Anomaly Detection based on Markov Data: A Statistical Depth Approach

It is the main purpose of this article to extend the notion of statistical depth to the case of sample paths of a Markov chain, a very popular probabilistic model to describe parsimoniously random phenomena with a temporal causality. 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, for (unsupervised) anomaly detection tasks in particular. In this paper, overcoming the lack of natural order on the torus composed of all possible trajectories of finite length, 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 its numerous applications, focusing particularly on anomaly detection, a key task in various fields involving the analysis of (supposedly) Markov time-series (\textit{e.g.} health monitoring of complex infrastructures, security). Beyond the description of the methodology promoted and the statistical analysis carried out to guarantee its validity, numerical experiments are displayed, providing strong empirical evidence of the relevance of the novel concept we introduce here to quantify the degree of abnormality of Markov path sequences of variable length.