principal_components

Computes principal components.

Synopsis

#include <imsls.h>

float *imsls_f_principal_components (int n_variables, float covariances[], ..., 0)

The type double function is imsls_d_principal_components.

Required Arguments

int n_variables (Input)
Order of the covariance matrix.

float covariances[] (Input)
Array of length n_variables by n_variables containing the covariance or correlation matrix.

Return Value

An array of length n_variables containing the eigenvalues of the matrix covariances ordered from largest to smallest.

Synopsis with Optional Arguments

#include <imsls.h>

float *imsls_f_principal_components (int n_variables, float covariances[],

IMSLS_COVARIANCE_MATRIX, or

IMSLS_CORRELATION_MATRIX,

IMSLS_CUM_PERCENT, float **cum_percent,

IMSLS_CUM_PERCENT_USER, float cum_percent[],

IMSLS_EIGENVECTORS, float **eigenvectors,

IMSLS_EIGENVECTORS_USER, float eigenvectors[],

IMSLS_CORRELATIONS, float **correlations,

IMSLS_CORRELATIONS_USER, float correlations[],

IMSLS_STD_DEV, int n_degrees_freedom, float **std_dev,

IMSLS_STD_DEV_USER, int n_degrees_freedom, float std_dev[],

IMSLS_COV_COL_DIM, int cov_col_dim,

IMSLS_RETURN_USER, float eigenvalues[],

Optional Arguments

IMSLS_COVARIANCE_MATRIX (Input)
Treat the input vector covariances as a covariance matrix.

Default = IMSLS_COVARIANCE_MATRIX.

IMSLS_CORRELATION_MATRIX (Input)
Treat the input vector covariances as a correlation matrix.

Default = IMSLS_COVARIANCE_MATRIX.

IMSLS_CUM_PERCENT, float **cum_percent (Output)
The address of a pointer to an internally allocated array of length n_variables containing the cumulative percent of the total variances explained by each principal component.

IMSLS_CUM_PERCENT_USER, float cum_percent[] (Output)
Storage for array cum_percent is provided by the user. See IMSLS_CUM_PERCENT.

IMSLS_EIGENVECTORS, float **eigenvectors (Output)
The address of a pointer to an internally allocated array of length n_variables by n_variables containing the eigenvectors of covariances, stored columnwise. Each vector is normalized to have Euclidean length equal to the value one. Also, the sign of each vector is set so that the largest component in magnitude (the first of the largest if there are ties) is made positive.

IMSLS_EIGENVECTORS_USER, float eigenvectors[] (Output)
Storage for array eigenvectors is provided by the user. See IMSLS_EIGENVECTORS.

IMSLS_CORRELATIONS, float **correlations (Output)
The address of a pointer to an internally allocated array of length n_variables by n_variables containing the correlations of the principal components (the columns) with the observed/standardized variables (the rows). If IMSLS_COVARIANCE_MATRIX is specified, then the correlations are with the observed variables. Otherwise, the correlations are with the standardized (to a variance of 1.0) variables. In the principal component model for factor analysis, matrix correlations is the matrix of unrotated factor loadings.

IMSLS_CORRELATIONS_USER, float correlations[] (Output)
Storage for array correlations is provided by the user. See IMSLS_CORRELATIONS.

IMSLS_STD_DEV, int n_degrees_freedom, float **std_dev (Input/Output)
Argument n_degrees_freedom contains the number of degrees of freedom in covariances. Argument std_dev is the address of a pointer to an internally allocated array of length n_variables containing the estimated asymptotic standard errors of the eigenvalues.

IMSLS_STD_DEV_USER, int n_degrees_freedom, float std_dev[] (Input/Output)
Storage for array std_dev is provided by the user. See IMSLS_STD_DEV.

IMSLS_COV_COL_DIM int cov_col_dim (Input)
Column dimension of covariances.
Default: cov_col_dim = n_variables

IMSLS_RETURN_USER, float eigenvalues[] (Output)
User-supplied array of length n_variables containing the eigenvalues of covariances ordered from largest to smallest.

Description

Function imsls_f_principal_components finds the principal components of a set of variables from a sample covariance or correlation matrix. The characteristic roots, characteristic vectors, standard errors for the characteristic roots, and the correlations of the principal component scores with the original variables are computed. Principal components obtained from correlation matrices are the same as principal components obtained from standardized (to unit variance) variables.

