Sequential detection of transient changes in stochastic-dynamical systems

Do, Van Long; Fillatre, Lionel; Nikiforov, Igor

Journal de la société française de statistique, Tome 156 (2015) no. 4, pp. 60-97.

Cet article s’intéresse au problème de détection de changements transitoires dans des systèmes stochastiques et dynamiques. Le modèle d’observation statistique étudié dépend de l’état inconnu du système considéré comme un paramètre de nuisance. Ce paramètre de nuisance est éliminé en utilisant la technique, bien connue dans la communauté du diagnostic automatique, de la projection des observations dans l’espace de parité. L’algorithme de la Somme Cumulée à Fenêtre Limitée et Seuils Variables (VTWL CUSUM) est adapté au modèle d’observation utilisé. Le critère de détection de changement transitoire étudié vise à minimiser la pire probabilité de détection manquée sous la contrainte que la pire probabilité de la fausse alarme soit bornée pendant une période de longueur donnée. Les seuils de l’algorithme sont optimisés pour obtenir la meilleure performance. Il est montré que l’algorithme VTWL CUSUM optimal est équivalent à l’algorithme de la Moyenne Glissante Finie (FMA). Une méthode numérique est proposée pour estimer les probabilités de fausse alarme et de détection manquée. Enfin, les résultats théoriques sont appliqués à la détection d’attaques cyber-physiques, dans un système de distribution d’eau potable, qui ont pour but de voler l’eau d’un réservoir.

This paper deals with the problem of detecting transient changes in stochastic-dynamical systems. A statistical observation model which depends on unknown system states (often regarded as the nuisance parameter) is developed. The negative impact of nuisance parameter is then eliminated from the observation model by utilizing the invariant statistics. The Variable Threshold Window Limited CUmulative SUM (VTWL CUSUM) test, previously developed for independent observations, is adapted to the novel observation model. Taking into account the transient change detection criterion, minimizing the worst-case probability of missed detection subject to an acceptable level of the worst-case probability of false alarm within a given time period, the thresholds of the VTWL CUSUM test are optimized. It is shown that the optimized VTWL CUSUM algorithm is equivalent to the Finite Moving Average (FMA) detection rule. A numerical method for estimating the probability of false alarm and missed detection is proposed. The theoretical results are applied to the problem of cyber/physical attack (stealing water from a reservoir) detection on a simple Supervisory Control and Data Acquisition (SCADA) water distribution system.

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