Robust dimension reduction based on canonical correlation

The canonical correlation (CANCOR) method for dimension reduction in a regression setting is based on the classical estimates of the first and second moments of the data, and therefore sensitive to outliers. In this paper, we study a weighted canonical correlation (WCANCOR) method, which captures a subspace of the central dimension reduction subspace, as well as its asymptotic properties. In the proposed WCANCOR method, each observation is weighted based on its Mahalanobis distance to the location of the predictor distribution. Robust estimates of the location and scatter, such as the minimum covariance determinant (MCD) estimator of Rousseeuw [P.J. Rousseeuw, Multivariate estimation with high breakdown point, Mathematical Statistics and Applications B (1985) 283-297], can be used to compute the Mahalanobis distance. To determine the number of significant dimensions in WCANCOR, a weighted permutation test is considered. A comparison of SIR, CANCOR and WCANCOR is also made through simulation studies to show the robustness of WCANCOR to outlying observations. As an example, the Boston housing data is analyzed using the proposed WCANCOR method.