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

Abstract

The curse of dimensionality is a commonly encountered problem in statistics and data analysis. Variable sensitivity analysis methods are a well studied and established set of tools designed to overcome these sorts of problems. However, as this work shows, these methods fail to capture relevant features and patterns hidden within the geometry of the enveloping manifold projected onto a variable. Here we propose an index that captures, reflects and correlates the relevance of distinct variables within a model by focusing on the geometry of their projections. We construct the 2-simplices of a Vietoris-Rips complex and then estimate the area of those objects from a data-set cloud. The analysis was made with an original R-package called TopSA, short for Topological Sensitivity Analysis. The TopSA R-package is available at the site https://github.com/maikol-solis/TopSA.