Contents

Generalized linear models extends ordinary linear regression by allowing the response variable to follow a more general set of distributions (rather than simply Gaussian distributions), and for a general family of functions of the response variable (the link function) to vary linearly with the predicted values (rather than assuming that the response itself must vary linearly).

For example, data of counts would typically be modeled with a Poisson distribution and a log link, while binary outcomes would typically be modeled with a Bernoulli distribution (or binomial distribution, depending on exactly how the problem is phrased) and a log-odds (or logit) link function.

Currently, the implemented distribution families are

Distribution Family	Link Functions
Binomial	logit, probit
Gamma	inverse, identity, log
Gaussian	identity, inverse, log
Inverse Gaussian	inverse of square, inverse, identity, log
Poisson	log, identity, square-root

Training Function

GLM training function has the following format:

glm(source_table,
    model_table,
    dependent_varname,
    independent_varname,
    family_params,
    grouping_col,
    optim_params,
    verbose
    )

Arguments

source_table

TEXT. The name of the table containing the training data.

model_table

TEXT. Name of the generated table containing the model.

The model table produced by glm contains the following columns:

<...>	Text. Grouping columns, if provided in input. This could be multiple columns depending on the `grouping_col` input.
coef	FLOAT8. Vector of the coefficients in linear predictor.
log_likelihood	FLOAT8. The log-likelihood \( l(\boldsymbol \beta) \). We use the maximum likelihood estimate of dispersion parameter to calculate the log-likelihood while R and Python use deviance estimate and Pearson estimate respectively.
std_err	FLOAT8[]. Vector of the standard error of the coefficients.
z_stats or t_stats	FLOAT8[]. Vector of the z-statistics (in Poisson distribtuion and Binomial distribution) or the t-statistics (in all other distributions) of the coefficients.
p_values	FLOAT8[]. Vector of the p-values of the coefficients.
dispersion	FLOAT8. The dispersion value (Pearson estimate). When family=poisson or family=binomial, the dispersion is always 1.
num_rows_processed	BIGINT. Numbers of rows processed.
num_rows_skipped	BIGINT. Numbers of rows skipped due to missing values or failures.
num_iterations	INTEGER. The number of iterations actually completed. This would be different from the `nIterations` argument if a `tolerance` parameter is provided and the algorithm converges before all iterations are completed.

A summary table named <model_table>_summary is also created at the same time, which has the following columns:

method	'glm'
source_table	The data source table name.
model_table	The model table name.
dependent_varname	The dependent variable.
independent_varname	The independent variables
family_params	A string that contains family parameters, and has the form of 'family=..., link=...'
grouping_col	Name of grouping columns.
optimizer_params	A string that contains optimizer parameters, and has the form of 'optimizer=..., max_iter=..., tolerance=...'
num_all_groups	Number of groups in glm training.
num_failed_groups	Number of failed groups in glm training.
total_rows_processed	BIGINT. Total numbers of rows processed in all groups.
total_rows_skipped	BIGINT. Total numbers of rows skipped in all groups due to missing values or failures.

dependent_varname

TEXT. Name of the dependent variable column.

independent_varname

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 1 term in the independent variable list.

family_params (optional)

TEXT, Parameters for distribution family. Currently, we support

(1) family=poisson and link=[log or identity or sqrt].

(2) family=gaussian and link=[identity or log or inverse]. And when family=gaussian and link=identity, the GLM model is exactly the same as the linear regression.

(3) family=gamma and link=[inverse or identity or log].

(4) family=inverse_gaussian and link=[sqr_inverse or log or identity or inverse].

(5) family=binomial and link=[probit or logit].

grouping_col (optional)

TEXT, default: NULL. An expression list used to group the input dataset into discrete groups, running one regression per group. Similar to the SQL "GROUP BY" clause. When this value is NULL, no grouping is used and a single model is generated.

optim_params (optional)

TEXT, default: 'max_iter=100,optimizer=irls,tolerance=1e-6'. Parameters for optimizer. Currently, we support tolerance=[tolerance for relative error between log-likelihoods], max_iter=[maximum iterations to run], optimizer=irls.

verbose (optional)

BOOLEAN, default: FALSE. Provides verbose output of the results of training.

