Fitting three-dimensional Laguerre tessellations by hierarchical marked point process models

TL;DR

This paper introduces a statistical framework for analyzing 3D Laguerre tessellations using hierarchical marked point process models, enabling efficient parameter estimation and model validation for complex microstructure data.

Contribution

It develops a novel two-step methodology combining multiscale point process models and exponential family models for tessellation marks, with tractable maximum pseudolikelihood estimation.

Findings

Effective model fitting for 3D Laguerre tessellations.

Model validation using global envelopes and characteristic comparisons.

Potential to replace costly laboratory experiments with simulations.

Abstract

We present a general statistical methodology for analysing a Laguerre tessellation data set viewed as a realization of a marked point process model. In the first step, for the points we use a nested sequence of multiscale processes which constitute a flexible parametric class of pairwise interaction point process models. In the second step, for the marks/radii conditioned on the points we consider various exponential family models where the canonical sufficient statistic is based on tessellation characteristics. For each step parameter estimation based on maximum pseudolikelihood methods is tractable. Model checking is performed using global envelopes and corresponding tests in the first step and by comparing observed and simulated tessellation characteristics in the second step. We apply our methodology for a 3D Laguerre tessellation data set representing the microstructure of a polycrystalline metallic material, where simulations under a fitted model may substitute expensive laboratory experiments.