A Studentized Permutation Test for the Comparison of Spatial Point Patterns

Abstract

In this study, a new test is proposed for the hypothesis that two (or more) observed point patterns are realizations of the same spatial point process model. To this end, the point patterns are divided into disjoint quadrats, on each of which an estimate of Ripley’s K-function is calculated. The two groups of empirical K-functions are compared by a permutation test using a Studentized test statistic. The proposed test performs convincingly in terms of empirical level and power in a simulation study, even for point patterns where the K-function estimates on neighboring subsamples are not strictly exchangeable. It also shows improved behavior compared with a test suggested by Diggle et al. for the comparison of groups of independently replicated point patterns. In an application to two point patterns from pathology that represent capillary positions in sections of healthy and cancerous tissue, our Studentized permutation test indicates statistical significance, although the patterns cannot be clearly distinguished by the eye.

Keywords:

• K-function
• Nonparametric test
• Quadrat
• Spatial point process
• Subsampling

Acknowledgments

This research has been financially supported by the Danish Council for Strategic Research and the Centre for Stochastic Geometry and Advanced Bioimaging, funded by a grant from the Villum Foundation. The author cordially thanks Professor Eva B. Vedel Jensen for her encouragement and support, as well as for helpful discussions and valuable comments on the article.

Notes

NOTE: The test was applied to two groups of nine realizations each of a Poisson point process; for details on the homoscedastic case (a) and the heteroscedastic cases (b) and (c), see text. Observed rejection rates are based on 10,000 replications of the simulation experiment.

NOTE: Large windows: two groups of nine replicates each on the unit square window; large samples: two groups of 18 replicates each on square windows with side length 0.5. Observed rejection rates are based on 10,000 replications of the simulation experiment.

NOTE: The test was applied to two groups of nine realizations of a Poisson point process; (a) both point processes with intensity λ=100 on a 0.5×0.5 square, (b) both on a 0.5×0.5 square but with different intensities (λ₁=100, λ₂=200), and (c) both with the same intensity (λ=100) but different windows (squares with side lengths 0.5 and 1.0, respectively). Observations are based on 10,000 replications of the simulation experiments.