High-dimensional computation of the deepest location

Abstract

The halfspace location depth of a point θ relative to a data set X n is defined as the smallest number of observations in any closed halfspace with boundary through θ. As such, halfspace depth can be seen as a kind of multivariate ranking. The deepest location, i.e. the θ with maximal halfspace depth, is a multivariate generalization of the median. Until now the deepest location could only be computed for bivariate data. In this paper, we construct an algorithm (called DEEPLOC) to approximate the deepest location in higher dimensions.