Pairwise interaction function estimation of Gibbs point processes using basis expansion

The class of Gibbs point processes (GPP) is a large class of spatial point processes in the sense that they can 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, natural and easiest to interpret 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. Estimating this function non parametrically has almost never been considered in the literature. We tackle this question and propose an orthogonal series estimation procedure of the log pairwise interaction function. Under some conditions provided on the spatial GPP and on the basis system, we show that this orthogonal series estimator is consistent and asymptotically normal. The estimation procedure is simple, fast and completely data-driven. We show its efficiency through simulation experiments and we apply it to three datasets.