On Hyperplane Alignment for Linear and Nonlinear Sufficient Dimension Reduction

• Activity Number: 635
• Type: Invited
• Date/Time: Thursday, August 4, 2011 : 10:30 AM to 12:20 PM
• Sponsor: Biometrics Section
• Author(s): Bing Li*+ and Andreas Artemiou
• Companies: Penn State University and Michigan Technological University
• Keywords: contour regression ; inverse regression ; principal components ; reproducing kernel Hilber space ; support vector machine

Abstract

We introduce a Hyperplane Alignment (HA) approach that can be used for both linear and nonlinear sufficient dimension reduction. The basic idea is to divide the response variables into slices and use a modified form of support vector machine to find the optimal hyperplanes that separate them. These optimal hyperplanes are then aligned by the principal components of their normal vectors. It is proved that the aligned normal vectors provide an unbiased, square root n consistent, and asymptotically normal estimator of the sufficient dimension reduction space. The method is then generalized to nonlinear sufficient dimension reduction using the reproducing kernel Hilbert space. In that context, the aligned normal vectors become functions and it is proved that they are unbiased in the sense that they are functions of the true nonlinear sufficient predictors. We compare HA with other sufficient dimension reduction methods by simulation and in real data analysis, and through both comparisons firmly establish its practical advantages.