Computes predicted values, confidence intervals, and diagnostics after fitting a polynomial regression model.
Synopsis
#include <imsls.h>
float *imsls_f_poly_prediction (Imsls_f_poly_regression *poly_info, int n_predict, float x[], ..., 0)
The type double function is imsls_d_poly_prediction.
Required Arguments
Imsls_f_poly_regression
*poly_info (Input)
Pointer to a structure of type
Imsls_f_poly_regression. See function imsls_f_poly_regression.
int n_predict
(Input)
Length of array x.
float x[]
(Input)
Array of length n_predict containing
the values of the independent variable for which calculations are to be
performed.
Return Value
A pointer to an internally allocated array of length n_predict containing the predicted values.
Synopsis with Optional Arguments
#include <imsls.h>
float
*imsls_f_poly_prediction (Imsls_f_poly_regression *poly_info,
int n_predict, float
x[],
IMSLS_CONFIDENCE, float
confidence,
IMSLS_WEIGHTS, float weights[],
IMSLS_SCHEFFE_CI, float
**lower_limit, float **upper_limit,
IMSLS_SCHEFFE_CI_USER, float
lower_limit[], float upper_limit[],
IMSLS_POINTWISE_CI_POP_MEAN, float
**lower_limit,
float **upper_limit,
IMSLS_POINTWISE_CI_POP_MEAN_USER, float
lower_limit[],
float upper_limit[],
IMSLS_POINTWISE_CI_NEW_SAMPLE, float
**lower_limit,
float **upper_limit,
IMSLS_POINTWISE_CI_NEW_SAMPLE_USER, float lower_limit[],
float upper_limit[],
IMSLS_LEVERAGE, float
**leverage,
IMSLS_LEVERAGE_USER, float
leverage[],
IMSLS_RETURN_USER, float
y_hat[],
IMSLS_Y, float
y[],
IMSLS_RESIDUAL, float
**residual,
IMSLS_RESIDUAL_USER, float
residual[],
IMSLS_STANDARDIZED_RESIDUAL, float **standardized_residual,
IMSLS_STANDARDIZED_RESIDUAL_USER,
float standardized_residual[],
IMSLS_DELETED_RESIDUAL, float
**deleted_residual,
IMSLS_DELETED_RESIDUAL_USER, float
deleted_residual[],
IMSLS_COOKSD, float
**cooksd,
IMSLS_COOKSD_USER, float
cooksd[],
IMSLS_DFFITS, float
**dffits,
IMSLS_DFFITS_USER, float
dffits[],
0)
Optional Arguments
IMSLS_CONFIDENCE, float
confidence (Input)
Confidence level for both two-sided
interval estimates on the mean and for two-sided prediction intervals in
percent. Argument confidence must be in
the range [0.0, 100.0). For one-sided intervals with confidence level onecl, where
50.0 ≤ onecl < 100.0,
set confidence = 100.0 – 2.0
*
(100.0 − onecl).
Default:
confidence = 95.0
IMSLS_WEIGHTS, float weights[]
(Input)
Array of length n_predict containing
the weight for each row of x. The computed
prediction interval uses SSE/(DFE*weights[i]) for the estimated
variance of a future response.
Default: weights[] = 1
IMSLS_SCHEFFE_CI, float **lower_limit, float
**upper_limit (Output)
Array lower_limit is the
address of a pointer to an internally allocated array of length n_predict containing
the lower confidence limits of Scheffé confidence intervals corresponding to the
rows of x. Array
upper_limit is
the address of a pointer to an internally allocated array of length n_predict containing
the upper confidence limits of Scheffé confidence intervals corresponding to the
rows of x.
IMSLS_SCHEFFE_CI_USER, float lower_limit[], float
upper_limit[] (Output)
Storage for arrays lower_limit and
upper_limit is provided by the user. See IMSLS_SCHEFFE_CI.
IMSLS_POINTWISE_CI_POP_MEAN, float **lower_limit,
float **upper_limit
(Output)
Array lower_limit is the
address of a pointer to an internally allocated array of length n_predict containing
the lower confidence limits of the confidence intervals for two-sided interval
estimates of the means, corresponding to the rows of x. Array upper_limit is the
address of a pointer to an internally allocated array of length n_predict containing
the upper confidence limits of the confidence intervals for two-sided interval
estimates of the means, corresponding to the rows
of x.
