Persistent homology based goodness-of-fit tests for spatial tessellations

TL;DR

This paper develops topological data analysis-based goodness-of-fit tests for spatial tessellations, specifically Voronoi and Laguerre types, using persistence diagrams and Gibbs point processes, with applications to material science data.

Contribution

It introduces a new asymptotic normality result for tessellation-adapted persistence diagram statistics and applies it to goodness-of-fit testing for complex spatial models.

Findings

Test statistics follow asymptotic normal distribution.

Method effectively distinguishes between different tessellation models.

Successful application to real foam data.

Abstract

Motivated by the rapidly increasing relevance of virtual material design in the domain of materials science, it has become essential to assess whether topological properties of stochastic models for a spatial tessellation are in accordance with a given dataset. Recently, tools from topological data analysis such as the persistence diagram have allowed to reach profound insights in a variety of application contexts. In this work, we establish the asymptotic normality of a variety of test statistics derived from a tessellation-adapted refinement of the persistence diagram. Since in applications, it is common to work with tessellation data subject to interactions, we establish our main results for Voronoi and Laguerre tessellations whose generators form a Gibbs point process. We elucidate how these conceptual results can be used to derive goodness of fit tests, and then investigate their power in a simulation study. Finally, we apply our testing methodology to a tessellation describing real foam data.