Pairwise interaction function estimation of Gibbs point processes using basis expansion

TL;DR

This paper introduces a simple, fast, and data-driven method for nonparametric estimation of pairwise interaction functions in Gibbs point processes using orthogonal series expansion, with proven asymptotic properties.

Contribution

It proposes a novel nonparametric estimation method for pairwise interaction functions in Gibbs point processes using orthogonal series expansion of the logarithm, improving simplicity and efficiency.

Findings

Method is consistent and asymptotically normal.

Estimation procedure is computationally efficient.

Validated through simulations and real datasets.

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 $\mathbf{x}$ 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 $\mathbf{x}$. 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 non parametrically. 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 datasets.