Hypothesis Testing for Shapes using Vectorized Persistence Diagrams

Abstract

Topological data analysis involves the statistical characterization of the shape of data. Persistent homology is a primary tool of topological data analysis, which can be used to analyze those topological features and perform statistical inference. In this paper, we present a two-stage hypothesis test for vectorized persistence diagrams. The first stage filters elements in the vectorized persistence diagrams to reduce false positives. The second stage consists of multiple hypothesis tests, with false positives controlled by false discovery rates. We demonstrate applications of the proposed procedure on simulated point clouds and three-dimensional rock image data. Our results show that the proposed hypothesis tests can provide flexible and informative inferences on the shape of data with lower computational cost compared to the permutation test.

Some recent trends in embeddings of time series and dynamic networks

“…One may want to introduce the block bootstrap to take better care of dependence structures. There are also recent contributions to hypothesis testing, Moon and Lazar (2020), sufficient statistics, Curry et al (2018), and Bayesian statistics for TDA, Maroulas et al (2020).…”