Fundamental problems in fitting spatial cluster process models

Abstract

Existing methods for fitting Neyman–Scott cluster process models to spatial point pattern data often fail to converge, or converge to implausible values of the parameters, or exhibit numerical instability. These failures have been viewed as weaknesses of the particular model-fitting method. However, we show that they are attributable to fundamental flaws in the model structure: The model is not closed under convergence in distribution; the Poisson process is not included in the model; cluster scale is unidentifiable when the model is close to a Poisson process. We obtain new results about properties of cluster processes, and about the distance from a cluster process to a Poisson process. We define an index of cluster strength φ which plays an important role in the analysis. Remedies for the fundamental problems are proposed: the model is extended to include the Poisson process by allowing φ=0; unidentifiability is remedied using shrinkage estimators involving a penalty on cluster scale. To improve understanding of the fitted model in applications we propose several derived parameters. The improved fitting methods are implemented in open source R code. Simulation experiments and real data examples demonstrate the improved performance.