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

Simon Schindler, Elias Steffen Reich, Saverio Messineo, Simon Hoher, Stefan Huber

TL;DR

This work tackles the challenge of identifying recurrence times in multi-variate time series, where traditional periodicity detectors falter under uneven sampling and variable cycle lengths. It introduces a persistent-homology-based framework that maps a time series to a scalar function and analyzes its sublevel-set filtration to recover recurrence times, validated by stability proofs. Three specialized methods address periodic, repetitive, and recurring behaviors, with Method 3 performing best for periodic signals, Method 2 offering strong overall performance for repetitive cases, and Method 1 providing a robust general baseline. The approach is demonstrated on a high-fidelity injection-molding dataset, highlighting practical applicability for industrial monitoring and control tasks, and underlying the potential for broader topology-based recurrence analysis in complex time-series domains.

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.

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

TL;DR

This work tackles the challenge of identifying recurrence times in multi-variate time series, where traditional periodicity detectors falter under uneven sampling and variable cycle lengths. It introduces a persistent-homology-based framework that maps a time series to a scalar function and analyzes its sublevel-set filtration to recover recurrence times, validated by stability proofs. Three specialized methods address periodic, repetitive, and recurring behaviors, with Method 3 performing best for periodic signals, Method 2 offering strong overall performance for repetitive cases, and Method 1 providing a robust general baseline. The approach is demonstrated on a high-fidelity injection-molding dataset, highlighting practical applicability for industrial monitoring and control tasks, and underlying the potential for broader topology-based recurrence analysis in complex time-series domains.

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.
Paper Structure (34 sections, 11 equations, 3 figures, 2 tables)

This paper contains 34 sections, 11 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Problem setting
  4. Basic definitions.
  5. Problem statement.
  6. Prior and related Work
  7. Limitations of classical periodicity detection methods.
  8. Topology of time series.
  9. Summary of the state of the art.
  10. Contribution
  11. Background
  12. Persistent homology
  13. Topology-driven identification of repetitions
  14. Determination of Recurrences (Method 1)
  15. Method.
  16. ...and 19 more sections

Figures (3)

  • Figure 1: Overview of the function spaces in our approach of first constructing a scalar-valued function $v$ from a multi-variate time series $x$ with the three different methods presented and subsequently applying the sublevel set filtration to estimate the times of recurrence.
  • Figure 2: From left to right: recurrent, repetitive, and periodic time series, each with the corresponding scalar-valued function $v_x, v_x, v_{x}$ constructed by the three methods and the resulting persistence diagram $D(v_x), D(v_x), D(v_{x})$.
  • Figure 3: Distribtion of the absolute error of predicted recurrence times relative to the true cycle duration for each method and dataset section.

Theorems & Definitions (5)

  • definition thmcounterdefinition: periodic
  • definition thmcounterdefinition: repetitive
  • definition thmcounterdefinition: recurring
  • definition thmcounterdefinition: approximately I
  • definition thmcounterdefinition: approximately II