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[],
0)
Optional Arguments
IMSLS_COVARIANCE_MATRIX
Treat
the input vector covariances as a
covariance matrix. This option is the default.
or
IMSLS_CORRELATION_MATRIX
Treat
the input vector covariances as a
correlation 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 <stdio.h>
#include
<imsls.h>
#include <stdlib.h>
int main()
{
#define
N_VARIABLES 9
float
*values;
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,
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 <stdio.h>
#include
<imsls.h>
#include <stdlib.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
A
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.