IMSLS_POINTWISE_CI_POP_MEAN_USER, float lower_limit[],
float upper_limit[]
(Output)
Storage for arrays lower_limit and upper_limit is
provided by the user. See IMSLS_POINTWISE_CI_POP_MEAN.
IMSLS_POINTWISE_CI_NEW_SAMPLE, float **lower_limit,
float **upper_limit
(Output)
Array lower_limit is the
address of a pointer to an internally allocated array of length n_predict containing
the lower confidence limits of the confidence intervals for two-sided prediction
intervals, corresponding to the rows of x. Array upper_limit is the
address of a pointer to an internally allocated array of length n_predict containing
the upper confidence limits of the confidence intervals for two-sided prediction
intervals, corresponding to the rows of x.
IMSLS_POINTWISE_CI_NEW_SAMPLE_USER, float lower_limit[],
float upper_limit[]
(Output)
Storage for arrays lower_limit and upper_limit is
provided by the user. See IMSLS_POINTWISE_CI_NEW_SAMPLE.
IMSLS_LEVERAGE, float
**leverage (Output)
Address of a pointer to an internally
allocated array of length n_predict containing
the leverages.
IMSLS_LEVERAGE_USER, float
leverage[] (Output)
Storage for array leverage is provided
by the user. See IMSLS_LEVERAGE.
IMSLS_RETURN_USER, float y_hat[]
(Output)
Storage for array y_hat is provided by
the user. The length n_predict array
contains the predicted values.
IMSLS_Y
float
y[] (Input)
Array of length n_predict containing
the observed responses.
Note: IMSLS_Y must be specified if any of the following optional arguments are specified.
IMSLS_RESIDUAL, float
**residual (Output)
Address of a pointer to an internally
allocated array of length n_predict containing
the residuals.
IMSLS_RESIDUAL_USER, float
residual[] (Output)
Storage for array residual is provided
by the user. See IMSLS_RESIDUAL.
IMSLS_STANDARDIZED_RESIDUAL, float
**standardized_residual (Output)
Address of a pointer to
an internally allocated array of length n_predict containing
the standardized residuals.
IMSLS_STANDARDIZED_RESIDUAL_USER,
float standardized_residual[]
(Output)
Storage for array standardized_residual
is provided by the user. See IMSLS_STANDARDIZED_RESIDUAL.
IMSLS_DELETED_RESIDUAL, float
**deleted_residual (Output)
Address of a pointer to an
internally allocated array of length n_predict containing
the deleted residuals.
IMSLS_DELETED_RESIDUAL_USER, float
deleted_residual[] (Output)
Storage for array deleted_residual is
provided by the user. See IMSLS_DELETED_RESIDUAL.
IMSLS_COOKSD, float **cooksd
(Output)
Address of a pointer to an internally allocated array of length
n_predict
containing the Cook’s D statistics.
IMSLS_COOKSD_USER, float cooksd[]
(Output)
Storage for array cooksd is provided by
the user. See IMSLS_COOKSD.
IMSLS_DFFITS, float **dffits
(Output)
Address of a pointer to an internally allocated array of length
n_predict
containing the DFFITS statistics.
IMSLS_DFFITS_USER, float dffits[]
(Output)
Storage for array dffits is provided by
the user. See IMSLS_DFFITS.
Description
Function imsls_f_poly_prediction assumes a polynomial model
where the observed values of the yi’s constitute the response, the xi’s are the settings of the independent variable, the βj’s are the regression coefficients and the ɛi’s are the errors that are independently distributed normal with mean 0 and the following variance:
Given the results of a polynomial regression, fitted using orthogonal polynomials and weights wi, function imsls_f_poly_prediction produces predicted values, residuals, confidence intervals, prediction intervals, and diagnostics for outliers and in influential cases.
Often, a predicted value and confidence interval are desired for a setting of the independent variable not used in computing the regression fit. This is accomplished by simply using a different x matrix when calling imsls_f_poly_prediction than was used for the fit (function imsls_f_poly_regression). See Example 1.
Results from function imsls_f_poly_regression, which produces the fit using orthogonal polynomials, are used for input by the structure poly_info. The fitted model from imsls_f_poly_regression is
where the zi’s are settings of the
independent variable x scaled to the interval
[−2, 2] and the
pj (z)’s are the orthogonal polynomials. The
XTX matrix for this
model is a diagonal matrix with elements dj. The case
statistics are easily computed from this model and are equal to those from the
original polynomial model with βj’s as the
regression coefficients.
