Pairwise interaction function estimation of stationary Gibbs point processes using basis expansion
Ismaïla Ba, Jean-François Coeurjolly, Francisco Cuevas-Pacheco
The Annals of Statistics · 2023 · Volume 51 · Issue 3
Abstract
The class of Gibbs point processes (GPP) is a large class of spatial point processes able to model both clustered and repulsive point patterns. They are specified by their conditional intensity, which for a point pattern x and a location u, is roughly speaking the probability that an event occurs in an infinitesimal ball around u given the rest of the configuration is x. The most simple and natural class of models is the class of pairwise interaction point processes where the conditional intensity depends on the number of points and pairwise distances between them. This paper is concerned with the problem of estimating the pairwise interaction function nonparametrically. We propose to estimate it using an orthogonal series expansion of its logarithm. Such an approach has numerous advantages compared to existing ones. The estimation procedure is simple, fast and completely data-driven. We provide asymptotic properties such as consistency and asymptotic normality and show the efficiency of the procedure through simulation experiments and illustrate it with several data sets.
Keywords
60G55, 62H11, 62J07, 65C60, 97K80, Gibbs point process, orthogonal series estimator, pairwise interaction function, spatial statistics
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