Extension of One-Dimensional Proximity Regions to Higher Dimensions

TL;DR

This paper extends the concept of spherical proximity maps from one-dimensional intervals to higher-dimensional regions using Delaunay triangulation, introducing new proximity maps and analyzing their properties for uniform data.

Contribution

It introduces new higher-dimensional proximity maps based on spherical regions and Delaunay triangulation, expanding the applicability of proximity catch digraphs beyond one dimension.

Findings

Properties of spherical proximity maps on the real line are characterized.

New proximity maps in higher dimensions are defined and illustrated.

Geometry invariance of PCDs for uniform data is established.

Abstract

Proximity maps and regions are defined based on the relative allocation of points from two or more classes in an area of interest and are used to construct random graphs called proximity catch digraphs (PCDs) which have applications in various fields. The simplest of such maps is the spherical proximity map which maps a point from the class of interest to a disk centered at the same point with radius being the distance to the closest point from the other class in the region. The spherical proximity map gave rise to class cover catch digraph (CCCD) which was applied to pattern classification. Furthermore for uniform data on the real line, the exact and asymptotic distribution of the domination number of CCCDs were analytically available. In this article, we determine some appealing properties of the spherical proximity map in compact intervals on the real line and use these properties as a guideline for defining new proximity maps in higher dimensions. Delaunay triangulation is used to partition the region of interest in higher dimensions. Furthermore, we introduce the auxiliary tools used for the construction of the new proximity maps, as well as some related concepts that will be used in the investigation and comparison of them and the resulting graphs. We characterize the geometry invariance of PCDs for uniform data. We also provide some newly defined proximity maps in higher dimensions as illustrative examples.