A Subsequence Approach to Topological Data Analysis for Irregularly-Spaced Time Series

TL;DR

This paper introduces a subsequence embedding method for irregularly-spaced time series that preserves topology and reduces noise-induced artifacts, extending topological data analysis techniques beyond uniform sampling.

Contribution

It proposes a novel subsequence embedding approach for irregular time series, with theoretical stability guarantees and demonstrated effectiveness on real data.

Findings

Preserves original topology of time series

Reduces spurious homological features

Shows stability under noise and irregularity

Abstract

A time-delay embedding (TDE), grounded in the framework of Takens's Theorem, provides a mechanism to represent and analyze the inherent dynamics of time-series data. Recently, topological data analysis (TDA) methods have been applied to study this time series representation mainly through the lens of persistent homology. Current literature on the fusion of TDE and TDA are adept at analyzing uniformly-spaced time series observations. This work introduces a novel {\em subsequence} embedding method for irregularly-spaced time-series data. We show that this method preserves the original state space topology while reducing spurious homological features. Theoretical stability results and convergence properties of the proposed method in the presence of noise and varying levels of irregularity in the spacing of the time series are established. Numerical studies and an application to real data illustrates the performance of the proposed method.