Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - Université de Rouen Normandie

The detection of disruptions in stochastic processes has evolved from early control charts to rigorous optimal stopping frameworks and modern adaptive systems. The classical procedures like CUSUM and Shiryaev’s rule set performance benchmarks and continue to inspire extensions to dependent, high-dimensional, and uncertain settings. The numerous application areas span industry, finance, engineering, cybersecurity, environmental protection, and healthcare, with a particularly significant impact on epidemiological surveillance. The ongoing research blends statistical optimality with machine learning and computation, promising faster, more reliable, and more automated detection. The key challenge remains responding as quickly as possible to changes while controlling the false alarms, and this goal will continue to guide research efforts.

This thesis addresses the online detection problem in a truncated formulation, i.e., when the observations are accessible only within bounded time intervals. Such situations are typical in numerous applications. For example, in the public health area devoted to the epidemiological analysis, epidemic outbreaks typically occur unexpectedly within two to three weeks, especially in large industrial and student cities. Furthermore, a similar situation arises in detection problems of signals, which may only appear over short time intervals (transient change detection problem). Unfortunately, the use of asymptotic methods in such a setting is impractical, since there is no way to consider an infinite or at least sufficiently large time interval, as suggested by the corresponding asymptotic optimal solutions. Moreover, it is important not to neglect the probability of a false alarm, as it can be costly.

The general goal of this thesis is to develop non-asymptotically optimal statistical sequential methods for the quickest online change-point detection in observations over bounded time intervals, while simultaneously controlling the false-alarm probabilities. Moreover, in this thesis, these problems are studied for dependent data represented by time series defined by Markov chains. To do this, two risks are used: the average delay risk and the probability criterion. The first means the expectation of the detection delay, and the second one is the probability that the detection delay exceeds a fixed admissible level. The goal of detection is to minimise these risks over all possible stopping times subject to a fixed upper bound for the false-alarm probabilities. In the thesis these problems are studied in a Bayesian setting for dependent data defined by Markov chains. To achieve these goals, one develops new sequential detection methods based on the optimal stopping theory for homogeneous Markov chains. The main difference between the proposed detection procedures and the usual ones is that they are based not on the posterior probabilities, but on the weighted Shiryaev-Roberts statistics.