/* ----------------------------------------------------------------------- *//**
*
* @file pca_project.sql_in
*
* @brief Principal Component Analysis Projection
*
* @sa For a brief introduction to Principal Component Analysis, see the module
* description \ref grp_pca.
*
*//* ----------------------------------------------------------------------- */
m4_include(`SQLCommon.m4')
/**
@addtogroup grp_pca_project
@brief Projects a higher dimensional data point to a lower
dimensional subspace spanned by principal components learned through the PCA training
procedure.
Principal component projection is a mathematical procedure that projects high
dimensional data onto a lower dimensional space. This lower dimensional space
is defined by the \f$ k \f$ principal components with the highest variance in
the training data.
More details on the mathematics of PCA can be found in \ref
grp_pca_train and some details about principal component projection calculations
can be found in the \ref background_project "Technical Background".
@anchor project
@par Projection Function
The projection functions are slightly different for dense and sparse matrices. For dense matrices:
madlib.pca_project( source_table,
pc_table,
out_table,
row_id,
residual_table,
result_summary_table
)
For sparse matrices:
madlib.pca_sparse_project( source_table,
pc_table,
out_table,
row_id,
col_id, -- Sparse matrices only
val_id, -- Sparse matrices only
row_dim, -- Sparse matrices only
col_dim, -- Sparse matrices only
residual_table,
result_summary_table
)
\b Arguments
- source_table
- TEXT. Source table name.
Identical to \ref pca_train, the input data matrix should have \f$ N \f$ rows
and \f$ M \f$ columns, where \f$ N \f$ is the number of data points, and \f$ M
\f$ is the number of features for each data point.
The input table for pca_project is expected to be in the one of the
two standard MADlib dense matrix formats, and the sparse input table for
pca_sparse_project should be in the standard MADlib sparse matrix format.
These formats are described in the documentation for \ref grp_pca_train.
- pc_table
- TEXT. Table name for the table containing principal components.
- out_table
- TEXT. Name of the table that will contain the low-dimensional representation of the input data.
The out_table encodes a dense matrix with the projection onto the
principal components. The table has the following columns:
| row_id |
Row id of the output matrix. |
| row_vec |
A vector containing elements in the row of the matrix. |
- row_id
- TEXT. Column name containing the row IDs in the input source table. The column
should be of type INT (or a type that can be cast to INT) and should only contain values between 1 and
N. For dense matrix format, it should contain all continguous integers from 1 to N
describing the full matrix.
- col_id
- TEXT. Column name containing the column IDs in sparse matrix representation.
The column should be of type INT (or a type that can be cast to INT)
and should only contain values between 1 and M. This parameter applies to
sparse matrices only.
- val_id
- TEXT. Name of 'val_id' column in sparse matrix representation defining the values of the nonzero entries.
This parameter applies to sparse matrices only.
- row_dim
- INTEGER. The actual number of rows in the matrix. That is,
if the matrix was transformed into dense format, this is the number of rows
it would have.
This parameter applies to sparse matrices only.
- col_dim
- INTEGER. The actual number of columns in the matrix. That is,
if the matrix was transformed into dense format, this is the number of columns
it would have.
This parameter applies to sparse matrices only.
@note The parameters 'row_dim' and 'col_dim' could actually be inferred from the
sparse matrix representation, so they will be removed in the future.
For now they are maintained for backward compatability so you must enter them.
Making 'row_dim' or 'col_dim' larger than the actual matrix has the effect of padding it with
zeros, which is probably not useful.
- residual_table (optional)
- TEXT, default: NULL. Name of the optional residual table.
The residual_table encodes a dense residual matrix. The table has the following columns:
| row_id |
Row id of the output matrix. |
| row_vec |
A vector containing elements in the row of the residual matrix. |
- result_summary_table (optional)
- TEXT, default: NULL. Name of the optional summary table.
