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

Abstract

AbstractMotivated 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.Keywords: Tessellationtopological data analysisgoodness-of-fitpersistence diagram2010 Mathematics Subject Classifications: 60K3560F1082C22 AcknowledgmentsWe thank the two anonymous referees for their careful reading of the manuscript. Their comments and suggestions substantially improved the quality of the presentation. We thank Anne Jung (Helmut Schmidt University Hamburg) for providing the foam sample and Christian Jung (RPTU Kaiserslautern-Landau) for computing the Laguerre approximation.Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingJohannes Krebs was partially supported by the German Research Foundation (DFG), Grant Number KR-4977/2-1.