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Some Generalized Entropy Ergodic Theorems for Nonhomogeneous Hidden Markov Models

Author

Listed:
  • Qifeng Yao

    (School of Economics and Management, Nanjing University of Science and Technology, Nanjing 210094, China
    These authors contributed equally to this work.)

  • Longsheng Cheng

    (School of Economics and Management, Nanjing University of Science and Technology, Nanjing 210094, China
    These authors contributed equally to this work.)

  • Wenhe Chen

    (School of Economics and Management, Nanjing University of Science and Technology, Nanjing 210094, China)

  • Ting Mao

    (School of Economics and Management, Nanjing University of Science and Technology, Nanjing 210094, China)

Abstract

Entropy measures the randomness or uncertainty of a stochastic process, and the entropy rate refers to the limit of the time average of entropy. The generalized entropy rate in the form of delayed averages can overcome the redundancy of initial information while ensuring stationarity. Therefore, it has better practical value. A Hidden Markov Model (HMM) contains two stochastic processes, a stochastic process in which all states can be observed and a Markov chain in which all states cannot be observed. The entropy rate is an important characteristic of HMMs. The transition matrix of a homogeneous HMM is unique, while a Nonhomogeneous Hidden Markov Model (NHMM) requires the transition matrices to be dependent on time variables. From the perspective of model structure, NHMMs are novel extensions of homogeneous HMMs. In this paper, the concepts of the generalized entropy rate and NHMMs are defined and fully explained, a strong limit theorem and limit properties of a norm are presented, and then generalized entropy ergodic theorems with an almost surely convergence for NHMMs are obtained. These results provide concise formulas for the computation and estimation of the generalized entropy rate for NHMMs.

Suggested Citation

  • Qifeng Yao & Longsheng Cheng & Wenhe Chen & Ting Mao, 2024. "Some Generalized Entropy Ergodic Theorems for Nonhomogeneous Hidden Markov Models," Mathematics, MDPI, vol. 12(4), pages 1-15, February.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:4:p:605-:d:1340926
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    References listed on IDEAS

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    1. Guoqing Yang & Weiguo Yang & Xiaotai Wu, 2017. "The strong laws of large numbers for countable non homogeneous hidden Markov models," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 46(17), pages 8808-8819, September.
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