A Novel Method for Decoding Any High-Order Hidden Markov Model

This paper proposes a novelmethod for decoding any high-order hiddenMarkovmodel. First, the high-order hiddenMarkovmodel is transformed into an equivalent first-order hidden Markov model by Hadar’s transformation. Next, the optimal state sequence of the equivalent first-order hidden Markov model is recognized by the existing Viterbi algorithm of the first-order hidden Markov model. Finally, the optimal state sequence of the high-order hiddenMarkovmodel is inferred from the optimal state sequence of the equivalent first-order hidden Markov model. This method provides a unified algorithm framework for decoding hidden Markov models including the first-order hidden Markov model and any high-order hidden Markov model.


Introduction
Hidden Markov models are powerful tools for modeling and analyzing sequential data.For several decades, hidden Markov models have been used in many fields including handwriting recognition [1][2][3], speech recognition [4,5], computational biology [6,7], and longitudinal data analysis [8,9].Past and current developments on hidden Markov models are well documented in [10,11].A hidden Markov model comprises an underlying Markov chain and an observed process, where the observed process is a probabilistic function of the underlying Markov chain [12].Given a hidden Markov model, an efficient procedure for finding the optimal state sequence is of great interest in the real-world applications.In the traditional first-order hidden Markov model, the Viterbi algorithm is utilized to recognize the optimal state sequence [13].Like the Kalman filter, the Viterbi algorithm tracks the optimal state sequence with a recursive method.
In recent years, the theory and applications of high-order hidden Markov models have been substantially advanced, and high-order hidden Markov models are known to be more powerful than the first-order hidden Markov model.There are two basic approaches to study the algorithms of highorder hidden Markov models.The first one is called the extended approach, which is to extend directly the existing algorithms of the first-order hidden Markov model to highorder hidden Markov models [14][15][16].The second one is called the model reduction method, which is to transform a high-order hidden Markov model to an equivalent first-order hidden Markov model by some means and then to establish the algorithms of the high-order hidden Markov model by using standard techniques applicable to the first-order hidden Markov model [17][18][19][20].
In this paper, we propose a novel method for decoding any high-order hidden Markov model.First, the high-order hidden Markov model is transformed into an equivalent first-order hidden Markov model by Hadar's transformation.Next, the optimal state sequence of the equivalent first-order hidden Markov model is recognized by the existing Viterbi algorithm of the first-order hidden Markov model.Finally, the optimal state sequence of the high-order hidden Markov model is inferred from the optimal state sequence of the equivalent first-order hidden Markov model.
Definition 1 (see [18,20]).A high-order hidden Markov model is a doubly stochastic process with an underlying state process that is not directly observable but can be observed only through another stochastic process that is called the observation process.The observation process is governed by the hidden state process and produces the observation sequence.The state process and observation process, respectively, satisfy the following conditions.
(a) The hidden state process {q  } is a homogeneous Markov chain of order , that is, a stochastic process that satisfies (b) The observation process {  } is governed by the hidden state process according to a set of probability distributions that satisfy To model the high-order hidden Markov model, the following parameters are needed.
Definition 2 (see [18]).Let be the mapping of any base  number to its decimal value; that is, if Definition 3 (see [17]).Any two models  1 and  2 are defined as equivalent if for any arbitrary observation sequence .In other words, two models are only considered equivalent if they yield the same likelihood, regardless of the specific observation sequence.
Proof.Without loss of generality, we may assume that   =  and  +1 = , where ,  ∈ .
where  1 ,  2 , . . .,   ∈ S.Moreover, we have Analogously, it is easy to see that Combining these with Lemma 6, we prove that the two processes {  } and {  } form the first-order hidden Markov model.[18] had also mentioned the fact that the two processes {  } and {  } form the first-order hidden Markov model, but they did not discuss and prove it in detail.

Remark 8. Hadar and Messer
To model the first-order hidden Markov model {  ,   }, the following parameters are needed.
On the other hand, it is easy to see that Hence, by Lemma 10, we have Theorem 13.Let  =  1 ⋅ ⋅ ⋅   be any given observation sequence, and assume that the state sequence that is, the state sequence Q * = q * 2− ⋅ ⋅ ⋅ q *  is some optimal state sequence of the high-order hidden Markov model {q  ,   }.Let  *  = [q *  , . . ., q * −(−1) ] (1 ≤  ≤ ); then the state sequence That is, the state sequence  * =  * 1 ⋅ ⋅ ⋅  *  is some optimal state sequence of the first hidden Markov model {  ,   }.
Proof.By Theorem 12, it is easy to see that Meanwhile, we have the equation Hence, we drive that According to Theorem 13, we can know that some optimal state sequence of the high-order hidden Markov model {q  ,   } is mapped to some optimal state sequence of the firstorder hidden Markov model {  ,   }.Similarly, we can draw the following conclusion.
That is, the state sequence Q * = q * 1 ⋅ ⋅ ⋅ q *  is some optimal state sequence of the high-order hidden Markov model {q  ,   }.
Remark 15.Combining Theorem 13 with Theorem it is known that there exists a one to one correspondence between the optimal state sequence of the high-order hidden Markov model {q  ,   } and the optimal state sequence of the first-order hidden Markov model {  ,   }.
To decode any high-order hidden Markov model {q  ,   }, we transform it into an equivalent first-order hidden Markov model {  ,   } by Hadar's transformation.Then, do as follows.