Abstract
Novelty detection is a particular example of pattern recognition identifying patterns that departure from some model of “normal behaviour”. The classification of point patterns is considered that are defined as sets of N observations of a multivariate random variable X and where the value N follows a discrete stochastic distribution. The use of point process models is introduced that allow us to describe the length N as well as the geometrical configuration in data space of such patterns. It is shown that such infinite dimensional study can be translated into a one-dimensional study that is analytically tractable for a multivariate Gaussian distribution. Moreover, for other multivariate distributions, an analytic approximation is obtained, by the use of extreme value theory, to model point patterns that occur in low-density regions as defined by X. The proposed models are demonstrated on synthetic and real-world data sets.
Highlights
Novelty detection is the task of recognising test data that differ in some respect from the data that were available during training (Bishop, 1994); it is typically used when there is a large quantity of ‘‘normal’’ data available, but an insufficient quantity of ‘‘abnormal’’ data, preventing accurate estimation of the ‘‘abnormal’’ class in a two-class classification setting (Nitesh, 2005)
We provide an analytic approximation using extreme value theory (EVT) to model those point patterns that are situated in regions where the density defined by X is low
It can be shown that when the limit in (8) holds for the random variable Z for some scale parameter σ, the corresponding sequences of point process models (PPMs) of exceedancesZne will converge to a Poisson point process (PPP) for large u, meaning that the corresponding sequence of counting measures NAn associated withZne converge in distribution to a Poisson distribution: NAn d
Summary
Novelty detection is the task of recognising test data that differ in some respect from the data that were available during training (Bishop, 1994); it is typically used when there is a large quantity of ‘‘normal’’ data available, but an insufficient quantity of ‘‘abnormal’’ data, preventing accurate estimation of the ‘‘abnormal’’ class in a two-class classification setting (Nitesh, 2005). The method introduced in this article is based on the development of point process models (PPMs) and provides a valid probabilistic interpretation of the degree of novelty of an entire point pattern x. This approach enables us to prevent those. We provide an analytic approximation using extreme value theory (EVT) to model those point patterns that are situated in regions where the density defined by X is low (the ‘extremes’ of the pattern) These regions are of particular importance because the decision boundary for novelty detection is typically situated at the edge of the support of X , where the density is low.
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