A robust predictive approach for canonical correlation analysis

Canonical correlation analysis (CCA) is a dimension-reduction technique in which 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, retaining as much information as possible. The components of the transformed vectors are called canonical variables. One seeks linear combinations of the original vectors maximizing the correlation subject to the constraint that they are to be uncorrelated with the previous canonical variables within each vector. By these means one actually gets two transformed random vectors of lower dimension whose expected square distance has been minimized subject to have uncorrelated components of unit variance within each vector. Since the closeness between the two transformed vectors is evaluated through a highly sensitive measure to outlying observations as the mean square loss, the linear transformations we are seeking are also affected. In this paper we use a robust univariate dispersion measure (like an M-scale) based on the distance of the transformed vectors to derive robust S-estimators for canonical vectors and correlations. An iterative algorithm is performed by exploiting the existence of efficient algorithms for S-estimation in the context of Principal Component Analysis. Some convergence properties are analyzed for the iterative algorithm. A simulation study is conducted to compare the new procedure with some other robust competitors available in the literature, showing a remarkable performance. We also prove that the proposal is Fisher consistent.