The leverage is computed as follows:
The estimated variance of
is given by the following:
The computation of the remainder of the case statistics follows easily from the definitions. See Diagnostics for Individual Cases” for the definition of the case diagnostics.
Often, predicted values and confidence intervals are desired for combinations of settings of the independent variables not used in computing the regression fit. This can be accomplished by defining a new data matrix. Since the information about the model fit is input in poly_info, it is not necessary to send in the data set used for the original calculation of the fit, i.e., only variable combinations for which predictions are desired need be entered in x.
Examples
Example 1
A polynomial model is fit to the data discussed by
Neter and Wasserman
(1974, pp. 279–285). The data set contains the
response variable y measuring coffee sales (in hundred gallons) and the
number of self-service dispensers. Responses for 14 similar cafeterias are in
the data set.
#include
<imsls.h>
main()
{
Imsls_f_poly_regression *poly_info;
float *y_hat, *coefficients;
int n_observations =
14;
int degree =
2;
int n_predict =
8;
float x[] = {0.0, 0.0, 1.0,
1.0, 2.0, 2.0,
4.0,
4.0, 5.0, 5.0, 6.0, 6.0, 7.0, 7.0};
float y[] = {508.1, 498.4, 568.2, 577.3, 651.7, 657.0,
755.3,
758.9, 787.6, 792.1, 841.4, 831.8, 854.7, 871.4};
float x2[] = {0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0,
7.0};
/* Generate the polynomial regression
fit*/
coefficients = imsls_f_poly_regression
(n_observations, x, y,
degree,
IMSLS_POLY_REGRESSION_INFO, &poly_info, 0);
/*
Compute predicted values */
y_hat =
imsls_f_poly_prediction(poly_info, n_predict, x2,
0);
/* Print predicted values
*/
imsls_f_write_matrix("Predicted Values", 1, n_predict,
y_hat, 0);
free(coefficients);
free(y_hat);
return;
}
Output
Predicted Values
1
2
3
4
5
6
503.3
578.3
645.4
704.4
755.6
798.8
7
8
834.1
861.4
Example 2
Predicted values, confidence intervals, and diagnostics are computed for the data set described in the first example.
#include
<imsls.h>
main()
{
#define N_PREDICT
14
Imsls_f_poly_regression
*poly_info;
float *coefficients,
y_hat[N_PREDICT],
lower_ci[N_PREDICT],
upper_ci[N_PREDICT],
lower_pi[N_PREDICT],
upper_pi[N_PREDICT],
s_residual[N_PREDICT],
d_residual[N_PREDICT],
leverage[N_PREDICT], cooksd[N_PREDICT],
dffits[N_PREDICT], lower_scheffe[N_PREDICT],
upper_scheffe[N_PREDICT];
int n_observations =
N_PREDICT;
int degree
= 2;
float x[] = {0.0, 0.0, 1.0,
1.0, 2.0, 2.0,
4.0,
4.0, 5.0, 5.0, 6.0, 6.0, 7.0, 7.0};
float y[] = {508.1, 498.4, 568.2, 577.3, 651.7, 657.0,
755.3,
758.9, 787.6, 792.1, 841.4, 831.8, 854.7, 871.4};
/*
Generate the polynomial regression fit*/
coefficients =
imsls_f_poly_regression (n_observations, x, y,
degree,
IMSLS_POLY_REGRESSION_INFO, &poly_info, 0);
/*
Compute predicted values and case statistics */
imsls_f_poly_prediction(poly_info, N_PREDICT, x,
IMSLS_RETURN_USER, y_hat,
IMSLS_POINTWISE_CI_POP_MEAN_USER,
lower_ci, upper_ci,
IMSLS_POINTWISE_CI_NEW_SAMPLE_USER, lower_pi, upper_pi,