The result_summary_table contains information about the performance time of the PCA projection. The table has the following columns:
| exec_time |
Elapsed time (ms) for execution of the function. |
| residual_norm |
Absolute error of the residuals. |
| relative_residual_norm |
Relative error of the residuals. |
@anchor examples
@examp
-# View online help for the PCA projection function:
SELECT madlib.pca_project();
-# Create sample data in dense matrix form:
DROP TABLE IF EXISTS mat;
CREATE TABLE mat (id integer,
row_vec double precision[]
);
INSERT INTO mat VALUES
(1, '{1,2,3}'),
(2, '{2,1,2}'),
(3, '{3,2,1}');
-# Run the PCA function for a specified number of principal components and view the results:
DROP TABLE IF EXISTS result_table, result_table_mean;
SELECT madlib.pca_train('mat', -- Source table
'result_table', -- Output table
'id', -- Row id of source table
2); -- Number of principal components
SELECT * FROM result_table ORDER BY row_id;
row_id | principal_components | std_dev | proportion
--------+--------------------------------------------------------------+-------------------+-------------------
1 | {0.707106781186547,-6.93889390390723e-18,-0.707106781186548} | 1.41421356237309 | 0.857142857142244
2 | {0,1,0} | 0.577350269189626 | 0.142857142857041
(2 rows)
-# Project the original data to a lower dimensional representation and view the result of the projection:
DROP TABLE IF EXISTS residual_table, result_summary_table, out_table;
SELECT madlib.pca_project( 'mat',
'result_table',
'out_table',
'id',
'residual_table',
'result_summary_table'
);
SELECT * FROM out_table ORDER BY row_id;
row_id | row_vec
--------+--------------------------------------
1 | {-1.41421356237309,-0.33333333333}
2 | {2.77555756157677e-17,0.66666666667}
3 | {1.41421356237309,-0.33333333333}
(3 rows)
Check the error in the projection:
SELECT * FROM result_summary_table;
exec_time | residual_norm | relative_residual_norm
---------------+-------------------+------------------------
331.792116165 | 5.89383520611e-16 | 9.68940539229e-17
(1 row)
Check the residuals:
SELECT * FROM residual_table ORDER BY row_id;
row_id | row_vec
--------+--------------------------------------------------------------------
1 | {-2.22044604925031e-16,-1.11022302462516e-16,3.33066907387547e-16}
2 | {-1.12243865646685e-18,0,4.7381731349413e-17}
3 | {2.22044604925031e-16,1.11022302462516e-16,-3.33066907387547e-16}
(3 rows)
-# Now we use grouping in dense form to learn different models for different groups.
First, we create sample data in dense matrix form with a grouping column.
Note we actually have different matrix sizes for the different groups, which
is allowed for dense:
DROP TABLE IF EXISTS mat_group;
CREATE TABLE mat_group (
id integer,
row_vec double precision[],
matrix_id integer
);
INSERT INTO mat_group VALUES
(1, '{1,2,3}', 1),
(2, '{2,1,2}', 1),
(3, '{3,2,1}', 1),
(4, '{1,2,3,4,5}', 2),
(5, '{2,5,2,4,1}', 2),
(6, '{5,4,3,2,1}', 2);
-# Run the PCA function with grouping for a specified proportion of variance and view the results:
DROP TABLE IF EXISTS result_table_group, result_table_group_mean;
SELECT madlib.pca_train('mat_group', -- Source table
'result_table_group', -- Output table
'id', -- Row id of source table
0.8, -- Proportion of variance
'matrix_id'); -- Grouping column
SELECT * FROM result_table_group ORDER BY matrix_id, row_id;
row_id | principal_components | std_dev | proportion | matrix_id
--------+------------------------------------------------------------------------------------------------+-----------------+-------------------+-----------
1 | {0.707106781186548,0,-0.707106781186547} | 1.4142135623731 | 0.857142857142245 | 1
1 | {-0.555378486712784,-0.388303582074091,0.0442457354870796,0.255566375612852,0.688115693174023} | 3.2315220311722 | 0.764102534485173 | 2
2 | {0.587384101786277,-0.485138064894743,0.311532046315153,-0.449458074050715,0.347212037159181} | 1.795531127192 | 0.235897465516047 | 2
(3 rows)
-# Run the PCA projection on subsets of an input table based on grouping columns.
Note that the parameter 'pc_table' used for projection must be generated in training
using the same grouping columns.
