/350/lectures/17/lecture-17.pdf

[2] Wikipedia: Condition Number, http://en.wikipedia.org/wiki/Condition_number.

[3] Wikipedia: Spectral Norm,
    http://en.wikipedia.org/wiki/Spectral_norm#Spectral_norm

[4] Wikipedia: Breusch–Pagan test,
    http://en.wikipedia.org/wiki/Breusch%E2%80%93Pagan_test

[5] Wikipedia: Heteroscedasticity-consistent standard errors,
http://en.wikipedia.org/wiki/Heteroscedasticity-consistent_standard_errors


@anchor related
@par Related Topics

@ref grp_robust

@ref grp_clustered_errors

@ref grp_validation

File linear.sql_in, source file for the SQL functions


@internal
@sa Namespace \ref madlib::modules::regress
    documenting the implementation in C++
@endinternal
*/

---------------------------------------------------------------------------
-- User facing functions
---------------------------------------------------------------------------
/**
 * @brief Linear regression training function with grouping support.
 */
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.linregr_train(
    source_table                VARCHAR,    -- name of input  table
    out_table                   VARCHAR,    -- name of output table
    dependent_varname           VARCHAR,    -- name of dependent variable
    independent_varname         VARCHAR,    -- name of independent variable
    grouping_cols               VARCHAR,    -- names of columns to group-by
    heteroskedasticity_option   BOOLEAN     -- do heteroskedasticity test or not
) RETURNS VOID AS $$
PythonFunction(regress, linear, linregr_train)
$$ LANGUAGE plpythonu
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA', `');

CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.linregr_train(
    source_table                VARCHAR,    -- name of input  table
    out_table                   VARCHAR,    -- name of output table
    dependent_varname           VARCHAR,    -- name of dependent variable
    independent_varname         VARCHAR,    -- name of independent variable
    grouping_cols               VARCHAR     -- names of columns to group-by
) RETURNS VOID AS $$
    SELECT MADLIB_SCHEMA.linregr_train($1, $2, $3, $4, $5, FALSE);
$$ LANGUAGE sql
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA', `');

CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.linregr_train(
    source_table                VARCHAR,    -- name of input  table
    out_table                   VARCHAR,    -- name of output table
    dependent_varname           VARCHAR,    -- name of dependent variable
    independent_varname         VARCHAR     -- name of independent variable
) RETURNS VOID AS $$
    SELECT MADLIB_SCHEMA.linregr_train($1, $2, $3, $4, NULL, FALSE);
$$ LANGUAGE sql
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA', `');
---------------------------------------------------------------------------

-----------------------------------------------------------------------
-- Online help function
-----------------------------------------------------------------------

----------------------------------------------------------------------

CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.linregr_train()
RETURNS VARCHAR AS $$
BEGIN
    RETURN MADLIB_SCHEMA.linregr_train('');
END;
$$ LANGUAGE plpgsql IMMUTABLE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `CONTAINS SQL', `');

----------------------------------------------------------------------

CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.linregr_train(
     message VARCHAR     -- usage string
)
RETURNS VARCHAR AS $$
PythonFunction(regress, linear, linregr_help_message)
$$ LANGUAGE plpythonu IMMUTABLE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `CONTAINS SQL', `');

----------------------------------------------------------------------


-- Deprecated functions --------------------------------------------------------

---------------------------------------------------------------------------
-- Result Types
---------------------------------------------------------------------------

DROP TYPE IF EXISTS MADLIB_SCHEMA.linregr_result CASCADE;
CREATE TYPE MADLIB_SCHEMA.linregr_result AS (
    coef            DOUBLE PRECISION[],
    r2              DOUBLE PRECISION,
    std_err         DOUBLE PRECISION[],
    t_stats         DOUBLE PRECISION[],
    p_values        DOUBLE PRECISION[],
    condition_no    DOUBLE PRECISION,
    num_processed   BIGINT,
    vcov            DOUBLE PRECISION[]
);

DROP TYPE IF EXISTS MADLIB_SCHEMA.heteroskedasticity_test_result CASCADE;
CREATE TYPE MADLIB_SCHEMA.heteroskedasticity_test_result AS (
    bp_stats DOUBLE PRECISION,
    bp_p_value DOUBLE PRECISION
);
---------------------------------------------------------------------------