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 results may look different, they are in fact the same.

Prediction Function

The prediction function is provided to estimate the conditional mean given a new predictor. It has the following syntax:

glm_predict(coef,
            col_ind_var
            link)

Arguments

coef

DOUBLE PRECISION[]. Model coefficients obtained from glm().

col_ind_var

New predictor, as a DOUBLE array. This should be the same length as the array obtained by evaluation of the 'independent_varname' argument in glm().

link

link function, as a string. This should match the link function the user inputted in glm().

Examples

Create the training data table.

CREATE TABLE warpbreaks(
    id      serial,
    breaks  integer,
    wool    char(1),
    tension char(1)
);
INSERT INTO warpbreaks(breaks, wool, tension) VALUES
(26, 'A', 'L'),
(30, 'A', 'L'),
(54, 'A', 'L'),
(25, 'A', 'L'),
(70, 'A', 'L'),
(52, 'A', 'L'),
(51, 'A', 'L'),
(26, 'A', 'L'),
(67, 'A', 'L'),
(18, 'A', 'M'),
(21, 'A', 'M'),
(29, 'A', 'M'),
(17, 'A', 'M'),
(12, 'A', 'M'),
(18, 'A', 'M'),
(35, 'A', 'M'),
(30, 'A', 'M'),
(36, 'A', 'M'),
(36, 'A', 'H'),
(21, 'A', 'H'),
(24, 'A', 'H'),
(18, 'A', 'H'),
(10, 'A', 'H'),
(43, 'A', 'H'),
(28, 'A', 'H'),
(15, 'A', 'H'),
(26, 'A', 'H'),
(27, 'B', 'L'),
(14, 'B', 'L'),
(29, 'B', 'L'),
(19, 'B', 'L'),
(29, 'B', 'L'),
(31, 'B', 'L'),
(41, 'B', 'L'),
(20, 'B', 'L'),
(44, 'B', 'L'),
(42, 'B', 'M'),
(26, 'B', 'M'),
(19, 'B', 'M'),
(16, 'B', 'M'),
(39, 'B', 'M'),
(28, 'B', 'M'),
(21, 'B', 'M'),
(39, 'B', 'M'),
(29, 'B', 'M'),
(20, 'B', 'H'),
(21, 'B', 'H'),
(24, 'B', 'H'),
(17, 'B', 'H'),
(13, 'B', 'H'),
(15, 'B', 'H'),
(15, 'B', 'H'),
(16, 'B', 'H'),
(28, 'B', 'H');
SELECT create_indicator_variables('warpbreaks', 'warpbreaks_dummy', 'wool,tension');

Train a GLM model.

DROP TABLE IF EXISTS glm_model, glm_model_summary;
SELECT glm('warpbreaks_dummy',
           'glm_model',
           'breaks',
           'ARRAY[1.0,"wool_B","tension_M", "tension_H"]',
           'family=poisson, link=log');

View the regression results.

-- Set extended display on for easier reading of output
\x on
SELECT * FROM glm_model;

Result:

coef               | {3.69196314494079,-0.205988442638621,-0.321320431600611,-0.51848849651156}
log_likelihood     | -242.527983208979
std_err            | {0.04541079434248,0.0515712427835191,0.0602659166951256,0.0639595193956924}
z_stats            | {81.3014438174473,-3.99425011926316,-5.3317106786264,-8.10651020224019}
p_values           | {0,6.48993254938271e-05,9.72918600322907e-08,5.20943463005751e-16}
num_rows_processed | 54
num_rows_skipped   | 0
iteration          | 5

Alternatively, unnest the arrays in the results for easier reading of output:

\x off
SELECT unnest(coef) as coefficient,
       unnest(std_err) as standard_error,
       unnest(z_stats) as z_stat,
       unnest(p_values) as pvalue
FROM glm_model;

Predicting dependent variable using GLM model. (This example uses the original data table to perform the prediction. Typically a different test dataset with the same features as the original training dataset would be used for prediction.)

