Abstract
Multivariate data sources with components of different information value seem to appear frequently in practice. Models in which the components change their homogeneity at different times are of significant importance. The fact whether any changes are influential for the whole process is determined not only by the moments of the change, but also depends on which coordinates. This is particularly important in issues such as reliability analysis of complex systems and the location of an intruder in surveillance systems. In this paper we developed a mathematical model for such sources of signals with discrete time having the Markov property given the times of change. The research also comprises a multivariate detection of the transition probabilities changes at certain sensitivity level in the multidimensional process. Additionally, the observation of the random vector is depicted. Each chosen coordinate forms the Markov process with different transition probabilities before and after some unknown moment. The aim of statisticians is to estimate the moments based on the observation of the process. The Bayesian approach is used with the risk function depending on measure of chance of a false alarm and some cost of overestimation. The moment of the system’s disorder is determined by the detection of transition probabilities changes at some coordinates. The overall modeling of the critical coordinates is based on the simple game.
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References
Bojdecki T (1979) Probability maximizing approach to optimal stopping and its application to a disorder problem. Stochastics 3:61–71
Brodsky B, Darkhovsky B (1993) Nonparametric methods in change-point problems. Mathematics and its applications, vol 243. Kluwer Academic, Dordrecht
Ferguson TS (2005) Selection by committee. In: Nowak A, Szajowski K (eds) Advances in dynamic games. Ann Internat Soc Dynam Games, vol 7. Birkhäuser, Boston, pp 203–209
Gharehshiran ON, Krishnamurthy V (2010) Coalition formation for bearings-only localization in sensor networks—a cooperative game approach. IEEE Trans Signal Process 58(8):4322–4338
Kurano M, Yasuda M, Nakagami J (1980) Multi-variate stopping problem with a majority rule. J Oper Res Soc Jpn 23:205–223
Mei Y (2006) Comments on: “A note on optimal detection of a change in distribution”, by Benjamin Yakir. Ann Stat 34(3):1570–1576
Moustakides GV (1998) Quickest detection of abrupt changes for a class of random processes. IEEE Trans Inf Theory 44(5):1965–1968
Nash J (1951) Non-cooperative game. Ann Math 54(2):286–295
Owen G (1995) Game theory, 3rd edn. Academic Press, San Diego
Peskir G, Shiryaev A (2006) Optimal stopping and free-boundary problems. Lectures in mathematics. ETH, Zürich
Raghavan V, Veeravalli VV (2010) Quickest change detection of a Markov process across a sensor array. IEEE Trans Inf Theory 56(4):1961–1981
Sarnowski W, Szajowski K (2011) Optimal detection of transition probability change in random sequence. Stochastics 83(4–6):569–581
Shiryaev A (1961) The detection of spontaneous effects. Sov Math Dokl 2:740–743. Translation from Dokl Akad Nauk SSSR 138:799–801 (1961)
Shiryaev A (1978) Optimal stopping rules. Springer, New York
Szajowski K (1992) Optimal on-line detection of outside observations. J Stat Plan Inference 30:413–422
Szajowski K, Yasuda M (1996) Voting procedure on stopping games of Markov chain. In: Christer AH, Osaki S, Thomas LC (eds) UK-Japanese res workshop on stoch. Modelling in innovative manufacturing, Moller Centre, Churchill College, Univ. Cambridge, UK, 21–22 July, 1995. LN in Econ and Math Sys, vol 445. Springer, Cambridge, pp 68–80
Szajowski K (2011) Multi-variate quickest detection of significant change process. In: Baras JS, Katz J, Altman E (eds) Decision and game theory for security. Second int conf, GameSec 2011, College Park, MD, USA, 14–15 Nov., 2011. LN in comp sci, vol 7037. Springer, Berlin, pp 56–66
Tartakovsky AG, Rozovskii BL, Blažek RB, Kim H (2006) Detection of intrusions in information systems by sequential change-point methods. Stat Methodol 3(3):252–293
Tartakovsky AG, Veeravalli VV (2008) Asymptotically optimal quickest change detection in distributed sensor systems. Seq Anal 27(4):441–475
Yakir B (1992) Optimal detection of a change in distribution when the observations form a Markov chain with a finite state space. In: Carlstein E, Müller HG, Siegmund D (eds) Change-point problems. IMS LN-monograph series, vol 23. Papers from the AMS-IMS-SIAM Summer Res Conf held at Mt Holyoke College, South Hadley, MA, USA, 11–16 July, 1992. IMS, Hayward, pp 346–358
Yoshida M (1983) Probability maximizing approach for a quickest detection problem with complocated Markov chain. J Inf Optim Sci 4:127–145
Acknowledgements
The author would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme “Inference for Change-Point and Related Processes”, where a part of the work on this paper was undertaken.
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Szajowski, K. (2015). On Some Distributed Disorder Detection. In: Steland, A., Rafajłowicz, E., Szajowski, K. (eds) Stochastic Models, Statistics and Their Applications. Springer Proceedings in Mathematics & Statistics, vol 122. Springer, Cham. https://doi.org/10.1007/978-3-319-13881-7_21
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DOI: https://doi.org/10.1007/978-3-319-13881-7_21
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