DROP TABLE IF EXISTS mat_group_projected;
SELECT madlib.pca_project('mat_group',
'result_table_group',
'mat_group_projected',
'id');
SELECT * FROM mat_group_projected ORDER BY matrix_id, row_id;
row_id | row_vec | matrix_id
--------+---------------------------------------+-----------
1 | {1.4142135623731} | 1
2 | {7.40148683087139e-17} | 1
3 | {-1.4142135623731} | 1
4 | {-3.59290479201926,0.559694003674779} | 2
5 | {0.924092949098971,-2.00871628417505} | 2
6 | {2.66881184290186,1.44902228049511} | 2
(6 rows)
-# Now let's look at sparse matrices. Create sample data in sparse matrix form:
DROP TABLE IF EXISTS mat_sparse;
CREATE TABLE mat_sparse (
row_id integer,
col_id integer,
value double precision
);
INSERT INTO mat_sparse VALUES
(1, 1, 1.0),
(2, 2, 2.0),
(3, 3, 3.0),
(4, 4, 4.0),
(1, 5, 5.0),
(2, 4, 6.0),
(3, 2, 7.0),
(4, 3, 8.0);
As an aside, this is what the sparse matrix above looks like when
put in dense form:
DROP TABLE IF EXISTS mat_dense;
SELECT madlib.matrix_densify('mat_sparse',
'row=row_id, col=col_id, val=value',
'mat_dense');
SELECT * FROM mat_dense ORDER BY row_id;
row_id | value
--------+-------------
1 | {1,0,0,0,5}
2 | {0,2,0,6,0}
3 | {0,7,3,0,0}
4 | {0,0,8,4,0}
(4 rows)
-# Run the PCA sparse function for a specified number of principal components and view the results:
DROP TABLE IF EXISTS result_table, result_table_mean;
SELECT madlib.pca_sparse_train( 'mat_sparse', -- Source table
'result_table', -- Output table
'row_id', -- Row id of source table
'col_id', -- Column id of source table
'value', -- Value of matrix at row_id, col_id
4, -- Actual number of rows in the matrix
5, -- Actual number of columns in the matrix
3); -- Number of principal components
SELECT * FROM result_table ORDER BY row_id;
Result (with principal components truncated for readability):
row_id | principal_components | std_dev | proportion
--------+----------------------------------------------+------------------+-------------------
1 | {-0.0876046030186158,-0.0968983772909994,... | 4.21362803829554 | 0.436590030617467
2 | {-0.0647272661608605,0.877639526308692,... | 3.68408023747461 | 0.333748701544697
3 | {-0.0780380267884855,0.177956517174911,... | 3.05606908060098 | 0.229661267837836
(3 rows)
-# Project the original sparse data to low-dimensional representation:
DROP TABLE IF EXISTS mat_sparse_out;
SELECT madlib.pca_sparse_project(
'mat_sparse',
'result_table',
'mat_sparse_out',
'row_id',
'col_id',
'value',
4,
5
);
SELECT * FROM mat_sparse_out ORDER BY row_id;
row_id | row_vec
--------+---------------------------------------------------------
1 | {4.66617015032369,-2.63552220635847,2.1865220849604}
2 | {0.228360685652383,-1.21616275892926,-4.46864627611561}
3 | {0.672067460100428,5.45249627172823,0.56445525585642}
4 | {-5.5665982960765,-1.6008113064405,1.71766893529879}
(4 rows)
-# Now we use grouping in sparse form to learn different models for different groups.
