Aspects of robust canonical correlation analysis, principal components and association

Abstract

Principal component analysis (PCA) and canonical correlation analysis (CCA) are dimension-reduction techniques in which either a random vector is well approximated in a lower dimensional subspace or two random vectors from high dimensional spaces are reduced to a new pair of low dimensional vectors after applying linear transformations to each of them. In both techniques, the closeness between the higher dimensional vector and the lower representations is under concern, measuring the closeness through a robust function. Robust SM-estimation has been treated in the context of PCA and CCA showing an outstanding performance under casewise contamination, which encourages the study of asymptotic properties. We analyze consistency and asymptotic normality for the SM-canonical vectors. As a by-product of the CCA derivations, the asymptotics for PCA can also be obtained. A classical measure of robustness as the influence function is analyzed, showing the usual performance of S-estimation in different statistical models. The general ideas behind SM-estimation in either PCA or CCA are specially tailored to the context of association, rendering robust measures of association between random variables. By these means, a robust correlation measure is derived and the connection with the association measure provided by S-estimation for bivariate scatter is analyzed. On the other hand, we also propose a second robust correlation measure which is reminiscent of depth-based procedures.