# Testing goodness of fit for point processes via topological data analysis

## TL;DR

This paper develops new goodness-of-fit tests for point patterns using topological data analysis, specifically persistent Betti numbers, and demonstrates their effectiveness through simulations and a neuroscience application.

## Contribution

It introduces a novel statistical framework based on persistent Betti numbers for assessing point process models, including theoretical conditions and practical testing procedures.

## Key findings

- Persistent Betti numbers are asymptotically Gaussian for large observation windows.
- The proposed tests outperform global envelope tests in power.
- Application to neuroscience data demonstrates practical utility.

## 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.