Nonparametric PU Learning of State Estimation in Markov Switching Model

  • A. Dobrovidov
  • V. Vasilyev
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 250)


In this contribution, we develop methods of nonlinear filtering and prediction of an unobservable Markov chain which controls the states of observable stochastic process. This process is a mixture of two subsidiary stochastic processes, the switching of which is controlled by the Markov chain. Each of this subsidiary processes is described by conditional distribution density (cdd). The feature of the problem is that cdd’s and transition probability matrix of the Markov chain are unknown, but a training sample (positive labeled) from one of the two subsidiary processes and training sample (unlabeled) from the mixture process are available. Construction of process binary classifier using positive and unlabeled samples in machine learning is called PU learning. To solve this problem for stochastic processes, nonparametric kernel estimators based on weakly dependent observations are applied. We examine the novel method performance on simulated data and compare it with the same performance of the optimal Bayesian solution with known cdd’s and the transition matrix of the Markov chain. The modeling shows close results for the optimal task and the PU learning problem even in the case of a strong overlapping of the conditional densities of subsidiary processes.


  1. 1.
    Abramson, I. S. (1982). On bandwidth variation in kernel estimates-a square root law. Annals of Statistics, 10(4), 1217–1223.Google Scholar
  2. 2.
    Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31(3), 307–327.Google Scholar
  3. 3.
    Bowman, A. (1984). An alternative method of cross-validation for the smoothing of density estimates. Biometrika, 7, 353–360.Google Scholar
  4. 4.
    Breiman, L., Meisel, W., & Purcell, E. (1977). Variable kernel estimates of multivariate densities. Technometrics, 19(2), 135–144. ISSN 0040(1706)Google Scholar
  5. 5.
    Dobrovidov, A. V., Koshkin, G. M., & Vasiliev V. A. (2012). Non-parametric models and statistical inference from dependent observations. Heber: Kendrick Press.Google Scholar
  6. 6.
    Douc, R., Moulines, E., & Rydén, T. (2004). Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime. The Annals of Statistics, 32(5), 2254–2304.Google Scholar
  7. 7.
    Francq, C., & Zakoian, J.-M. (2005). The l2-structures of standard and switching-regime Garch models. Stochastic Processes and Their Applications, 115(9), 1557–1582.Google Scholar
  8. 8.
    Giraitis, L., Leipus, R., & Surgailis, D. (2007). Recent advances in arch modelling. In Long memory in economics (pp. 3–38). Berlin: Springer.Google Scholar
  9. 9.
    Hall, P., & Marron, J. (1991). Local minima in cross-validation functions. Journal of the Royal Statistical Society, Series B (Methodological), 53, 245–252.Google Scholar
  10. 10.
    Hamilton, J. D., & Susmel, R. (1994). Autoregressive conditional heteroskedasticity and changes in regime. Journal of Econometrics, 64(1), 307–333.Google Scholar
  11. 11.
    Klaassen, F. (2002). Improving Garch volatility forecasts with regime-switching Garch. Empirical Economics, 27(2), 363–394.Google Scholar
  12. 12.
    Lanchantin, P., & Pieczynski, W. (2005). Unsupervised restoration of hidden non stationary Markov chain using evidential priors. IEEE Transactions on Signal Processing, 53(8), 3091–3098.Google Scholar
  13. 13.
    Loftsgaarden, D. O., & Quesenberry, C. P. (1965). A nonparametric estimate of a multivariate density function. Annals of Mathematical Statistics, 36(3), 1049–1051.Google Scholar
  14. 14.
    Rudemo, M. (1982). Empirical choice of histograms and kernel density estimators. Scandinavian Journal of Statistics, 9, 65–78.Google Scholar
  15. 15.
    Silverman, B. (1986). Density estimation for statistics and data analysis. Monographs on statistics and applied probability. Boca Raton, London: Chapman and Hall.Google Scholar
  16. 16.
    Yao, J.-F., & Attali, J.-G. (2000). On stability of nonlinear ar processes with Markov switching. Advances in Applied Probability, 32(2), 394–407.Google Scholar
  17. 17.
    Zhang, M. Y., Russell, J. R., & Tsay, R. S. (2001). A nonlinear autoregressive conditional duration model with applications to financial transaction data. Journal of Econometrics, 104(1), 179–207.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • A. Dobrovidov
    • 1
  • V. Vasilyev
    • 2
  1. 1.V.A. Trapeznikov Institute of Control Sciences, Russian Academy of SciencesMoscowRussia
  2. 2.Moscow Institute of Physics and Technology (State University)MoscowRussia

Personalised recommendations