Intensity approximation for pairwise interaction Gibbs point processes using determinantal point processes

Jean-François Coeurjolly,Frédéric Lavancier

doi:10.1214/18-ejs1477

Jean-François Coeurjolly, Frédéric Lavancier

Open Access

https://doi.org/10.1214/18-ejs1477

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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 (2012) 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 determinan-tal 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.

Highlights

Due to their simple interpretation, Gibbs point processes and in particular pairwise interaction point processes play a central role in the analysis of spatial point patterns (see van Lieshout (2000); Møller and Waagepetersen (2004); Baddeley et al (2015))
Local finiteness of X means that XB = X∩B is finite almost surely (a.s.), that is the number of points N (B) of XB is finite imsart-ejs ver. 2014/10/16 file: appIntDPP.tex date: December 1, 2017
Following the same idea as the Poisson-saddlepoint approximation, for a repulsive stationary pairwise interaction point process with pairwise interaction function g ≤ 1 having a finite range R, we suggest to substitute the measure P involved in the expectation (2.7) by the measure Q corresponding to a Determinantal point processes (DPP) defined on B(0, R) with some kernel K and intensity λ, i.e. K(u, u) = λ

Summary

Introduction

Due to their simple interpretation, Gibbs point processes and in particular pairwise interaction point processes play a central role in the analysis of spatial point patterns (see van Lieshout (2000); Møller and Waagepetersen (2004); Baddeley et al (2015)). Baddeley and Nair (2012) suggest to evaluate the expectation with respect to a homogeneous Poisson point process with intensity λ This results in the Poisson-saddlepoint approximation, denoted by λps, obtained as the solution of log λps = log β − λps G where G = Rd (1 − g(u)) du (provided this integral is finite). This setting is considered by Baddeley and Nair (2012).

Background and Poisson point processes

Gibbs point processes

Determinantal point processes

Poisson-saddlepoint approximation

DPP approximation

Numerical study