\x off
-- Display predicted mean value on the original dataset
SELECT
    w.id,
    madlib.glm_predict(
        coef,
        ARRAY[1, "wool_B", "tension_M", "tension_H"]::float8[],
        'log') AS mu
FROM warpbreaks_dummy w, glm_model m
ORDER BY w.id;

-- Display predicted counts (which are predicted mean values rounded to the nearest integral value) on the original dataset
SELECT
    w.id,
    madlib.glm_predict_poisson(
        coef,
        ARRAY[1, "wool_B", "tension_M", "tension_H"]::float8[],
        'log') AS poisson_count
FROM warpbreaks_dummy w, glm_model m
ORDER BY w.id;

Example for Gaussian family:

Create a testing data table

CREATE TABLE abalone (
    id integer,
    sex text,
    length double precision,
    diameter double precision,
    height double precision,
    whole double precision,
    shucked double precision,
    viscera double precision,
    shell double precision,
    rings integer
);
INSERT INTO abalone VALUES
(3151, 'F', 0.655000000000000027, 0.505000000000000004, 0.165000000000000008, 1.36699999999999999, 0.583500000000000019, 0.351499999999999979, 0.396000000000000019, 10),
(2026, 'F', 0.550000000000000044, 0.469999999999999973, 0.149999999999999994, 0.920499999999999985, 0.381000000000000005, 0.243499999999999994, 0.267500000000000016, 10),
(3751, 'I', 0.434999999999999998, 0.375, 0.110000000000000001, 0.41549999999999998, 0.170000000000000012, 0.0759999999999999981, 0.14499999999999999, 8),
(720, 'I', 0.149999999999999994, 0.100000000000000006, 0.0250000000000000014, 0.0149999999999999994, 0.00449999999999999966, 0.00400000000000000008, 0.0050000000000000001, 2),
(1635, 'F', 0.574999999999999956, 0.469999999999999973, 0.154999999999999999, 1.1160000000000001, 0.509000000000000008, 0.237999999999999989, 0.340000000000000024, 10),
(2648, 'I', 0.5, 0.390000000000000013, 0.125, 0.582999999999999963, 0.293999999999999984, 0.132000000000000006, 0.160500000000000004, 8),
(1796, 'F', 0.57999999999999996, 0.429999999999999993, 0.170000000000000012, 1.47999999999999998, 0.65349999999999997, 0.32400000000000001, 0.41549999999999998, 10),
(209, 'F', 0.525000000000000022, 0.41499999999999998, 0.170000000000000012, 0.832500000000000018, 0.275500000000000023, 0.168500000000000011, 0.309999999999999998, 13),
(1451, 'I', 0.455000000000000016, 0.33500000000000002, 0.135000000000000009, 0.501000000000000001, 0.274000000000000021, 0.0995000000000000051, 0.106499999999999997, 7),
(1108, 'I', 0.510000000000000009, 0.380000000000000004, 0.115000000000000005, 0.515499999999999958, 0.214999999999999997, 0.113500000000000004, 0.166000000000000009, 8),
(3675, 'F', 0.594999999999999973, 0.450000000000000011, 0.165000000000000008, 1.08099999999999996, 0.489999999999999991, 0.252500000000000002, 0.279000000000000026, 12),
(2108, 'F', 0.675000000000000044, 0.550000000000000044, 0.179999999999999993, 1.68849999999999989, 0.562000000000000055, 0.370499999999999996, 0.599999999999999978, 15),
(3312, 'F', 0.479999999999999982, 0.380000000000000004, 0.135000000000000009, 0.507000000000000006, 0.191500000000000004, 0.13650000000000001, 0.154999999999999999, 12),
(882, 'M', 0.655000000000000027, 0.520000000000000018, 0.165000000000000008, 1.40949999999999998, 0.585999999999999965, 0.290999999999999981, 0.405000000000000027, 9),
(3402, 'M', 0.479999999999999982, 0.395000000000000018, 0.149999999999999994, 0.681499999999999995, 0.214499999999999996, 0.140500000000000014, 0.2495, 18),
(829, 'I', 0.409999999999999976, 0.325000000000000011, 0.100000000000000006, 0.394000000000000017, 0.20799999999999999, 0.0655000000000000027, 0.105999999999999997, 6),
