Samuel Hugueny

Novelty Detection with Multivariate Extreme Value Statistics

Novelty detection, or one-class classification, aims to determine if data are “normal” with respect to some model of normality constructed using examples of normal system behaviour. If that model is composed of generative probability distributions, the extent of “normality” in the data space can be described using Extreme Value Theory (EVT), a branch of statistics concerned with describing the tails of distributions. This paper demonstrates that existing approaches to the use of EVT for novelty detection are appropriate only for univariate, unimodal problems. We generalise the use of EVT for novelty detection to the analysis of data with multivariate, multimodal distributions, allowing a principled approach to the analysis of high-dimensional data to be taken. Examples are provided using vital-sign data obtained from a large clinical study of patients in a high-dependency hospital ward.

An Extreme Function Theory for Novelty Detection

We introduce an extreme function theory as a novel method by which probabilistic novelty detection may be performed with functions, where the functions are represented by time-series of (potentially multivariate) discrete observations. We set the method within the framework of Gaussian processes (GP), which offers a convenient means of constructing a distribution over functions. Whereas conventional novelty detection methods aim to identify individually extreme data points, with respect to a model of normality constructed using examples of “normal” data points, the proposed method aims to identify extreme functions, with respect to a model of normality constructed using examples of “normal” functions, where those functions are represented by time-series of observations. The method is illustrated using synthetic data, physiological data acquired from a large clinical trial, and a benchmark time-series dataset.

Probabilistic Patient Monitoring with Multivariate, Multimodal Extreme Value Theory

Conventional patient monitoring is performed by generating alarms when vital signs exceed pre-determined thresholds, but the false-alarm rate of such monitors in hospitals is so high that alarms are typically ignored. We propose a principled, probabilistic method for combining vital signs into a multivariate model of patient state, using extreme value theory (EVT) to generate robust alarms if a patient’s vital signs are deemed to have become sufficiently “extreme”. Our proposed formulation operates many orders of magnitude faster than existing methods, allowing on-line learning of models, leading ultimately to patient-specific monitoring.

Extending the Generalised Pareto Distribution for Novelty Detection in High-Dimensional Spaces

Novelty detection involves the construction of a “model of normality”, and then classifies test data as being either “normal” or “abnormal” with respect to that model. For this reason, it is often termed one-class classification. The approach is suitable for cases in which examples of “normal” behaviour are commonly available, but in which cases of “abnormal” data are comparatively rare. When performing novelty detection, we are typically most interested in the tails of the normal model, because it is in these tails that a decision boundary between “normal” and “abnormal” areas of data space usually lies. Extreme value statistics provides an appropriate theoretical framework for modelling the tails of univariate (or low-dimensional) distributions, using the generalised Pareto distribution (GPD), which can be demonstrated to be the limiting distribution for data occurring within the tails of most practically-encountered probability distributions. This paper provides an extension of the GPD, allowing the modelling of probability distributions of arbitrarily high dimension, such as occurs when using complex, multimodel, multivariate distributions for performing novelty detection in most real-life cases. We demonstrate our extension to the GPD using examples from patient physiological monitoring, in which we have acquired data from hospital patients in large clinical studies of high-acuity wards, and in which we wish to determine “abnormal” patient data, such that early warning of patient physiological deterioration may be provided.

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 “normal” 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 “normal” areas of data space from “abnormal” 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.

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 “normal” 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 “normal” areas from “abnormal” 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.

A comparison of approaches to multivariate extreme value theory for novelty detection

Novelty detection, one-class classification, or outlier detection, is typically employed for analysing signals when few examples of “abnormal” data are available, such that a multiclass approach cannot be taken. Multivariate, multimodal density estimation can be used to construct a model of the distribution of normal data. However, setting a decision boundary such that test data can be classified “normal” or “abnormal” with respect to the model of normality is typically performed using heuristic methods, such as thresholding the unconditional data density, p(x). This paper describes two principled methods of setting a decision boundary based on extreme value statistics: (i) a numerical method that produces an “optimal” solution, and (ii) an analytical approximation in closed form. We compare the performance of both approaches using large datasets from biomedical patient monitoring and jet engine health monitoring, and conclude that the analytical approach performs novelty detection as successfully as the “optimal” numerical approach, both of which outperform the conventional method.

Pinning the tail on the distribution: A multivariate extension to the generalised Pareto distribution

Novelty detection is often used for analysis where there are insufficient examples of “abnormal” data to take a multi-class approach to classification. Models of normality are constructed from commonly-available examples of “normal” behaviour, and we then reason about the presence of abnormalities with respect to this normal model. Extreme value theory (EVT) is a branch of statistics that is concerned with modelling extremal events, and is therefore appealing for use with novelty detection. However, conventional existing EVT approaches are limited to the analysis of univariate or low-dimension data. This paper considers the peaks-over-threshold method of EVT, in which exceedances over a (typically univariate) threshold can be shown to tend towards the generalised Pareto distribution (GPD). We extend this method for use with high-dimensional data, allowing us to reason about the “extreme” data lying in the tails of the distributions of complex, real-world datasets, which are typically multivariate and multimodal. Illustrations are provided from the analysis of large clinical studies of hospital patient vital-sign data.