Pseudolikelihood inference for Gibbsian T-tessellations ... and point processes

TL;DR

This paper develops a pseudolikelihood inference method for Gibbsian T-tessellations, extending point process techniques, and demonstrates its effectiveness through simulations and theoretical connections.

Contribution

It introduces a novel pseudolikelihood estimation approach for T-tessellations based on Campbell measures and KL divergence, linking it to existing point process methods.

Findings

Bias and variability decrease with larger tessellation domains

The approach generalizes pseudo-likelihood methods for point processes

Simulation results support the method's effectiveness

Abstract

Recently a new class of planar tessellations, named T-tessellations, was introduced. Splits, merges and a third local modification named flip where shown to be sufficient for exploring the space of T-tessellations. Based on these local transformations and by analogy with point process theory, tools Campbell measures and a general simulation algorithm of Metropolis-Hastings-Green type were translated for random T-tessellations.The current report is concerned with parametric inference for Gibbs models of T-tessellations. The estimation criterion referred to as the pseudolikelihood is derived from Campbell measures of random T-tessellations and the Kullback-Leibler divergence. A detailed algorithm for approximating the pseudolikelihood maximum is provided. A simulation study seems to show that bias and variability of the pseudolikelihood maximum decrease when the tessellated domain grows in size.In the last part of the report, it is shown that an analogous approach based on the Campbell measure and the KL divergence when applied to point processes leads to the well-known pseudo-likelihood introduced by Besag. More surprisingly, the binomial regression method recently proposed by Baddeley and his co-authors for computing the pseudolikelihood maximum can be derived using the same approach starting from a slight modification of the Campbell measure.