Common Principal Components inkGroups

1 December 1984

journal article
research article
Published by Taylor & Francis in Journal of the American Statistical Association

Vol. 79 (388) , 892-898
https://doi.org/10.1080/01621459.1984.10477108

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.

Keywords

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