---------------------------------------------------------------------------
-- Functions for user-defined aggregates
---------------------------------------------------------------------------
-- normal
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.linregr_transition(
    state MADLIB_SCHEMA.bytea8,
    y DOUBLE PRECISION,
    x DOUBLE PRECISION[])
RETURNS MADLIB_SCHEMA.bytea8
AS 'MODULE_PATHNAME'
LANGUAGE C
IMMUTABLE STRICT
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `NO SQL', `');


CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.linregr_merge_states(
    state1 MADLIB_SCHEMA.bytea8,
    state2 MADLIB_SCHEMA.bytea8)
RETURNS MADLIB_SCHEMA.bytea8
AS 'MODULE_PATHNAME'
LANGUAGE C
IMMUTABLE STRICT
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `NO SQL', `');

CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.linregr_final(
    state MADLIB_SCHEMA.bytea8)
RETURNS MADLIB_SCHEMA.linregr_result
AS 'MODULE_PATHNAME'
LANGUAGE C IMMUTABLE STRICT
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `NO SQL', `');

-- hetero
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.hetero_linregr_transition(
    state MADLIB_SCHEMA.bytea8,
    y DOUBLE PRECISION,
    x DOUBLE PRECISION[],
    coef DOUBLE PRECISION[])
RETURNS MADLIB_SCHEMA.bytea8
AS 'MODULE_PATHNAME'
LANGUAGE C
IMMUTABLE STRICT
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `NO SQL', `');


CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.hetero_linregr_merge_states(
    state1 MADLIB_SCHEMA.bytea8,
    state2 MADLIB_SCHEMA.bytea8)
RETURNS MADLIB_SCHEMA.bytea8
AS 'MODULE_PATHNAME'
LANGUAGE C
IMMUTABLE STRICT
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `NO SQL', `');


-- Final functions
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.hetero_linregr_final(
    state MADLIB_SCHEMA.bytea8)
RETURNS MADLIB_SCHEMA.heteroskedasticity_test_result
AS 'MODULE_PATHNAME'
LANGUAGE C IMMUTABLE STRICT
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `NO SQL', `');
---------------------------------------------------------------------------


---------------------------------------------------------------------------
-- User-defined aggregates
---------------------------------------------------------------------------
/**
 * @brief Compute linear regression coefficients and diagnostic statistics.
 *
 * @param dependentVariable Column containing the dependent variable
 * @param independentVariables Column containing the array of independent variables
 *
 * @par
 * To include an intercept in the model, set one coordinate in the
 * <tt>independentVariables</tt> array to 1.
 *
 * @return A composite value:
 *  - <tt>coef FLOAT8[]</tt> - Array of coefficients, \f$ \boldsymbol c \f$
 *  - <tt>r2 FLOAT8</tt> - Coefficient of determination, \f$ R^2 \f$
 *  - <tt>std_err FLOAT8[]</tt> - Array of standard errors,
 *    \f$ \mathit{se}(c_1), \dots, \mathit{se}(c_k) \f$
 *  - <tt>t_stats FLOAT8[]</tt> - Array of t-statistics, \f$ \boldsymbol t \f$
 *  - <tt>p_values FLOAT8[]</tt> - Array of p-values, \f$ \boldsymbol p \f$
 *  - <tt>condition_no FLOAT8</tt> - The condition number of matrix
 *    \f$ X^T X \f$.
 *
 * @usage
 *  - Get vector of coefficients \f$ \boldsymbol c \f$ and all diagnostic
 *    statistics:\n
 *    <pre>SELECT (linregr(<em>dependentVariable</em>,
 *        <em>independentVariables</em>)).*
 *FROM <em>sourceName</em>;</pre>
 *  - Get vector of coefficients \f$ \boldsymbol c \f$:\n
 *    <pre>SELECT (linregr(<em>dependentVariable</em>,
 *        <em>independentVariables</em>)).coef
 *FROM <em>sourceName</em>;</pre>
 *  - Get a subset of the output columns, e.g., only the array of coefficients
 *    \f$ \boldsymbol c \f$, the coefficient of determination \f$ R^2 \f$, and
 *    the array of p-values \f$ \boldsymbol p \f$:
 *    <pre>SELECT (lr).coef, (lr).r2, (lr).p_values
 *FROM (
 *    SELECT linregr( <em>dependentVariable</em>,
 *                    <em>independentVariables</em>) AS lr
 *    FROM <em>sourceName</em>
 *) AS subq;</pre>
 */
DROP AGGREGATE IF EXISTS MADLIB_SCHEMA.linregr(DOUBLE PRECISION, DOUBLE PRECISION []);
CREATE AGGREGATE MADLIB_SCHEMA.linregr(
    /*+ "dependentVariable" */ DOUBLE PRECISION,
    /*+ "independentVariables" */ DOUBLE PRECISION[]) (

