Testing goodness of fit for point processes via topological data analysis

Christophe A. N. Biscio*, Nicolas Chenavier, Christian Hirsch, Anne Marie Svane

*Corresponding author for this work

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Abstract

We introduce tests for the goodness of fit of point patterns via methods from topological data analysis. More precisely, the persistent Betti numbers give rise to a bivariate functional summary statistic for observed point patterns that is asymptotically Gaussian in large observation windows. We analyze the power of tests derived from this statistic on simulated point patterns and compare its performance with global envelope tests. Finally, we apply the tests to a point pattern from an application context in neuroscience. As the main methodological contribution, we derive sufficient conditions for a functional central limit theorem on bounded persistent Betti numbers of point processes with exponential decay of correlations.

Original languageEnglish
Pages (from-to)1024-1074
Number of pages51
JournalElectronic journal of statistics
Volume14
Issue number1
DOIs
Publication statusPublished - 2020

Keywords

  • Point processes
  • goodness-of-fit tests
  • central limit theorem
  • topological data analysis
  • persistent Betti number
  • GAUSSIAN LIMITS
  • CONVERGENCE

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