Description Details Author(s) References

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_i*s 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).

Package: | FGalgorithm |

Type: | Package |

Version: | 1.0 |

Date: | 2012-11-14 |

License: | GPL (>= 2) |

Dariush Najarzadeh

Maintainer: Dariush Najarzadeh <[email protected]>

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.

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