XploRe Help : CPCprop

Keywords - Function groups - @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Library:	stats
See also:	draftman factor FluryStepUp pca CPC CPCp CPCFGalg Jacobirot

Quantlet:		CPCprop
Description:		CPCprop computes the common eigenmatrix, eigenvalues, correlation coefficients, their standard errors and their estimated population covariance matrices from sample covariances of k groups under the restriction that eigenvalues among groups are linked by a positive constant. Estimation is done using maximum likelihood.

Reference(s):

Flury (1988): Common Principle Components Analysis and Related Multivariate Models

Usage:	`{B, betaerror, lambda, lambdaerror, psi, rho, rhoerror}=CPCprop(A,N)`
Input:
	`A`	p x p x k array of k covariances
	`N`	k x 1 vector of weights, usually number of observations in group k
Output:
	`B`	p x p matrix, common orthogonal transformation matrix
	`betaerror`	p x p matrix, standard errors of B
	`lambda`	p x k matrix, p eigenvalues of each group k
	`lambdaerror`	p x k matrix, p standard errors of eigenvalues of each group k
	`psi`	p x p x k array of k estimated population covariance matrices
	`rho`	1 x k vector of proportional constants
	`rhoerror`	1 x k vector of standard errors of the proportional constants

Note:

The precision for maximum deviation of an element is currently set to eps=1e-4. Maximum number of iterations is 15. A warning is issued if convergence is not achieved within 15 steps. The example is taken from Flury (1988).

Example:

library("stats") covar1=#(1.1544,0.9109,1.0330,0.7993)~#(0.9109,2.0381,0.7056,1.4083)~#(1.0330,0.7056,1.2100,0.7958)~#(0.7993,1.4083,0.7958,2.0277) covar2=#(0.9617,0.2806,0.9841,0.6775)~#(0.2806,1.8475,0.3129,1.2960)~#(0.9841,0.3129,1.2804,0.7923)~#(0.6775,1.2960,0.7923,1.7819) covar=reshape(covar1~covar2,#(4,4,2)) ; Sample size n=#(92,47) ZZ=CPCprop(covar,n) ZZ

Result:

Contents of ZZ.B

[1,]  0.73052  0.39352  0.50947 -0.22786
[2,] -0.13376  0.57339  -0.5257 -0.61399
[3,] -0.66401  0.39587  0.63371 -0.028232
[4,]  0.08685  0.59971 -0.24999  0.75518

Contents of ZZ.betaerror

[1,]  0.020284  0.036369  0.041575  0.066943
[2,]  0.036647  0.041222  0.074681  0.059171
[3,]  0.020077  0.04306  0.033054  0.076083
[4,]  0.040951  0.031867  0.088426  0.034234

Contents of ZZ.lambda

[1,]  0.12931  0.12356
[2,]     4.34   4.1469
[3,]   1.2849   1.2277
[4,]  0.58001   0.5542

Contents of ZZ.lambdaerror

[1,]  0.016582  0.018251
[2,]  0.55653  0.61254
[3,]  0.16477  0.18135
[4,]  0.074376  0.081862

Contents of ZZ.psi

[,,1,1,1,1,1,1]
[1,]   1.1047  0.70365   1.0319  0.76898
[2,]  0.70365   2.0029  0.57863   1.3908
[3,]   1.0319  0.57863   1.2536  0.80699
[4,]  0.76898   1.3908  0.80699    1.973
[,,2,1,1,1,1,1]
[1,]   1.0556  0.67233  0.98602  0.73476
[2,]  0.67233   1.9138  0.55288   1.3289
[3,]  0.98602  0.55288   1.1978  0.77108
[4,]  0.73476   1.3289  0.77108   1.8852

Contents of ZZ.rho

[1,]        1   0.9555

Contents of ZZ.rhoerror

[1,]        0  0.12223

Author: M. R. Fengler, W. Haerdle, 20010717 license MD*Tech