@MISC{Bern00multivariateregression, author = {Marshall Bern and David Eppstein}, title = {Multivariate Regression Depth}, year = {2000} }

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Abstract

The regression depth of a hyperplane with respect to a set of n points in R d is the minimum number of points the hyperplane must pass through in a rotation to vertical. We generalize hyperplane regression depth to k-flats for any k between 0 and d - 1. The k = 0 case gives the classical notion of center points. We prove that for any k and d, deep k-flats exist, that is, for any set of n points there always exists a k-flat with depth at least a constant fraction of n. As a consequence, we derive a linear-time (1 + #)-approximation algorithm for the deepest flat. 1. INTRODUCTION Linear regression asks for an affine subspace (a flat) that fits a set of data points. The most familiar case assumes d-1 independent or explanatory variables and one dependent or response variable, and fits a hyperplane to explain the dependent variable as a linear function of the independent variables. Quite often, however, there may be more than one dependent variable, and the multivariate regression p...