Asymptotic theory for common principal component analysis
type: Item
instance of: scholarly article
title: Asymptotic theory for common principal component analysis
zbMATH Open document ID: 0613.62075
author: Bernard N. Flury
published in: The Annals of Statistics
publication date: 1986-01-01T00:00:00Z
review text: Common principal component analysis is a generalization of principal component analysis to k groups, and it is assumed that the \(p\times p\) covariance matrices \(\Sigma_ 1,...,\Sigma_ k\) of k populations can be diagonalized by the same orthogonal transformation, i.e., there exists an orthogonal matrix \(\beta\) such that \(\beta '\Sigma_ i\beta =\Lambda_ i\) is diagonal \((i=1,...,k)\). The asymptotic distribution of the maximum likelihood estimates of \(\beta\) and \(\Lambda_ i\) are derived. In particular, if \(\lambda_{ij}\) \((j=1,...,p)\) are the diagonal elements of \(\Lambda_ i\), and \({\hat \lambda}_{ij}\) are their maximum likelihood estimates, then the statistics \(\sqrt{n_ i}({\hat \lambda}_{ij}-\lambda_{ij})\) are asymptotically (min \(n_ i\to \infty)\) distributed as \(N(0,2\lambda^ 2_{ij})\), independent of the \(\beta_ j\) and independent of each other \((n_ i\) being the degrees of freedom of the sample covariance matrices). Using these results, tests are derived for (a) equality of q eigenvectors \(\beta_ 1,...,\beta_ q\) with a given set of orthonormal vectors \(\beta^ 0_ 1,...,\beta^ 0_ q\) (1\(\leq q\leq p)\), (b) neglecting p-q (out of p) principal components with relatively small variances, and (c) simultaneous sphericity of p-q common principal components. Some of the results are illustrated by a biometrical example (a three-dimensional case).
Mathematics Subject Classification ID: 62H25
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 62H15
zbMATH DE Number: 3992684
zbMATH Keywords: tests for equality of eigenvectors
zbMATH Keywords: likelihood-ratio test
zbMATH Keywords: asymptotic normality
zbMATH Keywords: principal component analysis
zbMATH Keywords: asymptotic distribution
zbMATH Keywords: maximum likelihood estimates
zbMATH Keywords: simultaneous sphericity
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176349930
OpenAlex ID: W2033265796
DOI: 10.1214/AOS/1176349930
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description: scientific article; zbMATH DE number 3992684