FGalgorithm-package.Rd: Execute the Flury and Gautschi diagonalisation algorithm,...

In FGalgorithm: Flury and Gautschi algorithms

Description

The minimization of the objective function

Φ (B) = {∏\limits_{i = 1}^k {≤ft[ {\frac{{\det (diag(B'{A_i}B))}}{{\det (B'{A_i}B)}}} \right]} ^{{n_i}}}

is required for a potpourri of statistical problems. This algorithm (Flury & Gautschi, 1984) is designed to find an orthogonal matrix B_0 of dimension p \times p such that

Φ (B) ≥ Φ (B_0)

for all orthogonal matrices B. The matrices A_1,...,A_k are positive-definite and are usually sample covariance matrices and n_is are positive real numbers.

It can be shown (Flury, 1983) that if B_0=[b_1, b_2,…, b_p ], then the following system of equations holds:

{b_l}'≤ft[{∑\limits_{i = 1}^k {{n_i}\frac{{{λ _{il}} - {λ _{ij}}}}{{{λ _{il}}{λ _{ij}}}}{A_i}} } \right]{b_j} = 0 \hspace{1cm} (l,j = 1, … ,p;l \not = j)

where

{λ _{ih}} = {b_h}^\prime {A_i}{b_h} \hspace{1cm} (i = 1, … ,k;h = 1, … ,p).

In other words, Flury and Gautschi algorithms find the solution B_0 of the above system of equations. Also, this algorithm can be used to find the maximum likelihood estimates of common principal components in k groups (Flury,1984).

Details

Package:	FGalgorithm
Type:	Package
Version:	1.0
Date:	2012-11-14
License:	GPL (>= 2)

Author(s)

Dariush Najarzadeh

Maintainer: Dariush Najarzadeh <[e-mail]>

References

Flury, B. N. (1983), "A generalization of principal component analysis to k groups", Technical Report No. 83-14, Dept. of Statistics, Purdue University.

Flury, B. N. (1984). Common principal components in k groups. Journal of the American Statistical Association, 79(388), 892-898.

Flury, B. N., & Gautschi, W. (1984). An algorithm for simultaneous orthogonal transformation of several positive definite symmetric matrices to nearly diagonal form. SIAM Journal on Scientific and Statistical Computing, 7(1), 169-184.

FGalgorithm documentation built on May 1, 2019, 10:55 p.m.