Anomaly Detection based on Markov Data: A Statistical Depth Approach

TL;DR

This work addresses anomaly detection in Markov data by extending statistical depth to finite Markov paths. It introduces Markov depth $D_{\text{Π}}(\mathbf{x})$, defined as the geometric mean of per-step depths along a path $\mathbf{x}=(x_0,\dots,x_n)$, providing a trajectory-wise center-outward ordering on the path space $\mathbb{T}$. The authors establish a limit distribution for $D_{\text{Π}}$ under positive recurrence and derive non-asymptotic bounds for sampling-based estimates when the base depth satisfies a Lipschitz condition, while demonstrating practical efficacy on ARCH(1) and GI/G/1 queuing models, particularly for dynamic anomalies. The method offers competitive or superior performance against standard baselines for variable-length trajectories and affords computational advantages for long paths, with potential extensions to clustering, homogeneity testing, and trajectory visualization.

Abstract

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.