Practical simulation and estimation for Gibbs Delaunay-Voronoi tessellations with geometric hardcore interaction

TL;DR

This paper develops practical methods for simulating and estimating Gibbs Delaunay-Voronoi tessellations with geometric hardcore interactions, addressing theoretical challenges and providing tools for real-world applications.

Contribution

It introduces new simulation and estimation techniques for complex tessellations with non-hereditary interactions, expanding practical capabilities.

Findings

Established existence conditions for the models

Developed algorithms for simulation and parameter estimation

Validated methods with detailed examples

Abstract

General models of Gibbs Delaunay-Voronoi tessellations, which can be viewed as extensions of Ord's process, are considered. The interaction may occur on each cell of the tessellation and between neighbour cells. The tessellation may also be subjected to a geometric hardcore interaction, forcing the cells not to be too large, too small, or too flat. This setting, natural for applications, introduces some theoretical difficulties since the interaction is not necessarily hereditary. Mathematical results available for studying these models are reviewed and further outcomes are provided. They concern the existence, the simulation and the estimation of such tessellations. Based on these results, tools to handle these objects in practice are presented: how to simulate them, estimate their parameters and validate the fitted model. Some examples of simulated tessellations are studied in details.