CONTROLO 2016 pp 253-263 | Cite as

Construction of Confidence Sets for Markov Chain Model

  • Dmitry ZavalishchinEmail author
  • Galina Timofeeva
Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 402)


The problem of forecasting the state probabilities vector for a stationary Markov chain with discrete time in case of transition probabilities are not exactly known and measured during the system operation is investigated. An auxiliary dynamic system with uncertainty and observation showing the dynamics of the state probabilities vector is constructed. Two approaches to the determination of the guaranteed with a given probability estimates are proposed. The first approach uses confidence sets for rows of the matrix of transition probabilities and the ellipsoidal calculus. The second approach is based on simulation and finding a sample quantile of an objective function, which determines the form of a confidence set for the system state.


Markov chain Incomplete information Confidence estimate 


  1. 1.
    Anderson, T.W., Goodman, L.A.: Statistical inference about Markov chains. Anal. Math. Stat. 28, 89–110 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bellman, R.: Introduction to Matrix Analysis, 2nd edn. Society for Industrial and Applied Mathematics (1987)Google Scholar
  3. 3.
    Chernousko, F.L., Ovseevich, A.I.: Properties of the optimal ellipsoids approximating the reachable sets of uncertain systems. J. Optim. Theory Appl. 120(2), 223–246 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Filippova, T.F., Matviychuk, O.G.: Estimates of reachable sets of control systems with bilinear quadratic nonlinearities. Ural Math. J. 1(1), 45–54 (2015)Google Scholar
  5. 5.
    Hanson, S., Schuermann, T.: Confidence intervals for probabilities of default. J. Bank. Finan. 30(8), 2281–2301 (2006). ElsevierGoogle Scholar
  6. 6.
    Jafry, Y., Schuermann. T.: Measurement, estimation and comparison of credit migration matrices. J. Bank. Finan. 28 (2004)Google Scholar
  7. 7.
    Kibzun, A.I., Kan, Yu. S.: Stochastic Programming Problem with Probability and Quantile Functions. Wiley, Chichester (1996)Google Scholar
  8. 8.
    Kurzhanski, A.B., Vályi, I.: Ellipsoidal Calculus for Estimation and Control. Birkhäuser, Boston (1997)CrossRefzbMATHGoogle Scholar
  9. 9.
    Precopa, A.: Stochastic Programming. Kluwer Acad. Publ., Dordrecht (1995)Google Scholar
  10. 10.
    Timofeeva, G.A., Timofeev, N.A.: Predicting the components of credit portfolio based on a Markov chain model. Autom. Remote Control 73(4), 637–651 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Timofeev, N., Timofeeva, G.: Estimation of loan portfolio risk on the basis of Markov chain model. In: Hömberg, D., Tröltzsch, F. (eds.) IFIP Advances in Information and Communication Technology. Volume 391: System Modeling and Optimization. IFIP TC7 Conference: CSMO 2011, pp. 207–216. Springer (2013)Google Scholar
  12. 12.
    Vishnyakov, B.V., Kibzun, A.I.: Application of the bootstrap method for estimation of the quantile function. Autom. Remote Control 68(11), 1931–1944 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.Department of Optimal ControlInstitute of Mathematics and Mechanics, Ural Branch of the Russian Academy of SciencesEkaterinburgRussia
  2. 2.Ural State University of Railway TransportEkaterinburgRussia
  3. 3.Ural Federal UniversityEkaterinburgRussia

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