Topology-driven identification of repetitions in multi-variate time series

TL;DR

This paper introduces a persistent homology framework to identify recurrence times in multi-variate time series exhibiting cyclic behaviors, validated on real-world industrial data.

Contribution

It presents three novel, stable methods within a topological framework for detecting repetitions in complex multi-variate time series.

Findings

Methods are provably stable.

Validated on real-world data.

Effective in diverse cyclic behaviors.

Abstract

Many multi-variate time series obtained in the natural sciences and engineering possess a repetitive behavior, as for instance state-space trajectories of industrial machines in discrete automation. Recovering the times of recurrence from such a multi-variate time series is of a fundamental importance for many monitoring and control tasks. For a periodic time series this is equivalent to determining its period length. In this work we present a persistent homology framework to estimate recurrence times in multi-variate time series with different generalizations of cyclic behavior (periodic, repetitive, and recurring). To this end, we provide three specialized methods within our framework that are provably stable and validate them using real-world data, including a new benchmark dataset from an injection molding machine.