An estimation method of the pair potential function for Gibbs point processes on spheres (Q2722309)

Spatial point processes are models for random distributions of points in a given space. They form the foundations of stochastic geometry. The theory concerns a large variety of areas such as physics or biology. Among these processes Gibbs point processes are particularly interesting because they allow the introduction and study of interactions between points through an associated potential function. In this paper stationary pairwise interaction point processes are considered. An estimation method for pairwise interaction potentials of stationary Gibbs point processes is introduced by considering the case of observations located on a sphere. It is based both on Fourier decomposition of the potential and on minimum contrast estimation. It is defined when many independent realizations of the process are available.NEWLINENEWLINENEWLINEConsistency and asymptotic normality are proved for the resulting estimators. The method enables derivation of the choice of the potential function by embedded hypotheses testing. The method is applied to independent observations of root locations on internodes around stem of maize roots. The internodes are described as circles and the attention is focused on the interaction function associated with the potential. Since a model with too many components seems to fail, a sequential procedure based on embedded hypotheses testing to build a simpler model is chosen. The two-step method presented in the paper requires the computation of the normalizing constants only in the second step and this is done only once.