The principal component scores are the elements of the vector y = ΓTx, where Γ is the matrix whose columns are the characteristic vectors (eigenvectors) of the sample covariance (or correlation) matrix and x is the vector of observed (or standardized) random variables. The variances of the principal component scores are the characteristic roots (eigenvalues) of the covariance (correlation) matrix.

Asymptotic variances for the characteristic roots were first obtained by Girschick (1939) and are given more recently by Kendall et al. (1983, p. 331). These variances are computed either for covariance matrices or for correlation matrices.

The correlations of the principal components with the observed (or standardized) variables are given in the matrix correlations. When the principal components are obtained from a correlation matrix, correlations is the same as the matrix of unrotated factor loadings obtained for the principal components model for factor analysis.

Examples

Example 1

In this example, eigenvalues of the covariance matrix are computed.

#include <imsls.h>

int main()

{

#define N_VARIABLES 9

float *values;

float covariances[N_VARIABLES * N_VARIABLES] = {

1.0, 0.523, 0.395, 0.471, 0.346, 0.426, 0.576, 0.434, 0.639,

0.523, 1.0, 0.479, 0.506, 0.418, 0.462, 0.547, 0.283, 0.645,

0.395, 0.479, 1.0, 0.355, 0.27, 0.254, 0.452, 0.219, 0.504,

0.471, 0.506, 0.355, 1.0, 0.691, 0.791, 0.443, 0.285, 0.505,

0.346, 0.418, 0.27, 0.691, 1.0, 0.679, 0.383, 0.149, 0.409,

0.426, 0.462, 0.254, 0.791, 0.679, 1.0, 0.372, 0.314, 0.472,

0.576, 0.547, 0.452, 0.443, 0.383, 0.372, 1.0, 0.385, 0.68,

0.434, 0.283, 0.219, 0.285, 0.149, 0.314, 0.385, 1.0, 0.47,

0.639, 0.645, 0.504, 0.505, 0.409, 0.472, 0.68, 0.47, 1.0

};

/* Perform analysis */

values = imsls_f_principal_components(N_VARIABLES, covariances,

0);

/* Print results. */

imsls_f_write_matrix("Eigenvalues", 1, N_VARIABLES, values,

0);

/* Free allocated memory. */

imsls_free(values);

}

Output

Eigenvalues

1 2 3 4 5 6

4.677 1.264 0.844 0.555 0.447 0.429

7 8 9

0.310 0.277 0.196

Example 2

In this example, principal components are computed for a nine-variable correlation matrix.

#include <imsls.h>

int main()

{

#define N_VARIABLES 9

float *values, *eigenvectors, *std_dev, *cum_percent, *a;

static float covariances[N_VARIABLES * N_VARIABLES] = {

1.0, 0.523, 0.395, 0.471, 0.346, 0.426, 0.576, 0.434, 0.639,

0.523, 1.0, 0.479, 0.506, 0.418, 0.462, 0.547, 0.283, 0.645,

0.395, 0.479, 1.0, 0.355, 0.27, 0.254, 0.452, 0.219, 0.504,

0.471, 0.506, 0.355, 1.0, 0.691, 0.791, 0.443, 0.285, 0.505,

0.346, 0.418, 0.27, 0.691, 1.0, 0.679, 0.383, 0.149, 0.409,

0.426, 0.462, 0.254, 0.791, 0.679, 1.0, 0.372, 0.314, 0.472,

0.576, 0.547, 0.452, 0.443, 0.383, 0.372, 1.0, 0.385, 0.68,

0.434, 0.283, 0.219, 0.285, 0.149, 0.314, 0.385, 1.0, 0.47,

0.639, 0.645, 0.504, 0.505, 0.409, 0.472, 0.68, 0.47, 1.0

};

/* Perform analysis */

values = imsls_f_principal_components(N_VARIABLES, covariances,

IMSLS_CORRELATION_MATRIX,

IMSLS_EIGENVECTORS, &eigenvectors,

IMSLS_STD_DEV, 100, &std_dev,

IMSLS_CUM_PERCENT, &cum_percent,

IMSLS_CORRELATIONS, &a,

0);