    SFUNC=MADLIB_SCHEMA.linregr_transition,
    STYPE=MADLIB_SCHEMA.bytea8,
    FINALFUNC=MADLIB_SCHEMA.linregr_final,
    m4_ifdef(`__POSTGRESQL__', `', `prefunc=MADLIB_SCHEMA.linregr_merge_states,')
    INITCOND=''
);

/**
 * @brief Compute studentized Breuch-Pagan heteroskedasticity test for
 * linear regression.
 *
 * @param dependentVariable Column containing the dependent variable
 * @param independentVariables Column containing the array of independent variables
 * @param olsCoefficients Column containing the array of the OLS coefficients (as obtained by linregr)
 *
 * @par
 * To include an intercept in the model, set one coordinate in the
 * <tt>independentVariables</tt> array to 1.
 *
 * @return A composite value:
 *  - <tt>test_statistic FLOAT8[]</tt> - Prob > test_statistc
 *  - <tt>p_value FLOAT8[]</tt> - Prob > test_statistc
 *
 * @usage
 *  <pre> SELECT (heteoskedasticity_test_linregr(<em>dependentVariable</em>,
 *  <em>independentVariables</em>, coef)).*
 *  FROM (
 *    SELECT linregr(<em>dependentVariable</em>, <em>independentVariables</em>).coef
 *  ) AS ols_coef, <em>sourceName</em> as src;
 * </pre>
 */
DROP AGGREGATE IF EXISTS MADLIB_SCHEMA.heteroskedasticity_test_linregr(
    DOUBLE PRECISION, DOUBLE PRECISION [], DOUBLE PRECISION []);
CREATE AGGREGATE MADLIB_SCHEMA.heteroskedasticity_test_linregr(
    /*+ "dependentVariable" */ DOUBLE PRECISION,
    /*+ "independentVariables" */ DOUBLE PRECISION[],
    /*+ "olsCoefficients" */ DOUBLE PRECISION[]) (

    SFUNC=MADLIB_SCHEMA.hetero_linregr_transition,
    STYPE=MADLIB_SCHEMA.bytea8,
    FINALFUNC=MADLIB_SCHEMA.hetero_linregr_final,
    m4_ifdef(`__POSTGRESQL__', `', `prefunc=MADLIB_SCHEMA.hetero_linregr_merge_states,')
    INITCOND=''
);
---------------------------------------------------------------------------

/**
 * @brief Predict the boolean value of a dependent variable for a specific
 *        independent variable value in a linear regression model
 *
 * @param coef Coefficients obtained by running linear regression.
 * @param col_ind Independent variable array
 * @returns DOUBLE PRECISION Predicted value
 *
 * This function computes the dot product of the independent variables and the
 * coefficients. This requires the length of the two vectors to be the same.
 *
 */
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.linregr_predict(
    coef            DOUBLE PRECISION[],
    col_ind_var     DOUBLE PRECISION[]
) RETURNS DOUBLE PRECISION
AS 'MODULE_PATHNAME', 'array_dot'
LANGUAGE c IMMUTABLE STRICT;

-- Help messages -------------------------------------------------------
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.linregr_predict(
    message    TEXT
) RETURNS TEXT AS $$
    PythonFunction(regress, linear, linregr_predict_help_message)
$$ LANGUAGE plpythonu IMMUTABLE;

CREATE OR REPLACE FUNCTION MADLIB_SCHEMA.linregr_predict()
RETURNS TEXT
AS $$
    SELECT MADLIB_SCHEMA.linregr_predict(''::TEXT);
$$
LANGUAGE sql IMMUTABLE;
-------------------------------------------------------------