Parallel computation of bivariate point data depths and display of intrinsic depth segments

Abstract

This paper presents a new way to compute simplicial and Tukey data depths using Open Multi‐Processing parallelization, which makes it practical to compute point depths for tens of thousands of points. The definition of point depth is the order statistic depth of a single point, here in two dimensions. Using the point depths, the regional depth characteristics of the dataset as a whole can be explored. Using this new methodology, fast parallel computation of both simplicial depth and Tukey depth for a dataset of n points has time complexity O(n2logn) with O(n) space, which is practical for n up to 100,000. Obtaining depths for a large number of points in a faster manner by parallel computation supports identifying the central region quickly, because the points of maximum depth are known. The point depth computation identifies the depths of selected spoke segments around each origin point. These spoke depths are used to create new visualizations of depth characteristics and contour depths without adding virtual points.