/* Print results */

imsls_f_write_matrix("Eigenvalues", 1, N_VARIABLES, values,

0);

imsls_f_write_matrix("Eigenvectors", N_VARIABLES, N_VARIABLES,

eigenvectors,

0);

imsls_f_write_matrix("STD", 1, N_VARIABLES, std_dev,

0);

imsls_f_write_matrix("PCT", 1, N_VARIABLES, cum_percent,

0);

imsls_f_write_matrix("A", N_VARIABLES, N_VARIABLES, a,

0);

/* Free allocated memory */

imsls_free(values);

imsls_free(eigenvectors);

imsls_free (cum_percent);

imsls_free (std_dev);

imsls_free(a);

}

Output

Eigenvalues

1 2 3 4 5 6

4.677 1.264 0.844 0.555 0.447 0.429

7 8 9

0.310 0.277 0.196

Eigenvectors

1 2 3 4 5 6

1 0.3462 -0.2354 0.1386 -0.3317 -0.1088 0.7974

2 0.3526 -0.1108 -0.2795 -0.2161 0.7664 -0.2002

3 0.2754 -0.2697 -0.5585 0.6939 -0.1531 0.1511

4 0.3664 0.4031 0.0406 0.1196 0.0017 0.1152

5 0.3144 0.5022 -0.0733 -0.0207 -0.2804 -0.1796

6 0.3455 0.4553 0.1825 0.1114 0.1202 0.0697

7 0.3487 -0.2714 -0.0725 -0.3545 -0.5242 -0.4355

8 0.2407 -0.3159 0.7383 0.4329 0.0861 -0.1969

9 0.3847 -0.2533 -0.0078 -0.1468 0.0459 -0.1498

7 8 9

1 0.1735 -0.1240 -0.0488

2 0.1386 -0.3032 -0.0079

3 0.0099 -0.0406 -0.0997

4 -0.4022 -0.1178 0.7060

5 0.7295 0.0075 0.0046

6 -0.3742 0.0925 -0.6780

7 -0.2854 -0.3408 -0.1089

8 0.1862 -0.1623 0.0505

9 -0.0251 0.8521 0.1225

STD

1 2 3 4 5 6

0.6498 0.1771 0.0986 0.0879 0.0882 0.0890

7 8 9

0.0944 0.0994 0.1113

PCT

1 2 3 4 5 6

0.520 0.660 0.754 0.816 0.865 0.913

7 8 9

0.947 0.978 1.000

1 2 3 4 5 6

1 0.7487 -0.2646 0.1274 -0.2471 -0.0728 0.5224

2 0.7625 -0.1245 -0.2568 -0.1610 0.5124 -0.1312

3 0.5956 -0.3032 -0.5133 0.5170 -0.1024 0.0990

4 0.7923 0.4532 0.0373 0.0891 0.0012 0.0755

5 0.6799 0.5646 -0.0674 -0.0154 -0.1875 -0.1177

6 0.7472 0.5119 0.1677 0.0830 0.0804 0.0456

7 0.7542 -0.3051 -0.0666 -0.2641 -0.3505 -0.2853

8 0.5206 -0.3552 0.6784 0.3225 0.0576 -0.1290

9 0.8319 -0.2848 -0.0071 -0.1094 0.0307 -0.0981

7 8 9

1 0.0966 -0.0652 -0.0216

2 0.0772 -0.1596 -0.0035

3 0.0055 -0.0214 -0.0442

4 -0.2240 -0.0620 0.3127

5 0.4063 0.0039 0.0021

6 -0.2084 0.0487 -0.3003

7 -0.1589 -0.1794 -0.0482

8 0.1037 -0.0854 0.0224

9 -0.0140 0.4485 0.0543

Warning Errors

IMSLS_100_DF	Because the number of degrees of freedom in “covariances” and “n_degrees_freedom” is less than or equal to 0, 100 degrees of freedom will be used.
IMSLS_COV_NOT_NONNEG_DEF	“eigenvalues[#]” = #. One or more eigenvalues much less than zero are computed. The matrix “covariances” is not nonnegative definite. In order to continue computations of “eigenvalues” and “correlations,” these eigenvalues are treated as 0.
IMSLS_FAILED_TO_CONVERGE	The iteration for the eigenvalue failed to converge in 100 iterations before deflating.