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.
Highlights
Tracing when topological features appear and disappear
A central limit theorem for persistent Betti numbers has been established in the Poisson setting [31, 26], but for general point processes the long-range interactions pose a formidable obstacle towards proving a fully-fledged functional CLT
In order to derive a functional CLT for the persistent Betti numbers, we add a further constraint on P, which is needed to establish a lower bound on the variance via a conditioning argument in the vein of [40, Lemma 4.3]
Summary
For a locally finite point set X ⊂ R2, persistent Betti numbers provide refined measures for the amount of clusters and voids on varying length scales. A central limit theorem for persistent Betti numbers has been established in the Poisson setting [31, 26], but for general point processes the long-range interactions pose a formidable obstacle towards proving a fully-fledged functional CLT. We concentrate on features whose spatial diameter does not exceed a large deterministic threshold M. To define these M -bounded features, we introduce the Gilbert graph Gr(X ) on the vertex set X. The Gilbert graph Gr(X ) has for vertices the points in X and two points are connected by an edge if the distance between them is at most 2r or, equivalently, if the two disks of radius r centered at the points intersect
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