Topology-driven goodness-of-fit tests in arbitrary dimensions

Abstract

This paper adopts a tool from computational topology, the Euler characteristic curve (ECC) of a sample, to perform one- and two-sample goodness of fit tests. We call our procedure TopoTests. The presented tests work for samples of arbitrary dimension, having comparable power to the state-of-the-art tests in the one-dimensional case. It is demonstrated that the type I error of TopoTests can be controlled and their type II error vanishes exponentially with increasing sample size. Extensive numerical simulations of TopoTests are conducted to demonstrate their power for samples of various sizes.

On the shape that matters – topology and geometry in data science

“…While providing slightly weaker invariants, they do not encounter the same problems as traditional methods including CDF. In our recent work [7], the ECC of a sample is utilised as a surrogate for the cumulative distribution function, yielding an efficient statistical test that surpasses the state of the art. This new family of tests, referred to as TopoTests, has proven to outperform existing methods even in low-dimensional and small data samples scenarios.…”

“…In this instance, TopoTests consistently outperform the standard Kolmogorov-Smirnov test, illustrating the potential applicability of topological tools in statistical analysis. As described in [7], TopoTests can be easily accessed and utilised through a public domain implementation.…”

“…These occasional limitations are a trade-off for the additional advantages offered by the proposed TopoTests. Please consult [7] for further details.…”