poly_regression

Performs a polynomial least-squares regression.

Synopsis

#include <imsls.h>

float *imsls_f_poly_regression (int n_observations, float x[], float y[], int degree, ..., 0)

The type double function is imsls_d_poly_regression.

Required Arguments

int n_observations (Input)
Number of observations.

float x[] (Input)
Array of length n_observations containing the independent variable.

float y[] (Input)
Array of length n_observations containing the dependent variable.

int degree (Input)
Degree of the polynomial.

Return Value

A pointer to the array of size degree + 1 containing the coefficients of the fitted polynomial. If a fit cannot be computed, NULL is returned.

Synopsis with Optional Arguments

#include <imsls.h>

float *imsls_f_poly_regression (int n_observations, float x[], float y[], int degree,

IMSLS_WEIGHTS, float weights[],

IMSLS_SSQ_POLY, float **ssq_poly,

IMSLS_SSQ_POLY_USER, float ssq_poly[],

IMSLS_SSQ_POLY_COL_DIM, int ssq_poly_col_dim,

IMSLS_SSQ_LOF, float **ssq_lof,

IMSLS_SSQ_LOF_USER, float ssq_lof[],

IMSLS_SSQ_LOF_COL_DIM, int ssq_lof_col_dim,

IMSLS_X_MEAN, float *x_mean,

IMSLS_X_VARIANCE, float *x_variance,

IMSLS_ANOVA_TABLE, float **anova_table,

IMSLS_ANOVA_TABLE_USER, float anova_table[]

IMSLS_DF_PURE_ERROR, int *df_pure_error,

IMSLS_SSQ_PURE_ERROR, float *ssq_pure_error,

IMSLS_RESIDUAL, float **residual,

IMSLS_RESIDUAL_USER, float residual[],

IMSLS_POLY_REGRESSION_INFO, Imsls_f_poly_regression **poly_info,

IMSLS_RETURN_USER, float coefficients[],

0)

Optional Arguments

IMSLS_WEIGHTS, float weights[] (Input)
Array with n_observations components containing the array of weights for the observation.
Default: weights[] = 1

IMSLS_SSQ_POLY, float **ssq_poly (Output)
Address of a pointer to the internally allocated array containing the sequential sums of squares and other statistics. Row i corresponds to xi, i = 0, ..., degree − 1, and the columns are described as follows:

Column

Description

0

degrees of freedom

1

Sums of squares

2

F-statistic

3

p-value

IMSLS_SSQ_POLY_USER, float ssq_poly[] (Output)
Storage for array ssq_poly is provided by the user. See IMSLS_SSQ_POLY.

IMSLS_SSQ_POLY_COL_DIM, int ssq_poly_col_dim (Input)
Column dimension of ssq_poly.
Default: ssq_poly_col_dim = 4

IMSLS_SSQ_LOF, float **ssq_lof (Output)
Address of a pointer to the internally allocated array containing the lack-of-fit statistics. Row icorresponds to xi, i = 0, ..., degree − 1, and the columns are described in the following table:

Column

Description

0

degrees of freedom

1

lack-of-fit sums of squares

2

F-statistic for testing lack-of-fit for a polynomial model of degree i

3

p-value for the test

IMSLS_SSQ_LOF_USER, float ssq_lof[] (Output)
Storage for array ssq_lof is provided by the user. See IMSLS_SSQ_LOF.

IMSLS_SSQ_LOF_COL_DIM, int ssq_lof_col_dim (Input)
Column dimension of ssq_lof.
Default: ssq_lof_col_dim = 4

IMSLS_X_MEAN, float *x_mean (Output)
Mean of x.

IMSLS_X_VARIANCE, float *x_variance (Output)
Variance of x.

IMSLS_ANOVA_TABLE, float **anova_table (Output)
Address of a pointer to the array containing the analysis of variance table.