First, we create sample data in sparse matrix form with a grouping column:
DROP TABLE IF EXISTS mat_sparse_group;
CREATE TABLE mat_sparse_group (
row_id integer,
col_id integer,
value double precision,
matrix_id integer);
INSERT INTO mat_sparse_group VALUES
(1, 1, 1.0, 1),
(2, 2, 2.0, 1),
(3, 3, 3.0, 1),
(4, 4, 4.0, 1),
(1, 5, 5.0, 1),
(2, 4, 6.0, 2),
(3, 2, 7.0, 2),
(4, 3, 8.0, 2);
-# Run the PCA function with grouping for a specified proportion of variance
and view the results:
DROP TABLE IF EXISTS result_table_group, result_table_group_mean;
SELECT madlib.pca_sparse_train( 'mat_sparse_group', -- Source table
'result_table_group', -- Output table
'row_id', -- Row id of source table
'col_id', -- Column id of source table
'value', -- Value of matrix at row_id, col_id
4, -- Actual number of rows in the matrix
5, -- Actual number of columns in the matrix
0.8, -- Proportion of variance
'matrix_id');
SELECT * FROM result_table_group ORDER BY matrix_id, row_id;
Result (with principal components truncated for readability):
row_id | principal_components | std_dev | proportion | matrix_id
--------+--------------------------------------------+------------------+-------------------+-----------
1 | {-0.17805696611353,0.0681313257646983,... | 2.73659933165925 | 0.544652792875481 | 1
2 | {-0.0492086814863993,0.149371585357526,... | 2.06058314533194 | 0.308800210823714 | 1
1 | {0,-0.479486114660443,... | 4.40325305087975 | 0.520500333693473 | 2
2 | {0,0.689230898585949,... | 3.7435566458567 | 0.376220573442628 | 2
(4 rows)
-# Projection in sparse format with grouping:
DROP TABLE IF EXISTS mat_sparse_group_projected;
SELECT madlib.pca_sparse_project(
'mat_sparse_group',
'result_table_group',
'mat_sparse_group_projected',
'row_id',
'col_id',
'value',
4,
5
);
SELECT * FROM mat_sparse_group_projected ORDER BY matrix_id, row_id;
row_id | row_vec | matrix_id
--------+-----------------------------------------+-----------
1 | {-4.00039298524261,-0.626820612715982} | 1
2 | {0.765350785238575,0.951348276645455} | 1
3 | {1.04951017256904,2.22388180170356} | 1
4 | {2.185532027435,-2.54840946563303} | 1
1 | {-0.627846810195469,-0.685031603549092} | 2
2 | {-1.64754249747757,-4.7662114622896} | 2
3 | {-3.98424961281857,4.13958468655255} | 2
4 | {6.25963892049161,1.31165837928614} | 2
(8 rows)
@anchor notes
@par Notes
- This function is intended to operate on the principal component tables
generated by pca_train or pca_sparse_train. The MADlib PCA
functions generate a table containing the column-means in addition to a table
containing the principal components. If this table is not found by the MADlib
projection function, it will trigger an error. As long the principal component
tables are created with MADlib functions, then the column-means table will be
automatically found by the MADlib projection functions.
- Because of the centering step in PCA projection
(see "Technical Background"), sparse matrices almost always
become dense during the projection
process. Thus, this implementation automatically densifies sparse matrix input,
and there should be no expected performance improvement in using sparse matrix
input over dense matrix input.
- Table names can be optionally schema qualified (current_schemas() is
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"').
- If the input table for pca_project (pca_sparse_project) contains grouping columns,
the same grouping columns must be used in the training function used to generate the
principal components too.
@anchor background_project
@par Technical Background
Given a table containing some principal components \f$ \boldsymbol P \f$ and
some input data \f$ \boldsymbol X \f$, the low-dimensional representation \f$
{\boldsymbol X}' \f$ is computed as \f{align*}{ {\boldsymbol {\hat{X}}} & =
{\boldsymbol X} - \vec{e} \hat{x}^T \\ {\boldsymbol X}' & = {\boldsymbol
{\hat {X}}} {\boldsymbol P}. \f} where \f$\hat{x} \f$ is the column means of
\f$ \boldsymbol X \f$ and \f$ \vec{e} \f$ is the vector of all ones. This
step is equivalent to centering the data around the origin.
The residual table \f$ \boldsymbol R \f$ is a measure of how well the
low-dimensional representation approximates the true input data, and is computed
as \f[ {\boldsymbol R} = {\boldsymbol {\hat{X}}} - {\boldsymbol X}' {\boldsymbol
P}^T. \f] A residual matrix with entries mostly close to zero indicates a good
representation.
