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`Journal of Probability and StatisticsVolume 2018, Article ID 8068196, 15 pageshttps://doi.org/10.1155/2018/8068196`
Research Article

## A Novel Entropy-Based Decoding Algorithm for a Generalized High-Order Discrete Hidden Markov Model

1Ted Rogers School of Management, Ryerson University, 350 Victoria St., Toronto, ON, Canada M5B 2K3
2School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Gelugor, Penang, Malaysia

Correspondence should be addressed to Jason Chin-Tiong Chan; ac.nosreyr@nahc.nosajgnoitnihc

Received 15 December 2017; Revised 12 February 2018; Accepted 27 February 2018; Published 2 May 2018

Academic Editor: Steve Su

Copyright © 2018 Jason Chin-Tiong Chan and Hong Choon Ong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The optimal state sequence of a generalized High-Order Hidden Markov Model (HHMM) is tracked from a given observational sequence using the classical Viterbi algorithm. This classical algorithm is based on maximum likelihood criterion. We introduce an entropy-based Viterbi algorithm for tracking the optimal state sequence of a HHMM. The entropy of a state sequence is a useful quantity, providing a measure of the uncertainty of a HHMM. There will be no uncertainty if there is only one possible optimal state sequence for HHMM. This entropy-based decoding algorithm can be formulated in an extended or a reduction approach. We extend the entropy-based algorithm for computing the optimal state sequence that was developed from a first-order to a generalized HHMM with a single observational sequence. This extended algorithm performs the computation exponentially with respect to the order of HMM. The computational complexity of this extended algorithm is due to the growth of the model parameters. We introduce an efficient entropy-based decoding algorithm that used reduction approach, namely, entropy-based order-transformation forward algorithm (EOTFA) to compute the optimal state sequence of any generalized HHMM. This EOTFA algorithm involves a transformation of a generalized high-order HMM into an equivalent first-order HMM and an entropy-based decoding algorithm is developed based on the equivalent first-order HMM. This algorithm performs the computation based on the observational sequence and it requires calculations, where is the number of states in an equivalent first-order model and is the length of observational sequence.