Functional central limit theorems for persistent Betti numbers on cylindrical networks

Abstract

We study functional central limit theorems for persistent Betti numbers obtained from networks defined on a Poisson point process. The limit is formed in large volumes of cylindrical shape stretching only in one dimension. The results cover a directed sublevel‐filtration for stabilizing networks and the Čech and Vietoris–Rips complex on the random geometric graph. The presented functional central limit theorems open the door to a variety of statistical applications in topological data analysis and we consider goodness‐of‐fit tests in a simulation study.

An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists

“…Robinson and Turner (2017) focused on pairwise distances of persistence diagrams, whereas Berry et al (2020) studied more general functional summaries. Hypothesis tests based on kernel approaches have been proposed in the study by Kusano (2019) . A two-stage hypothesis test of filtering and testing for persistent images was also presented in the study by Moon and Lazar (2020) .…”

Topology-based goodness-of-fit tests for sliced spatial data

“…However, the general methodology is sufficiently flexible to deal also with the present situation. In fact, the necessary modifications are essentially described in [16]. Nevertheless, to make the manuscript self-contained, we reproduce here the most important steps.…”

“…The proof of Theorem 2.5 combines arguments appearing in [5,16]. On the one hand, similarly to [16] our null hypothesis is based on a Poisson point process, so that we can build on the martingale approach from [22].…”

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 Polonik [25] established the strong stabilizing property of persistent Betti numbers and extended the validity of the central limit theorem to the binomial point process with a non-constant density. 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].…”