Topological data analysis for random sets and its application in detecting outliers and goodness of fit testing

TL;DR

This paper introduces a topological data analysis approach for detecting outliers and testing the goodness-of-fit of random sets, utilizing persistence homology and functional depths, validated through simulation studies.

Contribution

It develops a novel methodology combining topological data analysis with statistical testing for random sets, including outlier detection and goodness-of-fit testing.

Findings

Effective outlier detection using functional depths.

Successful goodness-of-fit tests via global envelope methods.

Validated approach through simulation with germ-grain models.

Abstract

In this paper we present the methodology for detecting outliers and testing the goodness-of-fit of random sets using topological data analysis. We construct the filtration from level sets of the signed distance function and consider various summary functions of the persistence diagram derived from the obtained persistence homology. The outliers are detected using functional depths for the summary functions. Global envelope tests using the summary statistics as test statistics were used to construct the goodness-of-fit test. The procedures were justified by a simulation study using germ-grain random set models.