regression_selection

Selects the best multiple linear regression models.

Synopsis

#include <imsls.h>

void imsls_f_regression_selection (int n_rows, int n_candidate, float x[], float y[], ..., 0)

The type double function is imsls_d_regression_selection.

Required Arguments

int n_rows (Input)
Number of observations or rows in x and y.

int n_candidate (Input)
Number of candidate variables (independent variables) or columns in x. n_candidate must be greater than 2.

float x[] (Input)
Array of size n_rows × n_candidate containing the data for the candidate variables.

float y[] (Input)
Array of length n_rows containing the responses for the dependent variable.

Synopsis with Optional Arguments

#include <imsls.h>

void imsls_f_regression_selection (int n_rows, int n_candidate, float x[], float y[],

IMSLS_X_COL_DIM, int x_col_dim,

IMSLS_PRINT, or

IMSLS_NO_PRINT,

IMSLS_WEIGHTS, float weights[],

IMSLS_FREQUENCIES, float frequencies[],

IMSLS_R_SQUARED, int max_subset_size, or

IMSLS_ADJ_R_SQUARED, or

IMSLS_MALLOWS_CP,

IMSLS_MAX_N_BEST, int max_n_best,

IMSLS_MAX_N_GOOD_SAVED, int max_n_good_saved,

IMSLS_CRITERIONS, int **index_criterions, float **criterions,

IMSLS_CRITERIONS_USER, int index_criterions[], float criterions[],

IMSLS_INDEPENDENT_VARIABLES, int **index_variables, int **independent_variables,

IMSLS_INDEPENDENT_VARIABLES_USER, int index_variables[], int independent_variables[],

IMSLS_COEF_STATISTICS, int **index_coefficients,

IMSLS_COEF_STATISTICS_USER, int index_coefficients[],

IMSLS_INPUT_COV, int n_observations, float cov[],

Optional Arguments

IMSLS_X_COL_DIM, int x_col_dim (Input)
The column dimension of x.
Default: x_col_dim = n_candidate

IMSLS_PRINT
Printing is performed. This is the default.

IMSLS_NO_PRINT
Printing is not performed.

IMSLS_WEIGHTS, float weights[] (Input)
Array of length n_rows containing the weight for each row of x.
Default: weights[] = 1

IMSLS_FREQUENCIES, float frequencies[] (Input)
Array of length n_rows containing the frequency for each row of x.
Default: frequencies[] = 1

IMSLS_R_SQUARED, int max_subset_size (Input)
The R2 criterion is used, where subset sizes 1, 2, ..., max_subset_size are examined. This option is the default with max_subset_size = n_candidate.

IMSLS_ADJ_R_SQUARED
The adjusted R2 criterion is used, where subset sizes 1, 2, ..., n_candidate are examined.

IMSLS_MALLOWS_CP
Mallows Cp criterion is used, where subset sizes 1, 2, ..., n_candidate are examined.

IMSLS_MAX_N_BEST, int max_n_best (Input)
Number of best regressions to be found. If the R2 criterions are selected, the max_n_best best regressions for each subset size examined are found. If the adjusted R2 or Mallows Cp criterion is selected, the max_n_best overall regressions are found.
Default: max_n_best = 1

IMSLS_MAX_N_GOOD_SAVED, int max_n_good_saved (Input)
Maximum number of good regressions of each subset size to be saved in finding the best regressions. Argument max_n_good_saved must be greater than or equal to max_n_best. Normally, max_n_good_saved should be less than or equal to 10. It doesn't ever need to be larger than the maximum number of subsets for any subset size. Computing time required is inversely related to max_n_good_saved.
Default: max_n_good_saved = 10

IMSLS_CRITERIONS, int **index_criterions, float **criterions (Output)
Argument index_criterions is the address of a pointer to the internally allocated array of length nsize + 1(where nsize is equal to max_subset_size if optional argument IMSLS_R_SQUARED is specified; otherwise, nsize is equal to n_candidate) containing the locations in criterions of the first element for each subset size. For I = 0, 1, ..., nsize −1, element numbers index_criterions[I], index_criterions[I] + 1, ..., index_criterions[I + 1] − 1 of criterions correspond to the (I + 1)-st subset size. Argument criterions is the address of a pointer to the internally allocated array of length max (index_criterions [nsize] − 1 , n_candidate) containing in its first index_criterions [nsize] − 1 elements the criterion values for each subset considered, in increasing subset size order.

