On detecting dynamical regime change using a transformation cost metric between persistent homology diagrams

Abstract

This work outlines a pipeline for time series analysis that incorporates a measure of similarity not previously applied between homological summaries. Specifically, the well-established, but disparate, methods of persistent homology and TrAnsformation Cost Time Series (TACTS) are combined to provide a metric for tracking dynamics via changing homological features. TACTS allows subtle changes in dynamics to be accounted for, gives a quantitative output that can be directly interpreted, and is tunable to provide several complementary perspectives simultaneously. Our method is demonstrated first with known dynamical systems and then with a real-world electrocardiogram dataset. This paper highlights inadequacies in existing persistent homology metrics and describes circumstances where TACTS can be more sensitive and better suited to detecting a variety of regime changes.

A Case Study on Identifying Bifurcation and Chaos with CROCKER Plots

“…Of course, many options are available for such a question; in this work we focus on a data driven approach using topological data analysis (TDA) to measure the shape and structure of the attractor of the system as a proxy for behavior. The idea of combining methods from TDA with dynamical systems and/or time series analysis is not new [8,12,3,19,16,10,5,11]. In this work, we focus on a new way to encode information about the bifurcation, namely the CROCKER plot [17,20,21].…”