Testing randomness of spatial point patterns with the Ripley statistic

¹ AgroParisTech, UMR 518 Mathématique et Informatique Appliquées, 19 avenue du Maine, 75732 Paris Cedex 15, France
gabriel.lang@agroparistech.fr
² AgroParisTech, UMR 745 Ecologie des Forêts de Guyane, Campus agronomique BP 316, 97379 Kourou Cedex, France
eric.marcon@agroparistech.fr

Received: 22 June 2011
Revised: 5 November 2012

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 homogeneous 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.