IMSLS_CRITERIONS_USER, int index_criterions[], float criterions[] (Output)
Storage for arrays index_criterions and criterions is provided by the user. An upper bound on the length of criterions is max(max_n_good_saved × nsize, n_candidate). See IMSLS_CRITERIONS.

IMSLS_INDEPENDENT_VARIABLES, int **index_variables, int **independent_variables (Output)
Argument index_variables is the address of a pointer to the internally allocated array of length nsize + 1 (where nsize is equal to max_subset_size if optional argument IMSLS_R_SQUARED is specified; otherwise, nsize is equal to n_candidate) containing the locations in independent_variables of the first element for each subset size. For I = 0, 1, ..., nsize − 1, element numbers index_variables[I], index_variables[I] + 1, ..., index_variables[I + 1] − 1 of independent_variables correspond to the (I+1)-st subset size. Argument independent_variables is the address of a pointer to the internally allocated array of length index_variables [nsize] − 1 containing the variable numbers for each subset considered and in the same order as in criterions.

IMSLS_INDEPENDENT_VARIABLES_USER, int index_variables[], int independent_variables[] (Output)
Storage for arrays index_variables and independent_variables is provided by the user. An upper bound for the length of independent_variables is as follows:

where nsize is equal to max_subset_size.

See IMSLS_INDEPENDENT_VARIABLES.

IMSLS_COEF_STATISTICS, int **index_coefficients, float **coefficients (Output)
Argument index_coefficients is the address of a pointer to the internally allocated array of length ntbest + 1 containing the locations in coefficients or the first row for each of the best regressions. Here, ntbest is the total number of best regression found and is equal to max_subset_size × max_n_best if IMSLS_R_SQUARED is specified, equal to max_n_best if either IMSLS_MALLOWS_CP or IMSLS_ADJ_R_SQUARED is specified, and equal to max_n_best × n_candidate, otherwise. For I = 0, 1, ..., ntbest − 1, rows index_coefficients[I], index_coefficients[I] + 1, ..., index_coefficients[I + 1] – 1 of coefficients correspond to the (I + 1)-st regression. Argument coefficients is the address of a pointer to the internally allocated array of size (index_coefficients [ntbest] − 1) × 5 containing statistics relating to the regression coefficients of the best models. Each row corresponds to a coefficient for a particular regression. The regressions are in order of increasing subset size. Within each subset size, the regressions are ordered so that the better regressions appear first. The statistic in the columns are as follows (inferences are conditional on the selected model):

Column	Description
0	variable number
1	coefficient estimate.
2	estimated standard error of the estimate
3	t-statistic for the test that the coefficient is 0
4	p-value for the two-sided t test

IMSLS_COEF_STATISTICS_USER, int index_coefficients[], float coefficients[] (Output)
Storage for arrays index_coefficients and coefficients is provided by the user. See IMSLS_COEF_STATISTICS.

IMSLS_INPUT_COV, int n_observations, float cov[] (Input)
Argument n_observations is the number of observations associated with array cov. Argument cov is an (n_candidate + 1) by (n_candidate + 1) array containing a variance-covariance or sum of squares and crossproducts matrix, in which the last column must correspond to the dependent variable. Array cov can be computed using imsls_f_covariances. Arguments x and y, and optional arguments frequencies and weights are not accessed when this option is specified. Normally, imsls_f_regression_selection computes cov from the input data matrices x and y. However, there may be cases when the user will wish to calculate the covariance matrix and manipulate it before calling imsls_f_regression_selection. See the description section below for a discussion of such cases.

Description

Function imsls_f_regression_selection finds the best subset regressions for a regression problem with n_candidate independent variables. Typically, the intercept is forced into all models and is not a candidate variable. In this case, a sum of squares and crossproducts matrix for the independent and dependent variables corrected for the mean is computed internally. There may be cases when it is convenient for the user to calculate the matrix; see the description of optional argument IMSLS_INPUT_COV.

“Best” is defined, on option, by one of the following three criteria:

•

R2 (in percent)

•

(adjusted R2 in percent)

Note that maximizing the criterion is equivalent to minimizing the residual mean square:

•

Mallows’ Cp statistic

Here, n is equal to the sum of the frequencies (or n_rows if IMSLS_FREQUENCIES is not specified) and SST is the total sum of squares.