(1305, 'M', 0.535000000000000031, 0.434999999999999998, 0.149999999999999994, 0.716999999999999971, 0.347499999999999976, 0.14449999999999999, 0.194000000000000006, 9),
(3613, 'M', 0.599999999999999978, 0.46000000000000002, 0.179999999999999993, 1.1399999999999999, 0.422999999999999987, 0.257500000000000007, 0.364999999999999991, 10),
(1068, 'I', 0.340000000000000024, 0.265000000000000013, 0.0800000000000000017, 0.201500000000000012, 0.0899999999999999967, 0.0475000000000000006, 0.0550000000000000003, 5),
(2446, 'M', 0.5, 0.380000000000000004, 0.135000000000000009, 0.583500000000000019, 0.22950000000000001, 0.126500000000000001, 0.179999999999999993, 12),
(1393, 'M', 0.635000000000000009, 0.474999999999999978, 0.170000000000000012, 1.19350000000000001, 0.520499999999999963, 0.269500000000000017, 0.366499999999999992, 10),
(359, 'M', 0.744999999999999996, 0.584999999999999964, 0.214999999999999997, 2.49900000000000011, 0.92649999999999999, 0.471999999999999975, 0.699999999999999956, 17),
(549, 'F', 0.564999999999999947, 0.450000000000000011, 0.160000000000000003, 0.79500000000000004, 0.360499999999999987, 0.155499999999999999, 0.23000000000000001, 12),
(1154, 'F', 0.599999999999999978, 0.474999999999999978, 0.160000000000000003, 1.02649999999999997, 0.484999999999999987, 0.2495, 0.256500000000000006, 9),
(1790, 'F', 0.54500000000000004, 0.385000000000000009, 0.149999999999999994, 1.11850000000000005, 0.542499999999999982, 0.244499999999999995, 0.284499999999999975, 9),
(3703, 'F', 0.665000000000000036, 0.540000000000000036, 0.195000000000000007, 1.76400000000000001, 0.850500000000000034, 0.361499999999999988, 0.469999999999999973, 11),
(1962, 'F', 0.655000000000000027, 0.515000000000000013, 0.179999999999999993, 1.41199999999999992, 0.619500000000000051, 0.248499999999999999, 0.496999999999999997, 11),
(1665, 'I', 0.604999999999999982, 0.469999999999999973, 0.14499999999999999, 0.802499999999999991, 0.379000000000000004, 0.226500000000000007, 0.220000000000000001, 9),
(635, 'M', 0.359999999999999987, 0.294999999999999984, 0.100000000000000006, 0.210499999999999993, 0.0660000000000000031, 0.0524999999999999981, 0.0749999999999999972, 9),
(3901, 'M', 0.445000000000000007, 0.344999999999999973, 0.140000000000000013, 0.475999999999999979, 0.205499999999999988, 0.101500000000000007, 0.108499999999999999, 15),
(2734, 'I', 0.41499999999999998, 0.33500000000000002, 0.100000000000000006, 0.357999999999999985, 0.169000000000000011, 0.067000000000000004, 0.104999999999999996, 7),
(3856, 'M', 0.409999999999999976, 0.33500000000000002, 0.115000000000000005, 0.440500000000000003, 0.190000000000000002, 0.0850000000000000061, 0.135000000000000009, 8),
(827, 'I', 0.395000000000000018, 0.28999999999999998, 0.0950000000000000011, 0.303999999999999992, 0.127000000000000002, 0.0840000000000000052, 0.076999999999999999, 6),
(3381, 'I', 0.190000000000000002, 0.130000000000000004, 0.0449999999999999983, 0.0264999999999999993, 0.00899999999999999932, 0.0050000000000000001, 0.00899999999999999932, 5),
(3972, 'I', 0.400000000000000022, 0.294999999999999984, 0.0950000000000000011, 0.252000000000000002, 0.110500000000000001, 0.0575000000000000025, 0.0660000000000000031, 6),
(1155, 'M', 0.599999999999999978, 0.455000000000000016, 0.170000000000000012, 1.1915, 0.695999999999999952, 0.239499999999999991, 0.239999999999999991, 8),
(3467, 'M', 0.640000000000000013, 0.5, 0.170000000000000012, 1.4544999999999999, 0.642000000000000015, 0.357499999999999984, 0.353999999999999981, 9),
(2433, 'F', 0.609999999999999987, 0.484999999999999987, 0.165000000000000008, 1.08699999999999997, 0.425499999999999989, 0.232000000000000012, 0.380000000000000004, 11),
