Anomaly detection in extreme regions via empirical MV-sets on the sphere

Extreme regions in the feature space are of particular concern for anomaly detection: anomalies are likely to be located in the tails, whereas data scarcity in such regions makes it difficult to distinguish between large normal instances and anomalies. This paper presents an unsupervised algorithm for anomaly detection in extreme regions. We propose a Minimum Volume set (MV-set) approach relying on multivariate extreme value theory. This framework includes a canonical pre-processing step, which addresses the issue of output sensitivity to standardization choices. The resulting data representation on the sphere highlights the dependence structure of the extremal observations. Anomaly detection is then cast as a MV-set estimation problem on the sphere, where volume is measured by the spherical measure and mass refers to the angular measure. An anomaly then corresponds to an unusual observation given that one of its variables is large. A preliminary rate bound analysis is carried out for the learning method we introduce and its computational advantages are discussed and illustrated by numerical experiments.