Column

Description

0

degrees of freedom for the model

1

degrees of freedom for error

2

total (corrected) degrees of freedom

3

sum of squares for the model

4

sum of squares for error

5

total (corrected) sum of squares

6

model mean square

7

error mean square

8

overall F-statistic

9

p-value

10

R2 (in percent)

11

adjusted R2 (in percent)

12

estimate of the standard deviation

13

overall mean of y

14

coefficient of variation (in percent)

Note that the p‑value is returned as 0.0 when the value is so small that all significant digits have been lost.

IMSLS_ANOVA_TABLE_USER, float anova_table[] (Output)
Storage for anova_table is provided by the user. See IMSLS_ANOVA_TABLE.

IMSLS_DF_PURE_ERROR, int *df_pure_error (Output)
If specified, the degrees of freedom for pure error are returned in df_pure_error.

IMSLS_SSQ_PURE_ERROR, float *ssq_pure_error (Output)
If specified, the sums of squares for pure error are returned in ssq_pure_error.

IMSLS_RESIDUAL, float **residual (Output)
Address of a pointer to the array containing the residuals.

IMSLS_RESIDUAL_USER, float residual[] (Output)
Storage for array residual is provided by the user. See IMSLS_RESIDUAL.

IMSLS_POLY_REGRESSION_INFO, Imsls_f_poly_regression **poly_info (Output)
Address of a pointer to an internally allocated structure containing the information about the polynomial fit required as input for IMSL function imsls_f_poly_prediction.

IMSLS_RETURN_USER, float coefficients[] (Output)
If specified, the least-squares solution for the regression coefficients is stored in array coefficients of size degree + 1 provided by the user.

Description

Function imsls_f_poly_regression computes estimates of the regression coefficients in a polynomial (curvilinear) regression model. In addition to the computation of the fit, imsls_f_poly_regression computes some summary statistics. Sequential sums of squares attributable to each power of the independent variable (stored in ssq_poly) are computed. These are useful in assessing the importance of the higher order powers in the fit. Draper and Smith (1981, pp. 101−102) and Neter and Wasserman (1974, pp. 278−287) discuss the interpretation of the sequential sums of squares. The statistic R2 is the percentage of the sum of squares of y about its mean explained by the polynomial curve. Specifically,

 

where

 

is the fitted yvalue at xi and is the mean of y. This statistic is useful in assessing the overall fit of the curve to the data. R2 must be between 0 and 100 percent, inclusive. R2 = 100 percent indicates a perfect fit to the data.

Estimates of the regression coefficients in a polynomial model are computed using orthogonal polynomials as the regressor variables. This reparameterization of the polynomial model in terms of orthogonal polynomials has the advantage that the loss of accuracy resulting from forming powers of the x-values is avoided. All results are returned to the user for the original model (power form).

Function imsls_f_poly_regression is based on the algorithm of Forsythe (1957). A modification to Forsythe’s algorithm suggested by Shampine (1975) is used for computing the polynomial coefficients. A discussion of Forsythe’s algorithm and Shampine’s modification appears in Kennedy and Gentle (1980, pp. 342−347).

Examples

Example 1

A polynomial model is fitted to 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 coffee dispensers. Responses for 14 similar cafeterias are in the data set. A graph of the results is also given.

 

#include <imsls.h>

 

#define DEGREE 2

#define NOBS 14

 

int main()

{

float *coefficients;

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};

 

coefficients = imsls_f_poly_regression (NOBS, x, y, DEGREE, 0);

 

imsls_f_write_matrix("Least-Squares Polynomial Coefficients",

DEGREE + 1, 1, coefficients,

IMSLS_ROW_NUMBER_ZERO,

0);

}

Output

 

Least-Squares Polynomial Coefficients

0 503.3

1 78.9

2 -4.0

 

Figure 1,  A Polynomial Fit

Example 2

This example is a continuation of the initial example. Here, many optional arguments are used.