SSEp is the error sum of squares in a model containing p regression parameters including β0 (or p − 1 of the n_candidate candidate variables). Variable

is the error mean square from the model with all n_candidate variables in the model. Hocking (1972) and Draper and Smith (1981, pp. 296−302) discuss these criteria.

Function imsls_f_regression_selection is based on the algorithm of Furnival and Wilson (1974). This algorithm finds max_n_good_saved candidate regressions for each possible subset size. These regressions are used to identify a set of best regressions. In large problems, many regressions are not computed. They may be rejected without computation based on results for other subsets; this yields an efficient technique for considering all possible regressions.

There are cases when the user may want to input the variance-covariance matrix rather than allow the function imsls_f_regression_selection to calculate it. This can be accomplished using optional argument IMSLS_INPUT_COV. Three situations in which the user may want to do this are as follows:

The intercept is not in the model. A raw (uncorrected) sum of squares and crossproducts matrix for the independent and dependent variables is required. Argument n_observations must be set to 1 greater than the number of observations. Form ATA, where A = [A, Y], to compute the raw sum of squares and crossproducts matrix.

An intercept is a candidate variable. A raw (uncorrected) sum of squares and crossproducts matrix for the constant regressor (= 1.0), independent, and dependent variables is required for cov. In this case, cov contains one additional row and column corresponding to the constant regressor. This row/column contains the sum of squares and crossproducts of the constant regressor with the independent and dependent variables. The remaining elements in cov are the same as in the previous case. Argument n_observations must be set to 1 greater than the number of observations.

There are m variables to be forced into the models. A sum of squares and crossproducts matrix adjusted for the m variables is required (calculated by regressing the candidate variables on the variables to be forced into the model). Argument n_observations must be set to m less than the number of observations.

Programming Notes

Function imsls_f_regression_selection can save considerable CPU time over explicitly computing all possible regressions. However, the function has some limitations that can cause unexpected results for users who are unaware of the limitations of the software.

For n_candidate + 1 > −log2 (ɛ), where ɛ is imsls_f_machine(4) and (imsls_d_machine(4) for double precision; see Chapter 15,Utilities), some results can be incorrect. This limitation arises because the possible models indicated (the model numbers 1, 2, ..., 2n_candidate) are stored as floating-point values; for sufficiently large n_candidate, the model numbers cannot be stored exactly. On many computers, this means imsls_f_regression_selection (for n_candidate > 24) and imsls_d_regression_selection (for n_candidate > 49) can produce incorrect results.

Function imsls_f_regression_selection eliminates some subsets of candidate variables by obtaining lower bounds on the error sum of squares from fitting larger models. First, the full model containing all n_candidate is fit sequentially using a forward stepwise procedure in which one variable enters the model at a time, and criterion values and model numbers for all the candidate variables that can enter at each step are stored. If linearly dependent variables are removed from the full model, error IMSLS_VARIABLES_DELETED is issued. If this error is issued, some submodels that contain variables removed from the full model because of linear dependency can be overlooked if they have not already been identified during the initial forward stepwise procedure. If error IMSLS_VARIABLES_DELETED is issued and you want the variables that were removed from the full model to be considered in smaller models, you can rerun the program with a set of linearly independent variables.

Examples

Example 1

This example uses a data set from Draper and Smith (1981, pp. 629−630). Function imsls_f_regression_selection is invoked to find the best regression for each subset size using the R2 criterion. By default, the function prints the results.

#include <imsls.h>

#define N_OBSERVATIONS 13

#define N_CANDIDATE 4

int main()

{

float x[N_OBSERVATIONS][N_CANDIDATE] = {

7., 26., 6., 60.,

1., 29., 15., 52.,

11., 56., 8., 20.,

11., 31., 8., 47.,

7., 52., 6., 33.,

11., 55., 9., 22.,

3., 71., 17., 6.,

1., 31., 22., 44.,

2., 54., 18., 22.,

21., 47., 4., 26.,

1., 40., 23., 34.,

11., 66., 9., 12.,

10., 68., 8., 12.

