Testing Homological Equivalence Using Betti Numbers

TL;DR

This paper introduces new homological tests based on Betti numbers to determine if the supports of unknown distributions are topologically equivalent, with proven consistency and practical validation through simulations and real data.

Contribution

It develops novel Betti number-based tests for homological equivalence, establishing their consistency and demonstrating their effectiveness compared to existing methods.

Findings

Test statistics based on Betti numbers are consistent in critical regimes.

Simulations show the new tests outperform some existing methods.

Real data applications confirm the practicality of the proposed tests.

Abstract

In this article, we propose a one-sample test to check whether the support of the unknown distribution generating the data is homologically equivalent to the support of some specified distribution or not OR using the corresponding two-sample test, one can test whether the supports of two unknown distributions are homologically equivalent or not. In the course of this study, test statistics based on the Betti numbers are formulated, and the consistency of the tests is established under the critical and the supercritical regimes. Moreover, some simulation studies are conducted and results are compared with the existing methodologies such as Robinson's permutation test and test based on mean persistent landscape functions. Furthermore, the practicability of the tests is shown on two well-known real data sets also.