PARAMETER ESTIMATION OF EXPONENTIAL HIDDEN
MARKOV MODEL AND CONVERGENCE OF ITS
PARAMETER ESTIMATOR SEQUENCE
Muhammad Firmasyah1, Berlian Setiawaty2,
I. Gusti Putu Purnaba3
1,2,3 First Department of Mathematics
Bogor Agricultural University
Meranti Street, Bogor - 16680, INDONESIA

Abstract

An exponential hidden Markov model (EHMM) is a hidden Markov model which consists of a pair of stochastic processes $\{X_t,Y_t\}_{t\in N}. \{Y_t\}_{t_\in N}$ is influenced by $\{X_t\}_{t\in N}$, which is assumed to form a Markov chain. $\{X_t\}_{t\in N}$ is not observed. $\{Y_t\}_{t\in N}$ is an observation process and $Y_t$ given $X_t$ has exponential distribution. In this paper, we estimate the parameter of EHMM and study the convergence of the parameter estimator sequence. EHMM is characterized by a parameter $\phi=(A,\lambda)$ where $A$ is a transition matrix of $X_t$ and $\lambda$ is a vector of parameters of probability density function of $Y_t$ given $X_t$. To determine the parameter estimator, a maximum likelihood method is used. Numerical approximation is used through an Expectation Maximization (EM) algorithm. Under the continuous assumption, the sequence $\{ \phi^{(k)}\}$ obtained by the EM algorithm, converges to $\phi^*$ which is the stationary point of ln $L_t(\phi)$ and the sequence $\{\ln L_t(\phi^{(k)})\}$ increasingly converges to ln $L_t(\phi^*)$.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 31
Issue: 1
Year: 2018

DOI: 10.12732/ijam.v31i1.5

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