A Persistent Homology-Based Algorithm for Unsupervised Anomaly Detection in Time Series

Abstract

In this article, we propose a new algorithm for unsupervised anomaly detection in univariate time series, based on topological data analysis. It relies on delay embeddings and on the extraction of persistent cycles from the 1-dimensional persistent homology constructed from the distance to measure Rips filtration. This filtration makes it possible to identify 1-cycles (i.e. loops) corresponding to recurrent patterns by leveraging density information. Points in those cycles are considered as normal, and the algorithm can then assign an anomaly score to any point which is its distance to the normal set. In this paper, we describe the algorithm, make a theoretical study, and test it on several real-world and synthetic datasets, showing that it is competitive with state-of-the-art anomaly detection methods.