XploRe Help : CPCFGalg

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:	nummath
See also:	pca CPC CPCprop CPCp

Quantlet:		CPCFGalg
Description:		CPCFGalg implements the FG-Algorithm which finds a common orthogonal transformation matrix in order to simultaneously diagonalize several positive definite symmetric matrices.

Reference(s):

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

Usage:	`{B,Phi}=CPCFGalg(A,N)`
Input:
	`A`	p x p x k array of k positive definite symmetric matrices
	`N`	k x 1 vector of weights, usually number of observations in group k
Output:
	`B`	p x p matrix, common orthogonal transformation matrix
	`Phi`	scalar: Measure of deviation from diagonality. Phi is 1, if complete diagonalizability of the matrices in A was achieved. Phi increases monotonically in deviation from diagonality.

Note:

The FG-algorithm consists of two algorithms (F and G). Maximum number of iterations for the outer F algorithm is maxit=15 and the precision is eps=1e-10. In the inner G algorithm the maximum number of iterations is set to maxitGALG=10 and the precision is epsGALG=1e-10. A warning is displayed if the maximum number of iterations is violated for F. You may want to use a higher number of steps. Sometimes it will also be necessary to use a different initial (orthogonal) matrix for the F algorithm. Currently this is the unit matrix. Example is taken from Flury.

Example:

library("nummath") A11=#(45,10,0,5,0,0)~#(10,45,5,0,0,0)~#(0,5,45,10,0,0) A12=#(5,0,10,45,0,0)~#(0,0,0,0,13.6,4.8)~#(0,0,0,0,4.8,16.4) A21=#(27.5,-12.5,-0.5,-4.5,-2.04,3.72)~#(-12.5,27.5,-4.5,-0.5,2.04,-3.72)~#(-0.5,-4.5,24.5,-9.5,-3.72,-2.04) A22=#(-4.5,-0.5,-9.5,24.5,3.72,2.04)~#(-2.04,2.04,-3.72,3.72,54.76,-4.68)~#(3.72,-3.72,-2.04,2.04,-4.68,51.24) A1=A11~A12 A2=A21~A22 A=reshape(A1~A2, 6|6|2) {B,Phi}=CPCFGalg(A,1|1) B Phi

Result:

Contents of B

[1,]      0.5  -0.5783      0.5  0.40584 -0.018199  0.023238
[2,]      0.5   0.5783      0.5 -0.40584  0.018199 -0.023238
[3,]     -0.5 -0.40577      0.5 -0.57435 -0.0081685 -0.073535
[4,]     -0.5  0.40577      0.5  0.57435  0.0081685  0.073535
[5,] -1.7117e-18 -0.041925  4.8648e-18 -0.057801  0.80146  0.59377
[6,]  2.8062e-17 -0.0099156  8.3002e-18 -0.086566 -0.59738  0.79721

Contents of Phi
[1,]   1.0176

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