Novelty detection with multivariate Extreme Value Theory, part I: A numerical approach to multimodal estimation

Extreme Value Theory (EVT) describes the distribution of data considered extreme with respect to some generative distribution, effectively modelling the tails of that distribution. In novelty detection, or one-class classification, we wish to determine if data are ldquonormalrdquo with respect to some model of normality. If that model consists of generative distributions, then EVT is appropriate for describing the behaviour of extremes generated from the model, and can be used to determine the location of decision boundaries that separate ldquonormalrdquo areas of data space from ldquoabnormalrdquo areas in a principled manner. This paper introduces existing work in the use of EVT for novelty detection, shows that existing work does not accurately describe the extrema of multivariate, multimodal generative distributions, and proposes a novel method for overcoming such problems. The method is numerical, and provides optimal solutions for generative multivariate, multimodal distributions of arbitrary complexity. In a companion paper, we present analytical closed-form solutions which are currently limited to unimodal, multivariate generative distributions.