Computational Topology Techniques for Characterizing Time-Series Data

TL;DR

This paper explores how topological data analysis (TDA) can characterize complex time-series data from dynamical systems, enabling comparison, classification, and change detection despite computational challenges.

Contribution

It introduces the use of witness complexes and persistent homology to efficiently analyze and distinguish time-series data from different systems.

Findings

TDA effectively characterizes nonlinear dynamical systems.

Persistent homology captures shape changes across scales.

TDA can differentiate similar signals from different sources.

Abstract

Topological data analysis (TDA), while abstract, allows a characterization of time-series data obtained from nonlinear and complex dynamical systems. Though it is surprising that such an abstract measure of structure - counting pieces and holes - could be useful for real-world data, TDA lets us compare different systems, and even do membership testing or change-point detection. However, TDA is computationally expensive and involves a number of free parameters. This complexity can be obviated by coarse-graining, using a construct called the witness complex. The parametric dependence gives rise to the concept of persistent homology: how shape changes with scale. Its results allow us to distinguish time-series data from different systems - e.g., the same note played on different musical instruments.