Linear Contour Learning: A Method for Supervised Dimension Reduction

We propose a novel approach to sufficient di­ mension reduction in regression, based on es­ timating contour directions of negligible vari­ ation for the response surface. These di­ rections span the orthogonal complement of the minimal space relevant for the regression, and can be extracted according to a mea­ sure of the variation in the response, lead­ ing to General Contour Regression (GCR). In comparison to existing sufficient dimen­ sion reduction techniques, this contour-based methodology guarantees exhaustive estima­ tion of the central space under ellipticity of the predictor distribution and very mild ad­ ditional assumptions, while maintaining fo• consistency and computational ease. Moreover, it proves to be robust to departures from ellipticity. We also establish some use­ ful population properties for GCR. Simula­ tions to compare performance with that of standard techniques such as ordinary least squares, sliced inverse regression, principal hessian directions, and sliced average vari­ ance estimation confirm the advantages an­ ticipated by theoretical analyses. We also demonstrate the use of contour-based meth­ ods on a data set concerning grades of stu­ dents from Massachusetts colleges. Introduction and Background unsupervised approaches; here we consider dimen­ sion reduction for the regression of a continuous re­ sponse Y on a vector of continuous predictors X = (X 1 , . .. , X P) T E JRP . Our approach is based on suffi­ cient dimension reduction, a body of statistical theory and methods for reducing the dimension of X while preserving information on the regression; that is, on the conditional distribution of YjX. A dimension re­ duction subspace (Cook, 1998) is defined as the column span of any p x d ( d < p) matrix rJ such that