Testing goodness of fit for point processes via topological data analysis

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.

Functional limit theorems for the euler characteristic process in the critical regime

“…Within the context of geometric complexes-such as theČech and Vietoris-Rips complexes-few attempts have been made thus far at deriving limit theorems on the functional level for topological invariants (for some exceptions, see [3,19,20]). From the viewpoint of persistent homology, such functional information is crucial for the understanding of topological invariants in a filtered topological space.…”

On approximation theorems for the Euler characteristic with applications to the bootstrap

“…Further potential applications of the smooth EC-transform I(χ n ) are goodness-of-fit tests as an exploratory tool in topological data analysis. We refer to [5] and [24] for similar applications in the context of persistent Betti numbers.…”

“…Krebs and Hirsch [24] studied functional central limit theorems for persistent Betti numbers on a cylindrical domain. Betti numbers of B-bounded features have been studied by Biscio et al [5].…”

Functional central limit theorems for persistent Betti numbers on cylindrical networks

“…Although asymptotically precise tests decouple the computation of the test statistic under the null hypothesis from the size of the sampling window, a possible practical problem is that a robust estimation of the limiting variances could still entail simulations on relative large windows. An alternative could be to start from the precise integral expressions for the limiting variances and covariances derived by Biscio et al (2020) and devise reasonable strategies to approximate them.…”

“…By the continuous mapping theorem, the only requirement is that the summary statistic varies continuously in the persistence diagram. For instance, Biscio et al (2020) use the accumulated persistence function to derive a TDA‐based goodness‐of‐fit test, which is then applied to a dataset from neuroscience.…”