Multivariate Regression Depth

Abstract

The regression depth of a hyperplane with respect to a set of n points in \Real ^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+ɛ) -approximation algorithm for the deepest flat. We also show how to compute the regression depth in time O(n ^d-2 +nlog n) when 1≤ k≤ d-2.