Precision of sequential change point detection

Volume 44 / 2017

Aleksandra Ochman-Gozdek, Wojciech Sarnowski, Krzysztof Szajowski Applicationes Mathematicae 44 (2017), 267-280 MSC: Primary 60G40; Secondary 60K99, 90D60. DOI: 10.4064/am2278-5-2017 Published online: 15 September 2017

Abstract

A random sequence having two segments that are homogeneous Markov processes is registered. Each segment has its own transition probability law, and the length of the segment is unknown and random. The transition probabilities of each of the processes are known and the a priori distribution of the disorder time is given. The decision maker’s aim is to detect the time when the transition probabilities change. The detection of the disorder is rarely precise. The decision maker accepts some deviation in estimation of the disorder time. In the model we consider the aim is to indicate the change point with fixed, bounded error with maximal probability. The precision differs for over and under estimation of this point. The case when the disorder does not appear with positive probability is also included. The results extend significantly the range of applications, explain the structure of optimal detector in various circumstances and show new details of the solution construction. The motivation for this investigation is the modelling of attacks in a node of networks. The objective is to detect one of the attacks immediately or in very short time before or after its appearance with the highest probability. The problem is reformulated as optimal stopping of the observed sequences. The detailed analysis of the problem is presented to show the form of the optimal decision function.

Authors

  • Aleksandra Ochman-GozdekObjectivity Bespoke Software Specialists Ltd.
    Strzegomska 142a
    54-429 Wrocław, Poland
    e-mail
  • Wojciech SarnowskiBusiness Intelligence Team
    at Krajowy Rejestr Długów
    Armii Ludowej 21
    51-214 Wrocław, Poland
    e-mail
  • Krzysztof SzajowskiFaculty of Pure and Applied Mathematics
    Wrocław University of Science and Technology
    Wybrzeże Wyspiańskiego 27
    50-370 Wrocław, Poland
    e-mail

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