Point process models for novelty detection on spatial point patterns and their extremes

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.