 1.21.0 User Documentation for Apache MADlib Logistic Regression
Contents

Binomial logistic regression models the relationship between a dichotomous dependent variable and one or more predictor variables. The dependent variable may be a Boolean value or a categorial variable that can be represented with a Boolean expression. The probabilities describing the possible outcomes of a single trial are modeled, as a function of the predictor variables, using a logistic function.

Training Function
The logistic regression training function has the following format:
logregr_train( source_table,
out_table,
dependent_varname,
independent_varname,
grouping_cols,
max_iter,
optimizer,
tolerance,
verbose
)

Arguments
source_table

TEXT. Name of the table containing the training data.

out_table

TEXT. Name of the generated table containing the output model.

The output table produced by the logistic regression training function contains the following columns:

<...> TEXT. Grouping columns, if provided in input. This could be multiple columns depending on the grouping_cols input. FLOAT8. Vector of the coefficients of the regression. FLOAT8. The log-likelihood $$l(\boldsymbol c)$$. FLOAT8[]. Vector of the standard error of the coefficients. FLOAT8[]. Vector of the z-statistics of the coefficients. FLOAT8[]. Vector of the p-values of the coefficients. FLOAT8[]. The odds ratio, $$\exp(c_i)$$. FLOAT8[]. The condition number of the $$X^{*}X$$ 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 may be more appropriate. INTEGER. The number of rows actually processed, which is equal to the total number of rows in the source table minus the number of skipped rows. INTEGER. The number of rows skipped during the training. A row will be skipped if the independent_varname is NULL or contains NULL values. 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. FLOAT[]. Variance/covariance matrix.

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

method 'logregr' for logistic regression. The data source table name. The output table name. The dependent variable name. The independent variable names. A string that contains all the optimizer parameters, and has the form of 'optimizer=..., max_iter=..., tolerance=...' How many groups of data were fit by the logistic model. How many groups failed in training. The total number of rows used in the computation. The total number of rows skipped. Names of the grouping columns.

dependent_varname

TEXT. Name of the dependent variable column (of type BOOLEAN) in the training data, or an expression evaluating to a BOOLEAN.

independent_varname

TEXT. Expression list to evaluate for the independent variables. An intercept variable is not assumed so it is common to provide an explicit intercept term by including a single constant 1 term in the independent variable list.

grouping_cols (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 for the whole data set.

max_iter (optional)

INTEGER, default: 20. The maximum number of iterations allowed.

optimizer (optional)

TEXT, default: 'irls'. The name of the optimizer to use:

'newton' or 'irls' Iteratively reweighted least squares conjugate gradient incremental gradient descent.

tolerance (optional)

FLOAT8, default: 0.0001. The difference between log-likelihood values in successive iterations that indicate convergence. A zero disables the convergence criterion, so that execution stops after the maximum iterations have completed, as set in the 'max_iter' parameter above.

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

Prediction Function
Two prediction functions are provided. One predicts the boolean value of the dependent variable, and the other predicts the probability of the value of the dependent variable being 'True'. Syntax is the same for both functions.

The function to predict the boolean value (True/False) of the dependent variable has the following syntax:

logregr_predict(coefficients,
ind_var
)


The function to predict the probability of the dependent variable being 'True' has the following syntax:

logregr_predict_prob(coefficients,
ind_var
)


Arguments

coefficients

DOUBLE PRECISION[]. Model coefficients obtained from training logregr_train().

ind_var
Independent variables expressed as a DOUBLE array. This should be the same length as the array obtained by evaluation of the 'independent_varname' argument in logregr_train().

Examples
1. Create the training data table. This data set is related to predicting a second heart attack given treatment and health factors.
DROP TABLE IF EXISTS patients;
CREATE TABLE patients( id INTEGER NOT NULL,
second_attack INTEGER,
treatment INTEGER,
trait_anxiety INTEGER);
INSERT INTO patients VALUES
(1,  1, 1, 70),
(2,  1, 1, 80),
(3,  1, 1, 50),
(4,  1, 0, 60),
(5,  1, 0, 40),
(6,  1, 0, 65),
(7,  1, 0, 75),
(8,  1, 0, 80),
(9,  1, 0, 70),
(10, 1, 0, 60),
(11, 0, 1, 65),
(12, 0, 1, 50),
(13, 0, 1, 45),
(14, 0, 1, 35),
(15, 0, 1, 40),
(16, 0, 1, 50),
(17, 0, 0, 55),
(18, 0, 0, 45),
(19, 0, 0, 50),
(20, 0, 0, 60);

