Testing goodness of fit for point processes via topological data analysis. (English) Zbl 1440.62409
Summary: 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.
MSC:
| 62R40 | Topological data analysis |
| 62G10 | Nonparametric hypothesis testing |
| 62H20 | Measures of association (correlation, canonical correlation, etc.) |
| 60D05 | Geometric probability and stochastic geometry |
| 60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |
| 60F17 | Functional limit theorems; invariance principles |
| 55N20 | Generalized (extraordinary) homology and cohomology theories in algebraic topology |
Keywords:
point processes; goodness-of-fit tests; central limit theorem; topological data analysis; persistent Betti numberReferences:
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