Ray-shooting depth: Computing statistical data depth of point sets in the plane

Abstract

Over the past several decades, many combinatorial measures have been devised for capturing the statistical data depth of a set of n points in ℝ2. These include Tukey depth [15], Oja depth [12], Simplicial depth [10] and several others. Recently Fox et al. [7] have defined the Ray-Shooting depth of a point set, and given a topological proof for the existence of points with high Ray-Shooting depth in ℝ2. In this paper, we present an O(n 2 log 2 n)-time algorithm for computing a point of high Ray-Shooting depth. We also present a linear time approximation algorithm. © 2011 Springer-Verlag Berlin Heidelberg.