2. Train a regression model.
DROP TABLE IF EXISTS patients_logregr, patients_logregr_summary;
SELECT madlib.logregr_train( 'patients',                             -- Source table
'patients_logregr',                     -- Output table
'second_attack',                        -- Dependent variable
'ARRAY[1, treatment, trait_anxiety]',   -- Feature vector
NULL,                                   -- Grouping
20,                                     -- Max iterations
'irls'                                  -- Optimizer to use
);

Note that in the example above 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 typically pre-build the arrays and store them in a single column.
3. View the regression results.
-- Set extended display on for easier reading of output
\x on
SELECT * from patients_logregr;

Result:
coef                     | {-6.36346994178192,-1.02410605239327,0.119044916668607}
log_likelihood           | -9.41018298388876
std_err                  | {3.21389766375099,1.17107844860319,0.0549790458269317}
z_stats                  | {-1.97998524145757,-0.874498248699539,2.16527796868916}
p_values                 | {0.0477051870698145,0.381846973530455,0.0303664045046183}
odds_ratios              | {0.00172337630923221,0.359117354054956,1.12642051220895}
condition_no             | 326.081922791575
num_rows_processed       | 20
num_missing_rows_skipped | 0
num_iterations           | 5
variance_covariance      | {{10.329138193064,-0.474304665195738,-0.171995901260057}, ...

4. Alternatively, unnest the arrays in the results for easier reading of output:
\x off
SELECT unnest(array['intercept', 'treatment', 'trait_anxiety']) as attribute,
unnest(coef) as coefficient,
unnest(std_err) as standard_error,
unnest(z_stats) as z_stat,
unnest(p_values) as pvalue,
unnest(odds_ratios) as odds_ratio
FROM patients_logregr;

Result:
   attribute   |    coefficient    |   standard_error   |       z_stat       |       pvalue       |     odds_ratio
---------------+-------------------+--------------------+--------------------+--------------------+---------------------
intercept     | -6.36346994178192 |   3.21389766375099 |  -1.97998524145757 | 0.0477051870698145 | 0.00172337630923221
treatment     | -1.02410605239327 |   1.17107844860319 | -0.874498248699539 |  0.381846973530455 |   0.359117354054956
trait_anxiety | 0.119044916668607 | 0.0549790458269317 |   2.16527796868916 | 0.0303664045046183 |    1.12642051220895
(3 rows)

5. Predict the dependent variable using the logistic regression 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 prediction value along with the original value
SELECT p.id, madlib.logregr_predict(coef, ARRAY[1, treatment, trait_anxiety]),
p.second_attack::BOOLEAN
FROM patients p, patients_logregr m
ORDER BY p.id;

Result:
  id | logregr_predict | second_attack
----+-----------------+---------------
1 | t               | t
2 | t               | t
3 | f               | t
4 | t               | t
5 | f               | t
6 | t               | t
7 | t               | t
8 | t               | t
9 | t               | t
10 | t               | t
11 | t               | f
12 | f               | f
13 | f               | f
14 | f               | f
15 | f               | f
16 | f               | f
17 | t               | f
18 | f               | f
19 | f               | f
20 | t               | f
(20 rows)

6. Predict the probability of the dependent variable being TRUE.
\x off
-- Display prediction value along with the original value
SELECT p.id, madlib.logregr_predict_prob(coef, ARRAY[1, treatment, trait_anxiety]),
p.second_attack::BOOLEAN
FROM patients p, patients_logregr m
ORDER BY p.id;

Result:
 id | logregr_predict_prob | second_attack
----+----------------------+---------------
1 |    0.720223028941527 | t
2 |    0.894354902502048 | t
3 |    0.192269541755171 | t
4 |    0.685513072239347 | t
5 |    0.167747881508857 | t
6 |     0.79809810891514 | t
7 |    0.928568075752503 | t
8 |    0.959305763693571 | t
9 |    0.877576117431452 | t
10 |    0.685513072239347 | t
11 |    0.586700895943317 | f
12 |    0.192269541755171 | f
13 |    0.116032010632994 | f
14 |   0.0383829143134982 | f
15 |   0.0674976224147597 | f
16 |    0.192269541755171 | f
17 |    0.545870774302621 | f
18 |    0.267675422387132 | f
19 |    0.398618639285111 | f
20 |    0.685513072239347 | f
(20 rows)