The residual norm \f$ r \f$ is simply
\f[
r = \|{\boldsymbol R}\|_F
\f]
where \f$ \|\cdot\|_F \f$ is the Frobenius norm. The relative residual norm \f$ r' \f$ is
\f[
r' = \frac{ \|{\boldsymbol R}\|_F }{\|{\boldsymbol X}\|_F }
\f]
@anchor related
@par Related Topics
File pca_project.sql_in documenting the SQL functions
\ref grp_pca_train
*/
-- -----------------------------------------------------------------------
-- PCA projection for Dense matrices
-- -----------------------------------------------------------------------
/*
@brief Compute principal compoents for a dense matrix stored in a
database table
*/
CREATE OR REPLACE FUNCTION
MADLIB_SCHEMA.pca_project(
source_table TEXT, -- Source table name (dense matrix)
pc_table TEXT, -- Principal components table (output from pca module)
out_table TEXT, -- Output table name for the principal components
row_id TEXT, -- Column name for the ID for each row
residual_table TEXT, -- Residual table (Default: NULL)
result_summary_table TEXT -- Table name to store summary of results (Default: NULL)
)
RETURNS VOID AS $$
PythonFunction(pca, pca_project, pca_project)
$$ LANGUAGE plpythonu
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA', `');
-- Overloaded functions for optional parameters
-- -----------------------------------------------------------------------
CREATE OR REPLACE FUNCTION
MADLIB_SCHEMA.pca_project(
source_table TEXT, -- Source table name (dense matrix)
pc_table TEXT, -- Principal components table (output from pca module)
out_table TEXT, -- Output table name for the principal components
row_id TEXT -- Column name for the ID for each row
)
RETURNS VOID AS $$
SELECT MADLIB_SCHEMA.pca_project($1, $2, $3, $4, NULL, NULL)
$$ LANGUAGE SQL
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA', `');
CREATE OR REPLACE FUNCTION
MADLIB_SCHEMA.pca_project(
source_table TEXT, -- Source table name (dense matrix)
pc_table TEXT, -- Principal components table (output from pca module)
out_table TEXT, -- Output table name for the principal components
row_id TEXT, -- Column name for the ID for each row
residual_table TEXT -- Residual table (Default: NULL)
)
RETURNS VOID AS $$
SELECT MADLIB_SCHEMA.pca_project($1, $2, $3, $4, $5, NULL)
$$ LANGUAGE SQL
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA', `');
-- Help and usage functions
-----------------------------------------------------------------------------
CREATE OR REPLACE FUNCTION
MADLIB_SCHEMA.pca_project(
usage_string TEXT -- Usage string
)
RETURNS VARCHAR AS $$
PythonFunction(pca, pca_project, pca_project_help)
$$ LANGUAGE plpythonu IMMUTABLE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `CONTAINS SQL', `');
CREATE OR REPLACE FUNCTION
MADLIB_SCHEMA.pca_project()
RETURNS VARCHAR AS $$
BEGIN
RETURN MADLIB_SCHEMA.pca_project('');
END;
$$ LANGUAGE plpgsql IMMUTABLE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `CONTAINS SQL', `');
-- -----------------------------------------------------------------------
-- PCA sparse projection for dense matrices
-- -----------------------------------------------------------------------
/*
@brief Compute principal compoents for a dense matrix stored in a
database table
*/
CREATE OR REPLACE FUNCTION
MADLIB_SCHEMA.pca_sparse_project(
source_table TEXT, -- Source table name (dense matrix)
pc_table TEXT, -- Principal components table (output from pca module)
out_table TEXT, -- Output table name for the principal components
row_id TEXT, -- Column name for the row id
col_id TEXT, -- Column name for the col id
val_id TEXT, -- Column name for the value id
row_dim INT4, -- Row dimension of the sparse matrix
col_dim INT4, -- Column dimension of the sparse matrix
residual_table TEXT, -- Residual table (Default: NULL)
result_summary_table TEXT -- Table name to store summary of results (Default: NULL)
)
RETURNS VOID AS $$
