Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-27T19:44:34.746Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalized adaptive exponential smoothing of observations from an ergodic hidden Markov model

Part of: Markov processes Inference from stochastic processes

Published online by Cambridge University Press:  14 July 2016

Ulrich Herkenrath
Ulrich Herkenrath*
Affiliation:
Gerhard-Mercator-Universität Duisburg
*
∗Postal address: Fachbereich 11 Mathematik, Gerhard-Mercator-Universität Duisburg, Postfach 10 15 03, D-47048 Duisburg, Germany. Email address: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a sequence of observations which is generated by a so-called hidden Markov model. An exponential smoothing procedure applied to such an observation sequence generates an inhomogeneous Markov process as a sequence of smoothed values. If the state sequence of the underlying hidden Markov model is moreover ergodic, then for two classes of smoothing functions the strong ergodicity of the sequence of smoothed values is proved. As a consequence a central limit theorem and a law of large numbers hold true for the smoothed values. The proof uses general results for so-called convergent inhomogeneous Markov processes. The procedure proposed by the author can be applied to some time series discussed in the literature.

Keywords

Smoothing of non-linear time seriesgeneralized exponential smoothinghidden Markov modelsconvergent inhomogeneous Markov processes

MSC classification

Primary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Secondary: 62M10: Time series, auto-correlation, regression, etc.
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 4 , December 1999 , pp. 987 - 998
Copyright
Copyright © Applied Probability Trust 1999 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bonsdorff, H. (1989). A comparison of the ordinary and a varying parameter exponential smoothing. J. Appl. Prob. 26, 784–792.Google Scholar
Doukhan, P. (1994). Mixing: Properties and Examples (Lecture Notes in Statist. 85). Springer, New York.Google Scholar
Elliott, R. J., Aggoun, L., and Moore, J. B. (1995). Hidden Markov Models. Springer, New York.Google Scholar
Gardner, E. S. Jr (1985). Exponential smoothing: the state of the art. J. Forecasting 4, 1–28.Google Scholar
Herkenrath, U. (1994a). Generalized adaptive exponential smoothing procedures. J. Appl. Prob. 31, 673–690.Google Scholar
Herkenrath, U. (1994b). Premium adjustment by generalized adaptive exponential smoothing. Insurance: Mathematics and Economics 15, 203–217.Google Scholar
Herkenrath, U. (1998). On ergodic properties of inhomogeneous Markov processes. Rev. Roumaine Math. Pure Appl. 43, 375–392.Google Scholar
Iosifescu, M. (1980). Finite Markov Processes and Their Applications. Wiley, New York.Google Scholar
Iosifescu, M., and Grigorescu, S. (1990). Dependence with Complete Connections and its Applications. CUP, Cambridge.Google Scholar
Iosifescu, M., and Theodorescu, R. (1969). Random Processes and Learning. Springer, Berlin.Google Scholar
Meyn, S. P., and Tweedie, R. L. (1996). Markov Chains and Stochastic Stability. Springer, London.Google Scholar
Mott, J. L. (1959). The central limit theorem for a convergent nonhomogeneous finite Markov chain. Proc. Roy. Soc. Edinb. A 65, 109–120.Google Scholar
Norman, M. F. (1972). Markov Processes and Learning Models. Academic Press, New York.Google Scholar
Tjostheim, D. (1990). Non-linear time series and Markov chains. Adv. Appl. Prob. 22, 587–611.Google Scholar
Tong, H. (1990). Non-linear Time Series: A Dynamical System Approach. OUP, Oxford.Google Scholar