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

2025-05-15

AI Summary

This paper addresses the automatic identification of repetitive behaviors (e.g., periodicity or recurrence) and precise estimation of recurrence times in multivariate time series. We propose the first topology-driven framework based on persistent homology. Methodologically, we systematically introduce persistent homology to this task for the first time, designing three theoretically provably stable algorithms tailored to recurrence detection; these integrate time-delay embedding with stability analysis for robust performance. Our contributions are threefold: (1) a theoretical foundation establishing topological representations of repetitive patterns with rigorous stability guarantees; (2) the first annotated benchmark dataset derived from real-world industrial settings—specifically, injection molding machines; and (3) empirical validation on real data demonstrating significant improvements over classical baselines—including spectral analysis and autocorrelation—thereby confirming the method’s generalizability and practical utility.

Technology Category

• Machine Learning: Time-Series/Data Streams
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Intelligent Robots: Other Foundations and Applications

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• User Modeling, Personalization and Recommendation: Studies of user behavior, including longitudinal effects of personalized systems

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.

Problem

• Identify repetitive patterns in multi-variate time series
• Estimate recurrence times for periodic and non-periodic behaviors
• Develop stable topological methods for real-world industrial data

Innovation

• Persistent homology framework for recurrence times
• Three specialized stable methods
• Validation with real-world benchmark data