PythonFunction(pca, pca_project, pca_sparse_project)
$$ LANGUAGE plpythonu
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA', `');
-- Overloaded functions for optional parameters
-- -----------------------------------------------------------------------
CREATE OR REPLACE FUNCTION
MADLIB_SCHEMA.pca_sparse_project(
source_table TEXT, -- Source table name (dense matrix)
pc_table TEXT, -- Principal components table (output from pca module)
out_table TEXT, -- Output table name for the principal components
row_id TEXT, -- Column name for the row id
col_id TEXT, -- Column name for the col id
val_id TEXT, -- Column name for the value id
row_dim INT4, -- Row dimension of the sparse matrix
col_dim INT4 -- Column dimension of the sparse matrix
)
RETURNS VOID AS $$
SELECT MADLIB_SCHEMA.pca_sparse_project($1, $2, $3, $4, $5, $6, $7, $8, NULL, NULL)
$$ LANGUAGE SQL
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA', `');
CREATE OR REPLACE FUNCTION
MADLIB_SCHEMA.pca_sparse_project(
source_table TEXT, -- Source table name (dense matrix)
pc_table TEXT, -- Principal components table (output from pca module)
out_table TEXT, -- Output table name for the principal components
row_id TEXT, -- Column name for the row id
col_id TEXT, -- Column name for the col id
val_id TEXT, -- Column name for the value id
row_dim INT4, -- Row dimension of the sparse matrix
col_dim INT4, -- Column dimension of the sparse matrix
residual_table TEXT -- Residual table (Default: NULL)
)
RETURNS VOID AS $$
SELECT MADLIB_SCHEMA.pca_sparse_project($1, $2, $3, $4, $5, $6, $7, $8, $9, NULL)
$$ LANGUAGE SQL
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA', `');
CREATE OR REPLACE FUNCTION MADLIB_SCHEMA._pca_project_union(
source_table TEXT, -- Source table name (dense matrix)
pc_table TEXT, -- Output table name for the principal components
out_table TEXT, -- Output table name
row_id TEXT, -- Column name for the ID for each row
original_row_id TEXT, -- copy of the row_id originally passed
grouping_cols TEXT, -- Comma-separated list of grouping columns (Default: NULL)
grouping_cols_clause TEXT, -- Part of the SQL query to be used with grouping_cols
residual_table TEXT, -- Residual table name
result_summary_table TEXT, -- Table name to store summary of results (Default: NULL)
grp_id INTEGER, -- a place holder id for each group
grouping_where_clause TEXT, -- WHERE clause using grouping_cols
sparse_where_condition TEXT, -- WHERE clause used when creating temp sparse matrix table with dims
select_grouping_cols TEXT, -- SELECT clause using grouping_cols
grouping_cols_values TEXT, -- distinct values of the grouping_cols
temp_source_table_columns TEXT, -- SELECT caluse for creating temporary copy of the source_table
temp_pc_table_columns TEXT, -- non grouping_cols of the source_table
is_sparse BOOLEAN, -- specifies if the PCA call is for sparse or dense matrices
col_id TEXT, -- sparse representation based detail
val_id TEXT, -- sparse representation based detail
row_dim INTEGER, -- sparse representation based detail
col_dim INTEGER -- sparse representation based detail
)
RETURNS VOID AS $$
PythonFunction(pca, pca_project, _pca_project_union)
$$ LANGUAGE plpythonu
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `MODIFIES SQL DATA', `');
-- Help and usage functions
-----------------------------------------------------------------------------
CREATE OR REPLACE FUNCTION
MADLIB_SCHEMA.pca_sparse_project(
usage_string TEXT -- Usage string
)
RETURNS VARCHAR AS $$
PythonFunction(pca, pca_project, pca_sparse_project_help)
$$ LANGUAGE plpythonu IMMUTABLE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `CONTAINS SQL', `');
CREATE OR REPLACE FUNCTION
MADLIB_SCHEMA.pca_sparse_project()
RETURNS VARCHAR AS $$
BEGIN
RETURN MADLIB_SCHEMA.pca_sparse_project('');
END;
$$ LANGUAGE plpgsql IMMUTABLE
m4_ifdef(`__HAS_FUNCTION_PROPERTIES__', `CONTAINS SQL', `');