Relative Density of the Random $r$-Factor Proximity Catch Digraph for Testing Spatial Patterns of Segregation and Association

TL;DR

This paper introduces a new graph-based statistical method using relative arc density of random digraphs for testing spatial patterns like segregation and association, with advantages in analysis and applicability across dimensions.

Contribution

It proposes a parameterized family of proximity maps and uses relative arc density as a novel statistic, enabling asymptotic analysis and efficiency optimization in spatial pattern testing.

Findings

Relative arc density is a U-statistic, simplifying asymptotic analysis.

The method is valid in any data dimension.

Application demonstrates effective testing of spatial patterns.

Abstract

Statistical pattern classification methods based on data-random graphs were introduced recently. In this approach, a random directed graph is constructed from the data using the relative positions of the data points from various classes. Different random graphs result from different definitions of the proximity region associated with each data point and different graph statistics can be employed for data reduction. The approach used in this article is based on a parameterized family of proximity maps determining an associated family of data-random digraphs. The relative arc density of the digraph is used as the summary statistic, providing an alternative to the domination number employed previously. An important advantage of the relative arc density is that, properly re-scaled, it is a $U$-statistic, facilitating analytic study of its asymptotic distribution using standard $U$-statistic central limit theory. The approach is illustrated with an application to the testing of spatial patterns of segregation and association. Knowledge of the asymptotic distribution allows evaluation of the Pitman and Hodges-Lehmann asymptotic efficacies, and selection of the proximity map parameter to optimize efficiency. Furthermore the approach presented here also has the advantage of validity for data in any dimension.