 

#include <imsls.h>

 

#define DEGREE 2

#define NOBS 14

 

int main()

{

int iset = 1, dfpe;

float *coefficients, *anova_table, sspe, *ssqpoly, *ssqlof;

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

};

char *coef_rlab[2];

char *coef_clab[] = {" ", "intercept", "linear", "quadratic"};

char *stat_clab[] = {" ", "Degrees of\nFreedom", "Sum of\nSquares",

"\nF-Statistic", "\np-value"};

 

char *anova_rlab[] = {

"degrees of freedom for regression",

"degrees of freedom for error",

"total (corrected) degrees of freedom",

"sum of squares for regression",

"sum of squares for error",

"total (corrected) sum of squares",

"regression mean square",

"error mean square", "F-statistic",

"p-value", "R-squared (in percent)",

"adjusted R-squared (in percent)",

"est. standard deviation of model error",

"overall mean of y",

"coefficient of variation (in percent)"

};

 

coefficients = imsls_f_poly_regression(NOBS, x, y, DEGREE,

IMSLS_SSQ_POLY, &ssqpoly,

IMSLS_SSQ_LOF, &ssqlof,

IMSLS_ANOVA_TABLE, &anova_table,

IMSLS_DF_PURE_ERROR, &dfpe,

IMSLS_SSQ_PURE_ERROR, &sspe,

0);

 

imsls_write_options(-1, &iset);

 

imsls_f_write_matrix("Least Squares Polynomial Coefficients",

1, DEGREE + 1, coefficients,

IMSLS_COL_LABELS, coef_clab,

0);

 

coef_rlab[0] = coef_clab[2];

coef_rlab[1] = coef_clab[3];

 

imsls_f_write_matrix("Sequential Statistics", DEGREE, 4,

ssqpoly,

IMSLS_COL_LABELS, stat_clab,

IMSLS_ROW_LABELS, coef_rlab,

IMSLS_WRITE_FORMAT, "%3.1f%8.1f%6.1f%6.4f",

0);

 

imsls_f_write_matrix("Lack-of-Fit Statistics", DEGREE, 4,

ssqlof,

IMSLS_COL_LABELS, stat_clab,

IMSLS_ROW_LABELS, coef_rlab,

IMSLS_WRITE_FORMAT, "%3.1f%8.1f%6.1f%6.4f",

0);

 

imsls_f_write_matrix("* * * Analysis of Variance * * *\n", 15,

1, anova_table,

IMSLS_ROW_LABELS, anova_rlab,

IMSLS_WRITE_FORMAT, "%9.2f",

0);

}

Output

 

Least Squares Polynomial Coefficients

intercept linear quadratic

503.3 78.9 -4.0

 

Sequential Statistics

Degrees of Sum of

Freedom Squares F-Statistic p-value

linear 1.0 220644.2 3415.8 0.0000

quadratic 1.0 4387.7 67.9 0.0000

 

Lack-of-Fit Statistics

Degrees of Sum of

Freedom Squares F-Statistic p-value

linear 5.0 4793.7 22.0 0.0004

quadratic 4.0 405.9 2.3 0.1548

* * * Analysis of Variance * * *

 

degrees of freedom for regression 2.00

degrees of freedom for error 11.00

total (corrected) degrees of freedom 13.00

sum of squares for regression 225031.94

sum of squares for error 710.55

total (corrected) sum of squares 225742.48

regression mean square 112515.97

error mean square 64.60

F-statistic 1741.86

p-value 0.00

R-squared (in percent) 99.69

adjusted R-squared (in percent) 99.63

est. standard deviation of model error 8.04

overall mean of y 710.99

coefficient of variation (in percent) 1.13

Warning Errors

IMSLS_CONSTANT_YVALUES

The y values are constant. A zero-order polynomial is fit. High order coefficients are set to zero.

IMSLS_FEW_DISTINCT_XVALUES

There are too few distinct x values to fit the desired degree polynomial. High order coefficients are set to zero.

IMSLS_PERFECT_FIT

A perfect fit was obtained with a polynomial of degree less than degree. High order coefficients are set to zero.

Fatal Errors

IMSLS_NONNEG_WEIGHT_REQUEST_2

All weights must be nonnegative.

IMSLS_ALL_OBSERVATIONS_MISSING

Each (x, y) point contains NaN. There are no valid data.

IMSLS_CONSTANT_XVALUES

The x values are constant.