Testing randomness of spatial point patterns with the Ripley statistic

TL;DR

This paper develops a statistical test based on the Ripley K-statistic to determine whether spatial point patterns are random or exhibit clustering, with proven asymptotic properties and validated through simulations.

Contribution

It introduces an asymptotically Gaussian test for spatial randomness using the Ripley K-statistic, including exact moments and covariance calculations under the Poisson model.

Findings

Test is numerically tractable for large datasets

Test maintains accuracy with small sample sizes

Derived a chi-square distribution for the test statistic

Abstract

Aggregation patterns are often visually detected in sets of location data. These clusters may be the result of interesting dynamics or the effect of pure randomness. We build an asymptotically Gaussian test for the hypothesis of randomness corresponding to a Poisson point process. We first compute the exact first and second moment of the Ripley K-statistic under the homogeneous Poisson point process model. Then we prove the asymptotic normality of a vector of such statistics for different scales and compute its covariance matrix. From these results, we derive a test statistic that is chi-square distributed. By a Monte-Carlo study, we check that the test is numerically tractable even for large data sets and also correct when only a hundred of points are observed.