Geometric goodness of fit measure to detect patterns in data point clouds

TL;DR

This paper introduces a geometric goodness-of-fit index based on topological data analysis to detect patterns in data clouds, utilizing Vietoris-Rips complexes to measure data space emptiness.

Contribution

The paper presents a novel goodness-of-fit measure derived from topological data analysis, implemented in the TopSA package, for pattern detection in data clouds.

Findings

Effective detection of data patterns using the new index

Implementation of the method in the TopSA package

Potential applications in data analysis and pattern recognition

Abstract

We derived a geometric goodness-of-fit index, similar to $R^2$ using topological data analysis techniques. We build the Vietoris-Rips complex from the data-cloud projected onto each variable. Estimating the area of the complex and their domain, we create an index that measures the emptiness of the space with respect to the data. We made the analysis with an own package called TopSA (Topological Sensitivy Analysis).