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In this paper, we propose a dimensionless anomaly detection method formultivariate streams. Our method is independent of the unit of measurement forthe different stream channels, therefore dimensionless. We first propose thevariance norm, a generalisation of Mahalanobis distance to handleinfinite-dimensional feature space and singular empirical covariance matrixrigorously. We then combine the variance norm with the path signature, aninfinite collection of iterated integrals that provide global features ofstreams, to propose SigMahaKNN, a method for anomaly detection on(multivariate) streams. We show that SigMahaKNN is invariant to streamreparametrisation, stream concatenation and has a graded discrimination powerdepending on the truncation level of the path signature. We implementSigMahaKNN as an open-source software, and perform extensive numericalexperiments, showing significantly improved anomaly detection on streamscompared to isolation forest and local outlier factors in applications rangingfrom language analysis, hand-writing analysis, ship movement paths analysis andunivariate time-series analysis.

Dimensionless Anomaly Detection on Multivariate Streams with Variance Norm and Path Signature