IMSLS_Y, y,
IMSLS_STANDARDIZED_RESIDUAL_USER,
s_residual,
IMSLS_DELETED_RESIDUAL_USER, d_residual,
IMSLS_LEVERAGE_USER, leverage,
IMSLS_COOKSD_USER, cooksd,
IMSLS_DFFITS_USER,
dffits,
IMSLS_SCHEFFE_CI_USER,
lower_scheffe, upper_scheffe,
0);
/* Print results */
imsls_f_write_matrix("Predicted Values", 1, N_PREDICT, y_hat, 0);
imsls_f_write_matrix("Lower Scheffe CI", 1, N_PREDICT,
lower_scheffe, 0);
imsls_f_write_matrix("Upper Scheffe CI", 1, N_PREDICT,
upper_scheffe,
0);
imsls_f_write_matrix("Lower CI", 1, N_PREDICT,
lower_ci, 0);
imsls_f_write_matrix("Upper CI", 1,
N_PREDICT, upper_ci, 0);
imsls_f_write_matrix("Lower PI",
1, N_PREDICT, lower_pi, 0);
imsls_f_write_matrix("Upper
PI", 1, N_PREDICT, upper_pi, 0);
imsls_f_write_matrix("Standardized Residual", 1, N_PREDICT,
s_residual, 0);
imsls_f_write_matrix("Deleted Residual", 1, N_PREDICT,
d_residual, 0);
imsls_f_write_matrix("Leverage", 1, N_PREDICT, leverage,
0);
imsls_f_write_matrix("Cooks Distance", 1, N_PREDICT,
cooksd, 0);
imsls_f_write_matrix("DFFITS", 1, N_PREDICT,
dffits, 0);
free(coefficients);
return;
}
Output
Predicted Values
1
2
3
4
5
6
503.3
503.3
578.3
578.3
645.4
645.4
7
8
9
10
11
12
755.6
755.6
798.8 798.8
834.1
834.1
13
14
861.4
861.4
Lower Scheffe CI
1
2
3
4
5
6
489.8
489.8
569.5
569.5
636.5
636.5
7
8
9
10
11
12
745.7
745.7
790.2
790.2
825.5
825.5
13
14
847.7
847.7
Upper Scheffe CI
1
2
3
4
5
6
516.9
516.9
587.1
587.1
654.2
654.2
7
8
9
10
11
12
765.5
765.5
807.4
807.4
842.7
842.7
13
14
875.1
875.1
Lower CI
1
2
3
4
5
6
492.8
492.8
571.5
571.5
638.4
638.4
7
8
9
10
11
12
747.9
747.9
792.1
792.1
827.4
827.4
13
14
850.7
850.7
Upper CI
1
2
3
4
5
6
513.9
513.9
585.2
585.2
652.3
652.3
7
8
9
10
11
12
763.3
763.3
805.5
805.5
840.8
840.8
13
14
872.1
872.1
Lower PI
1
2
3
4
5
6
482.8
482.8
559.3
559.3
626.4
626.4
7 8
9
10
11
12
736.3
736.3
779.9
779.9
815.2
815.2
13
14
840.8
840.8
Upper PI
1
2
3 4
5
6
523.9
523.9
597.3
597.3
664.3
664.3
7
8
9
10
11
12
774.9
774.9
817.7
817.7
853.0
853.0
13
14
882.1
882.1
Standardized Residual
1
2
3
4
5
6
0.737
-0.766 -1.366
-0.137
0.859
1.575
7
8
9
10
11
12
-0.041
0.456 -1.507
-0.902 0.982
-0.308
13
14
-1.051
1.557
Deleted Residual
1
2
3
4
5
6
0.720
-0.751 -1.429
-0.131
0.848
1.707
7
8
9
10
11
12
-0.039
0.439 -1.613
-0.894 0.980
-0.295
13
14
-1.056
1.681
Leverage
1
2
3
4
5
6
0.3554
0.3554 0.1507
0.1507 0.1535
0.1535
7
8
9 10
11
12
0.1897
0.1897 0.1429
0.1429 0.1429
0.1429
13
14
0.3650
0.3650
Cooks Distance
1
2
3
4
5
6
0.0997
0.1080 0.1104
0.0011 0.0446
0.1500
7
8
9
10
11
12
0.0001
0.0162 0.1262
0.0452 0.0536
0.0053
13
14
0.2116
0.4644
DFFITS
1
2
3
4
5
6
0.535
-0.558 -0.602
-0.055
0.361
0.727
7
8
9
10 11
12
-0.019 0.212
-0.659 -0.365
0.400
-0.120
13
14
-0.801 1.274
Warning Errors
IMSLS_LEVERAGE_GT_1 A leverage (= #) much greater than one is computed. It is set to 1.0.
IMSLS_DEL_MSE_LT_0 A deleted residual mean square (= #) much less than zero is computed. It is set to zero.
Fatal Errors
IMSLS_NEG_WEIGHT “weights[#]” = #. Weights must be nonnegative.
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