Anomaly Detection based on Markov Data: A Statistical Depth Approach

The purpose of this article is to extend the notion of statistical depth to the case of sample paths of a Markov chain. Initially introduced to define a center-outward ordering of points in the support of a multivariate distribution, depth functions permit to generalize the notions of quantiles and (signed) ranks for observations in $\mathbb{R}^d$ with $d>1$, as well as statistical procedures based on such quantities. Here we develop a general theoretical framework for evaluating the depth of a Markov sample path and recovering it statistically from an estimate of its transition probability with (non-) asymptotic guarantees. We also detail some of its applications, focusing particularly on unsupervised anomaly detection. Beyond the theoretical analysis carried out, numerical experiments are displayed, providing empirical evidence of the relevance of the novel concept we introduce here to quantify the degree of abnormality of Markov paths of variable length.