Common Principal Components in k Groups

Abstract

This article generalizes the method of principal components to so-called “common principal components” as follows: Consider the hypothesis that the covariance matrices Σ _i for k populations are simultaneously diagonalizable. That is, there is an orthogonal matrix β such that β' Σ _i β is diagonal for i = 1, …, k. I derive the normal-theory maximum likelihood estimates of the common component Σ _i matrices and the log-likelihood-ratio statistics for testing this hypothesis. The solution has some favorable properties that do not depend on normality assumptions. Numerical examples illustrate the method. Applications to data reduction, multiple regression, and nonlinear discriminant analysis are sketched.

Key Words:

• Maximum likelihood
• Covariance matrices
• Discriminant analysis