A comparative study of approximations for perturbation analysis of principal components

Principal component analysis is a well known method for dimension reduction based on the covariance matrix associated to a multivariate data table. Therefore, a large amount of work has been devoted to analyzing the sensitivity of the eigenstructure of this matrix to influential observations. In order to evaluate the effect of deleting one or a small subset of observations, several approximations for the perturbed eigenelements 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 since they are under-utilized given their excellent performance. A general approach, using refined inequalities, is proposed in order to get a precise evaluation of 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.