Introductory Topological Data Analysis

Abstract

This paper introduces topological data analysis. Starting from notions of a metric space and some elementary graph theory, we take example sets of data and demonstrate some of their topological properties. We discuss simplicial complexes and how they relate to something called the Nerve Theorem. For this we introduce notions from the field of topology such as open covers, homeomorphism and homotopy equivalences. This leads us into discussing filtering data and deriving topologically invariant simplicial complexes from the underlying data set. There is then a small introduction to persistent homology and Betti numbers as these are useful analytical tools for TDA. An accompanying online appendix for the code producing the bulk of the figures in this paper is available at bit.ly/TDA_2020.