(552, 'I', 0.614999999999999991, 0.489999999999999991, 0.154999999999999999, 0.988500000000000045, 0.41449999999999998, 0.195000000000000007, 0.344999999999999973, 13),
(1425, 'F', 0.729999999999999982, 0.57999999999999996, 0.190000000000000002, 1.73750000000000004, 0.678499999999999992, 0.434499999999999997, 0.520000000000000018, 11),
(2402, 'F', 0.584999999999999964, 0.41499999999999998, 0.154999999999999999, 0.69850000000000001, 0.299999999999999989, 0.145999999999999991, 0.195000000000000007, 12),
(1748, 'F', 0.699999999999999956, 0.535000000000000031, 0.174999999999999989, 1.77299999999999991, 0.680499999999999994, 0.479999999999999982, 0.512000000000000011, 15),
(3983, 'I', 0.57999999999999996, 0.434999999999999998, 0.149999999999999994, 0.891499999999999959, 0.362999999999999989, 0.192500000000000004, 0.251500000000000001, 6),
(335, 'F', 0.739999999999999991, 0.599999999999999978, 0.195000000000000007, 1.97399999999999998, 0.597999999999999976, 0.408499999999999974, 0.709999999999999964, 16),
(1587, 'I', 0.515000000000000013, 0.349999999999999978, 0.104999999999999996, 0.474499999999999977, 0.212999999999999995, 0.122999999999999998, 0.127500000000000002, 10),
(2448, 'I', 0.275000000000000022, 0.204999999999999988, 0.0800000000000000017, 0.096000000000000002, 0.0359999999999999973, 0.0184999999999999991, 0.0299999999999999989, 6),
(1362, 'F', 0.604999999999999982, 0.474999999999999978, 0.174999999999999989, 1.07600000000000007, 0.463000000000000023, 0.219500000000000001, 0.33500000000000002, 9),
(2799, 'M', 0.640000000000000013, 0.484999999999999987, 0.149999999999999994, 1.09800000000000009, 0.519499999999999962, 0.222000000000000003, 0.317500000000000004, 10),
(1413, 'F', 0.67000000000000004, 0.505000000000000004, 0.174999999999999989, 1.01449999999999996, 0.4375, 0.271000000000000019, 0.3745, 10),
(1739, 'F', 0.67000000000000004, 0.540000000000000036, 0.195000000000000007, 1.61899999999999999, 0.739999999999999991, 0.330500000000000016, 0.465000000000000024, 11),
(1152, 'M', 0.584999999999999964, 0.465000000000000024, 0.160000000000000003, 0.955500000000000016, 0.45950000000000002, 0.235999999999999988, 0.265000000000000013, 7),
(2427, 'I', 0.564999999999999947, 0.434999999999999998, 0.154999999999999999, 0.782000000000000028, 0.271500000000000019, 0.16800000000000001, 0.284999999999999976, 14),
(1777, 'M', 0.484999999999999987, 0.369999999999999996, 0.154999999999999999, 0.967999999999999972, 0.418999999999999984, 0.245499999999999996, 0.236499999999999988, 9),
(3294, 'M', 0.574999999999999956, 0.455000000000000016, 0.184999999999999998, 1.15599999999999992, 0.552499999999999991, 0.242999999999999994, 0.294999999999999984, 13),
(1403, 'M', 0.650000000000000022, 0.510000000000000009, 0.190000000000000002, 1.54200000000000004, 0.715500000000000025, 0.373499999999999999, 0.375, 9),
(2256, 'M', 0.510000000000000009, 0.395000000000000018, 0.14499999999999999, 0.61850000000000005, 0.215999999999999998, 0.138500000000000012, 0.239999999999999991, 12),
(3984, 'F', 0.584999999999999964, 0.450000000000000011, 0.125, 0.873999999999999999, 0.354499999999999982, 0.20749999999999999, 0.225000000000000006, 6),
(1116, 'M', 0.525000000000000022, 0.405000000000000027, 0.119999999999999996, 0.755499999999999949, 0.3755, 0.155499999999999999, 0.201000000000000012, 9),
(1366, 'M', 0.609999999999999987, 0.474999999999999978, 0.170000000000000012, 1.02649999999999997, 0.434999999999999998, 0.233500000000000013, 0.303499999999999992, 10),
(3759, 'I', 0.525000000000000022, 0.400000000000000022, 0.140000000000000013, 0.605500000000000038, 0.260500000000000009, 0.107999999999999999, 0.209999999999999992, 9);