};

float y[N_OBSERVATIONS] = {78.5, 74.3, 104.3, 87.6, 95.9,

109.2, 102.7, 72.5, 93.1, 115.9, 83.8, 113.3, 109.4};

imsls_f_regression_selection(N_OBSERVATIONS, N_CANDIDATE,

&x[0][0], y, 0);

}

Output

Regressions with 1 variable(s) (R-squared)

Criterion Variables

67.5 4

66.6 2

53.4 1

28.6 3

Regressions with 2 variable(s) (R-squared)

Criterion Variables

97.9 1 2

97.2 1 4

93.5 3 4

68 2 4

54.8 1 3

Regressions with 3 variable(s) (R-squared)

Criterion Variables

98.2 1 2 4

98.2 1 2 3

98.1 1 3 4

97.3 2 3 4

Regressions with 4 variable(s) (R-squared)

Criterion Variables

98.2 1 2 3 4

Best Regression with 1 variable(s) (R-squared)

Variable Coefficient Standard Error t-statistic p-value

4 -0.7382 0.1546 -4.775 0.0006

Best Regression with 2 variable(s) (R-squared)

Variable Coefficient Standard Error t-statistic p-value

1 1.468 0.1213 12.10 0.0000

2 0.662 0.0459 14.44 0.0000

Best Regression with 3 variable(s) (R-squared)

Variable Coefficient Standard Error t-statistic p-value

1 1.452 0.1170 12.41 0.0000

2 0.416 0.1856 2.24 0.0517

4 -0.237 0.1733 -1.36 0.2054

Best Regression with 4 variable(s) (R-squared)

Variable Coefficient Standard Error t-statistic p-value

1 1.551 0.7448 2.083 0.0708

2 0.510 0.7238 0.705 0.5009

3 0.102 0.7547 0.135 0.8959

4 -0.144 0.7091 -0.203 0.8441

Example 2

This example uses the same data set as the first example, but Mallow’s Cp statistic is used as the criterion rather than R2. Note that when Mallow’s Cp statistic (or adjusted R2) is specified, the variable max_n_best indicates the total number of “best” regressions (rather than indicating the number of best regressions per subset size, as in the case of the R2 criterion). In this example, the three best regressions are found to be (1, 2), (1, 2, 4), and (1, 2, 3).

#include <imsls.h>

#define N_OBSERVATIONS 13

#define N_CANDIDATE 4

int main()

{

float x[N_OBSERVATIONS][N_CANDIDATE] =

{7., 26., 6., 60.,

1., 29., 15., 52.,

11., 56., 8., 20.,

11., 31., 8., 47.,

7., 52., 6., 33.,

11., 55., 9., 22.,

3., 71., 17., 6.,

1., 31., 22., 44.,

2., 54., 18., 22.,

21., 47., 4., 26.,

1., 40., 23., 34.,

11., 66., 9., 12.,

10., 68., 8., 12.};

float y[N_OBSERVATIONS] = {78.5, 74.3, 104.3, 87.6, 95.9,

109.2, 102.7, 72.5, 93.1, 115.9, 83.8, 113.3, 109.4};

int max_n_best = 3;

imsls_f_regression_selection(N_OBSERVATIONS, N_CANDIDATE,

(float *) x, y,

IMSLS_MALLOWS_CP,

IMSLS_MAX_N_BEST, max_n_best,

0);

}

Output

Regressions with 1 variable(s) (Mallows CP)

Criterion Variables

139 4

142 2

203 1

315 3

Regressions with 2 variable(s) (Mallows CP)

Criterion Variables

2.68 1 2

5.5 1 4

22.4 3 4

138 2 4

198 1 3

Regressions with 3 variable(s) (Mallows CP)

Criterion Variables

3.02 1 2 4

3.04 1 2 3

3.5 1 3 4

7.34 2 3 4

Regressions with 4 variable(s) (Mallows CP)

Criterion Variables

5 1 2 3 4

Best Regression with 2 variable(s) (Mallows CP)

Variable Coefficient Standard Error t-statistic p-value

1 1.468 0.1213 12.10 0.0000

2 0.662 0.0459 14.44 0.0000

Best Regression with 3 variable(s) (Mallows CP)

Variable Coefficient Standard Error t-statistic p-value

1 1.452 0.1170 12.41 0.0000

2 0.416 0.1856 2.24 0.0517

4 -0.237 0.1733 -1.36 0.2054

2nd Best Regression with 3 variable(s) (Mallows CP)

Variable Coefficient Standard Error t-statistic p-value

1 1.696 0.2046 8.29 0.0000

2 0.657 0.0442 14.85 0.0000

3 0.250 0.1847 1.35 0.2089

Warning Errors

IMSLS_VARIABLES_DELETED

At least one variable is deleted from the full model because the variance-covariance matrix “cov” is singular.

Fatal Errors

IMSLS_NO_VARIABLES

No variables can enter any model.