  In this paper, we propose a dimensionless anomaly detection method for
multivariate streams. Our method is independent of the unit of measurement for
the different stream channels, therefore dimensionless. We first propose the
variance norm, a generalisation of Mahalanobis distance to handle
infinite-dimensional feature space and singular empirical covariance matrix
rigorously. We then combine the variance norm with the path signature, an
infinite collection of iterated integrals that provide global features of
streams, to propose SigMahaKNN, a method for anomaly detection on
(multivariate) streams. We show that SigMahaKNN is invariant to stream
reparametrisation, stream concatenation and has a graded discrimination power
depending on the truncation level of the path signature. We implement
SigMahaKNN as an open-source software, and perform extensive numerical
experiments, showing significantly improved anomaly detection on streams
compared to isolation forest and local outlier factors in applications ranging
from language analysis, hand-writing analysis, ship movement paths analysis and
univariate time-series analysis.
