A Comparison of Two Proximity Catch Digraph Families in Testing Spatial Clustering

TL;DR

This paper compares two families of proximity catch digraphs for testing spatial clustering, analyzing their statistical properties, performance, and optimal parameters through theoretical derivations and simulations, with applications in ecology.

Contribution

It introduces a detailed comparison of proportional-edge and central similarity PCDs, extending their distributional theory and evaluating their effectiveness in spatial pattern testing.

Findings

Central similarity PCD performs better under segregation.

Proportional-edge PCD performs better under association.

Optimal parameters depend on the type of spatial pattern.

Abstract

We consider two parametrized random digraph families, namely, proportional-edge and central similarity proximity catch digraphs (PCDs) and compare the performance of these two PCD families in testing spatial point patterns. These PCD families are based on relative positions of data points from two classes and the relative density of the PCDs is used as a statistic for testing segregation and association against complete spatial randomness. When scaled properly, the relative density of a PCD is a U-statistic. We extend the distribution of the relative density of central similarity PCDs for expansion parameter being larger than one. We compare the asymptotic distribution of the statistic for the two PCD families, using the standard central limit theory of U-statistics. We compare finite sample performance of the tests by Monte Carlo simulations and prove the consistency of the tests under the alternatives. The asymptotic performance of the tests under the alternatives is assessed by Pitman's asymptotic efficiency. We find the optimal expansion parameters of the PCDs for testing each of the segregation and association alternatives in finite samples and in the limit. We demonstrate that in terms of empirical power (i.e., for finite samples) relative density of central similarity PCD has better performance (which occurs for expansion parameter values larger than one) under segregation alternative, while relative density of proportional-edge PCD has better performance under association alternative. The methods are illustrated in a real-life example from plant ecology.