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Construction of Confidence Sets for Markov Chain Model

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CONTROLO 2016

Part of the book series: Lecture Notes in Electrical Engineering ((LNEE,volume 402))

Abstract

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.

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References

  1. Anderson, T.W., Goodman, L.A.: Statistical inference about Markov chains. Anal. Math. Stat. 28, 89–110 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bellman, R.: Introduction to Matrix Analysis, 2nd edn. Society for Industrial and Applied Mathematics (1987)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Hanson, S., Schuermann, T.: Confidence intervals for probabilities of default. J. Bank. Finan. 30(8), 2281–2301 (2006). Elsevier

    Google Scholar 

  6. Jafry, Y., Schuermann. T.: Measurement, estimation and comparison of credit migration matrices. J. Bank. Finan. 28 (2004)

    Google Scholar 

  7. Kibzun, A.I., Kan, Yu. S.: Stochastic Programming Problem with Probability and Quantile Functions. Wiley, Chichester (1996)

    Google Scholar 

  8. Kurzhanski, A.B., Vályi, I.: Ellipsoidal Calculus for Estimation and Control. Birkhäuser, Boston (1997)

    Book  MATH  Google Scholar 

  9. Precopa, A.: Stochastic Programming. Kluwer Acad. Publ., Dordrecht (1995)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Optimal Control, Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, S. Kovalevskaya str., 16, 620990, Ekaterinburg, Russia

    Dmitry Zavalishchin

  2. Ural State University of Railway Transport, Kolmogorov str., 64, 620034, Ekaterinburg, Russia

    Dmitry Zavalishchin & Galina Timofeeva

  3. Ural Federal University, Mira str., 19, 62002, Ekaterinburg, Russia

    Dmitry Zavalishchin & Galina Timofeeva

Authors
  1. Dmitry Zavalishchin
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  2. Galina Timofeeva
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Corresponding author

Correspondence to Dmitry Zavalishchin .

Editor information

Editors and Affiliations

  1. School of Engineering,Campus de Couros, University of Minho School of Engineering,Campus de Couros, Guimarães, Portugal

    Paulo Garrido

  2. School of Engineering,Campus de Couros, University of Minho School of Engineering,Campus de Couros, Guimarães, Portugal

    Filomena Soares

  3. Faculty of Engineering, University of Porto Faculty of Engineering, Porto, Portugal

    António Paulo Moreira

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Zavalishchin, D., Timofeeva, G. (2017). Construction of Confidence Sets for Markov Chain Model. In: Garrido, P., Soares, F., Moreira, A. (eds) CONTROLO 2016. Lecture Notes in Electrical Engineering, vol 402. Springer, Cham. https://doi.org/10.1007/978-3-319-43671-5_22

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  • DOI: https://doi.org/10.1007/978-3-319-43671-5_22

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-43670-8

  • Online ISBN: 978-3-319-43671-5

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