A comparison of approximations for perturbation study of covariance matrices with application to principal component analysis
Résumé
Covariance matrices play a central role in a wide range of multivariate statistical methods including for example principal component analysis which is a well known dimensionality reduction technique. Therefore, a large amount of work has been devoted to analyzing the sensitivity of their eigenstructure to influential observations. In order to evaluate the effect of deleting one or a small subset of observations, several approximations to the eigenelements of the perturbed matrix have been proposed. This paper provides a theoretical and numerical
comparison of the main approximations. A special emphasis is given to those based on Rayleigh quotients which are seldom used. A general approach, using new inequalities in the study of covariance matrices, is proposed in order to evaluate their accuracy without
having to recompute the exact perturbed eigenvalues and eigenvectors. This approach is of specific interest from a computational standpoint. Theoretical developments are illustrated with a numerical study which emphasizes the accuracy of approximations based on Rayleigh quotients.
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