Train a model with family=gaussian and link=identity

DROP TABLE IF EXISTS abalone_out, abalone_out_summary;
SELECT madlib.glm(
    'abalone',
    'abalone_out',
    'rings',
    'ARRAY[1, length, diameter, height, whole, shucked, viscera, shell]',
    'family=gaussian, link=identity');

Example for Gamma family: (reuse the dataset in Gaussian case)

Reuse the test data set in Gaussian

Train a model with family=gamma and link=inverse

DROP TABLE IF EXISTS abalone_out, abalone_out_summary;
SELECT madlib.glm(
    'abalone',
    'abalone_out',
    'rings',
    'ARRAY[1, length, diameter, height, whole, shucked, viscera, shell]',
    'family=gamma, link=inverse');

Example for Inverse Gaussian family: (reuse the dataset in Gaussian case)

Reuse the test data set in Gaussian

Train a model with family=inverse_gaussian and link=sqr_inverse

DROP TABLE IF EXISTS abalone_out, abalone_out_summary;
SELECT madlib.glm(
    'abalone',
    'abalone_out',
    'rings',
    'ARRAY[1, length, diameter, height, whole, shucked, viscera, shell]',
    'family=inverse_gaussian, link=sqr_inverse');

Example for Binomial family: (reuse the dataset in Gaussian case)

Reuse the test data set in Gaussian

Train a model with family=binomial and link=probit

DROP TABLE IF EXISTS abalone_out, abalone_out_summary;
SELECT madlib.glm(
    'abalone',
    'abalone_out',
    'rings < 10',
    'ARRAY[1, length, diameter, height, whole, shucked, viscera, shell]',
    'family=binomial, link=probit');

Predict output probabilities

SELECT madlib.glm_predict(
    coef,
    ARRAY[1, length, diameter, height, whole, shucked, viscera, shell]::float8[],
    'probit')
FROM abalone_out, abalone;

Predict output categories

SELECT madlib.glm_predict(
SELECT madlib.glm_predict_binomial(
    coef,
    ARRAY[1, length, diameter, height, whole, shucked, viscera, shell]::float8[],
    'probit')
FROM abalone_out, abalone;

Notes: 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"').

Currently implementation uses Newton's method and, according to performance tests, when number of features are over 1000, this GLM function could be running slowly.

Functions in Linear Regression is prefered to GLM with family=gaussian,link=identity, as the former require only a single pass over the training data. In addition, if user expects to use robust variance, clustered variance, or marginal effects on top of the trained model, functions in Linear Regression and Logistic Regression should be used.

Related Topics

File glm.sql_in documenting the training function

Linear Regression

Logistic Regression

Multinomial Logistic Regression