| FGalgorithm-package {FGalgorithm} | R Documentation |
Execute the Flury and Gautschi diagonalisation algorithm, which tries to simultaneously diagonalize a set of symmetric positive definite matrices.
Description
The minimization of the objective function
\Phi (B) = {\prod\limits_{i = 1}^k {\left[ {\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
\Phi (B)
\ge
\Phi (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,\ldots, b_p ], then
the following system of equations holds:
{b_l}'\left[{\sum\limits_{i = 1}^k {{n_i}\frac{{{\lambda _{il}}
- {\lambda _{ij}}}}{{{\lambda _{il}}{\lambda _{ij}}}}{A_i}} } \right]{b_j} = 0
\hspace{1cm}
(l,j = 1, \ldots ,p;l \not = j)
where
{\lambda _{ih}} = {b_h}^\prime {A_i}{b_h}
\hspace{1cm}
(i = 1, \ldots ,k;h = 1, \ldots ,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 <D_Najarzadeh@sbu.ac.ir>
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.