Novelty detection with multivariate Extreme Value Theory, part II: An analytical approach to unimodal 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, 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 extrema generated from the model, and can be used to separate ldquonormalrdquo areas from ldquoabnormalrdquo areas of feature space in a principled manner. In a companion paper, we show that existing work in the use of EVT for novelty detection does not accurately describe the extrema of multimodal, multivariate distributions and propose a numerical method for overcoming such problems. In this paper, we introduce an analytical approach to obtain closed-form solutions for the extreme value distributions of multivariate Gaussian distributions and present an application to vital-sign monitoring.