Novel semi-metrics for multivariate change point analysis and anomaly detection

https://doi.org/10.1016/j.physd.2020.132636Get rights and content

Highlights

  • We introduce a new family of semi-metrics and provide analysis of their properties.

  • Semi-metrics measure distance between time series based on their structural breaks.

  • We introduce a computational method for analysing transitivity properties.

  • Eigenvalue analysis quickly determines the size of a majority cluster.

  • We analyse the cryptocurrency market and 19th century UK measles data.

Abstract

This paper proposes a new method for determining similarity and anomalies between time series, most practically effective in large collections of (likely related) time series, by measuring distances between structural breaks within such a collection. We introduce a class of semi-metric distance measures, which we term MJ distances. These semi-metrics provide an advantage over existing options such as the Hausdorff and Wasserstein metrics. We prove they have desirable properties, including better sensitivity to outliers, while experiments on simulated data demonstrate that they uncover similarity within collections of time series more effectively. Semi-metrics carry a potential disadvantage: without the triangle inequality, they may not satisfy a “transitivity property of closeness.” We analyse this failure with proof and introduce an computational method to investigate, in which we demonstrate that our semi-metrics violate transitivity infrequently and mildly. Finally, we apply our methods to cryptocurrency and measles data, introducing a judicious application of eigenvalue analysis.

Keywords

Semi-metrics
Change-point detection
Multivariate analysis
Time series
Anomaly detection

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