Robust canonical analysis

JORGE G. ADROVER AND STELLA M. DONATO

Congreso; International Congress on Robust Statistics; 2011

Canonical correlation analysis (CCA) is a dimension-reduction technique in which two random vectors from higher dimensional spaces are reduced to a new pair of low dimensional vectors after applying linear transformations to each of them while 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 same 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 difference 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. Rather, we suggest using a robust univariate dispersion measure (like an M-scale) based on the difference of the transformed vectors to derive robust S-estimators for canonical vectors and correlations. We also state that there is an equivalence between CCA and Principal Component Analysis (PCA) after concatening the standardized original vectors. This result allows us to perform an iterative algorithm by exploiting the existence of efficient algorithms for S-estimation in the context of PCA. 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.