Approximation intensity for pairwise interaction Gibbs point processes using determinantal point processes

TL;DR

This paper introduces a new approximation method for the intensity of repulsive pairwise interaction Gibbs point processes using determinantal point processes, which improves accuracy over traditional Poisson-saddlepoint methods.

Contribution

The paper develops a novel approximation technique based on determinantal point processes for calculating the intensity of Gibbs point processes, enhancing accuracy and computational efficiency.

Findings

The determinantal point process approximation outperforms the Poisson-saddlepoint method in accuracy.

The new approximation is efficiently implementable for basic models.

Numerical examples demonstrate improved precision of the proposed method.

Abstract

The intensity of a Gibbs point process is usually an intractable function of the model parameters. For repulsive pairwise interaction point processes, this intensity can be expressed as the Laplace transform of some particular function. Baddeley and Nair (2002) developped the Poisson-saddlepoint approximation which consists, for basic models, in calculating this Laplace transform with respect to a homogeneous Poisson point process. In this paper, we develop an approximation which consists in calculating the same Laplace transform with respect to a specific determinantal point process. This new approximation is efficiently implemented and turns out to be more accurate than the Poisson-saddlepoint approximation, as demonstrated by some numerical examples.