Efficient Algorithms for Maximum Regression Depth

Abstract

We investigate algorithmic questions that arise in the statistical problem of computing lines or hyperplanes of maximum regression depth among a set of n points. We work primarily with a dual representation and find points of maximum undirected depth in an arrangement of lines or hyperplanes. An O(n d ) time and O(n d−1 ) space algorithm computes undirected depth of all points in d dimensions. Properties of undirected depth lead to an O(n log 2 n) time and O(n) space algorithm for computing a point of maximum depth in two dimensions, which has been improved to an O(n log n) time algorithm by Langerman and Steiger (Discrete Comput. Geom. 30(2):299-309, 2003). Furthermore, we describe the structure of depth in the plane and higher dimensions, leading to various other geometric and algorithmic results.

Dynamic Maintenance of Half-Space Depth for Points and Contours

“…In this section, we turn this static algorithm into a dynamic algorithm. The static version of the data structure that we use was originally presented by [van Kreveld et al, 1999] for a different problem.…”

“…In Section 3, we present an algorithm for maintaining the half-space depth of a single point in O(log n) time per operation (insertion or deletion) and linear overall space. The data structure that we use was originally presented in [van Kreveld et al, 1999], but we augment it to enable dynamic updates. In Section 4, we expand the algorithm and augment the data structure in Section 3 not only to maintain the depth of a point, but also to maintain the cover-based contours near points that do not lie on degenerate contours, i.e., contours consisting of either a single point or two points and the segment between them.…”

Intersecting convex sets by rays

“…So far, the only improvement was made in by Kreveld et.al. in [21], where the space used in the former algorithm was reduced by linear factor to O(n d - 1 ) using standard E-cutting method, see e.g. [16].…”

“…Given a collection of n pairwise disjoint convex compact flat sets C in jRd such that we can compute 1i(C), as defined in Chapter 2 so that no k + 1 hyperplanes in 1i(C) has non-empty intersection, in fen) deterministic time. By the previously mentioned algorithms from [13,21] we have. Given a finite set of points P in jRd, Tukey median is a point p in jRd, which maximize the minimum number d t (p) of points of P belonging to a closed half-space defined by a hyperplane through p. Formally, d t (p) = min{IP n 1'1: where I' is a halfspace defined by a hyperplane through p}.…”

Algorithms for bivariate zonoid depth

“…The design of efficient algorithms for these problems is essential for these depth measures to become useful statistical analysis tools. Computational geometry [32] has been of great help in this respect, and there are many results in the computational geometry literature [1][2][3][4][5][9][10][11][12]14,17,19,[21][22][23]26,29,31,33,34,36,37].…”