^{1}

^{2}

^{1}

^{2}

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

State sequence for the Hidden Markov Model (HMM) is invisible but we can track the most likelihood state sequence based on the model parameter and a given observational sequence. The restored state has many applications especially when the hidden state sequence has meaningful interpretations for making predictions. For example, Ciriza et al. [

This paper is organized as follows. In Section

The uncertainty appearing in a HHMM can be quantified by entropy. This concept is applied for quantifying the uncertainty of the state sequence tracked from a single observational sequence and model parameters. The entropy of the state sequence equals 0 if there is only one possible state sequence that could have generated the observation sequence as there is no uncertainty in the solution. The higher this entropy the higher the uncertainty involved in tracking the hidden state sequence. We extend the entropy-based Viterbi algorithm developed by Hernando et al. [

Before introducing the extended entropy-based Viterbi algorithm, we define a generalized high-order HMM, that is,

HHMM involves two stochastic processes, namely, hidden state process and observation process. The hidden state process cannot be directly observed. However, it can be observed through the observation process. The observational sequence is generated by the observation process incorporated with the hidden state process. For a discrete HHMM, it must satisfy the following conditions.

The hidden state process

The observation process

The elements for the

Number of distinct hidden states,

Number of distinct observed symbols,

Length of observational sequence,

Observational sequence,

Hidden state sequence,

Possible values for each state,

Possible symbols per observation,

Initial hidden state probability vector,

where

State transition probability matrix,

where

Emission probability matrix,

where

where

For the

Note that throughout this paper, we will use the following notations.

The entropy-based algorithm proposed by Hernando et al. [

The forward variable

The backward probability variable

We obtain the initial backward probability variable as

The probability of the observational sequence given the model parameter for the first-order HMM can be represented by using the classical forward probability and backward probability variables [

Let

The normalized forward probability variable

The normalized backward probability variable

From (

Hernando et al. [

This algorithm performs the computation linearly with respect to the length of the observation sequence with computational complexity

The extended classical Viterbi algorithm is commonly used for computing the optimal state sequence for HHMM. This algorithm provides the solution along with its likelihood. This likelihood probability can be determined as follows.

We define entropy of a discrete random variable as follows [

The entropy

From (

For this extended algorithm, we require an intermediate state entropy variable,

We define the state entropy variable for the

The state entropy variable,

We analyse the state entropy for the

From (

(i) According to the generalized high-order HMM, state

(ii) According to the basic property of entropy [

We now introduce the following lemma for the

For the

Our extended entropy-based algorithm for computing the optimal state sequence is based on normalized forward recursion variable, state entropy recursion variable, and auxiliary probability. From (

This extended algorithm performs the computation of the optimal state sequence linearly with respect to the length of observational sequence which requires

We consider a second-order HMM for illustrating our extended entropy-based algorithm in computing the optimal state sequence. Let us assume that this second-order HMM has the state space

The graphical representation of the first-order HMM that is used for the numerical example in this section is given in Figure

The graphical diagram shows a first-order HMM with 2 states and 3 observational symbols.

The graphical diagram shows a second-order HMM with 2 states and 3 observational symbols.

The initial state probability vectors for the first-order and second-order HMM are shown as follows:

The state transition probability matrices for the first-order and second-order HMMs are shown as follows:

The emission probability matrices for the first-order and second-order HMMs are shown as follows:

The following is the observational sequence that we used for illustrating our extended algorithm:

The evolution of the trellis structure of the second-order HMM with the observation sequence

The total entropy after each time step is displayed at the bottom of Figure

The extended entropy-based Viterbi algorithm in Section

This EOTFA algorithm involves a transformation of a generalized high-order HMM into an equivalent first-order HMM and an algorithm is developed based on the equivalent first-order HMM. This algorithm performs the computation based on the observational sequence and it requires

The transformation of a generalized high-order HMM into an equivalent first-order HMM is based on Hadar and Messer’s method [

Suppose

The observation process

(i)

(ii)

Note that we assume

The elements for the transformation of a high-order into an equivalent first-order discrete HMM are as follows:

Number of distinct hidden states,

Number of distinct observed symbols,

Length of observational sequence,

Observational sequence,

Hidden state sequence,

Possible values for each state,

Possible symbols per observation,

Initial hidden state probability vector,

State transition probability matrix,

where the first

Emission probability matrix,

In this subsection, we omit the derivations for forward and backward probability variables since the derivations are similar to the derivations in Section

The forward recursion variable for the transformed model at time

The normalized backward variables at time

For EOFTA algorithm, we require state entropy variable,

We define the state entropy variable as follows.

The state entropy variable,

The following proof is due to Hernando et al. [

For the transformation of a high-order into an equivalent first-order HMM, the entropy of the state sequence up to time

Our EOTFA algorithm for computing the optimal state sequence is based on the normalized forward recursion variable, state entropy recursion variable, and auxiliary probability. From (

The direct evaluation algorithm, Hernando et al.’s algorithm, and our algorithm perform the computation of state entropy exponentially with respect to the order of HMM. Our algorithm proposes the transformation of a generalized high-order into an equivalent first-order HMM and then compute the state entropy based on the equivalent first-order model; hence our algorithm is the most efficient in which it requires

We consider the second-order HMM in Section

Note that the above state transition probability and the emission probability matrices whose components are indicated as

The graphical diagram shows an equivalent first-order HMM.

The state space for the equivalent first-order HMM is

The equivalent first-order HMM was developed based on Hadar and Messer’s method [

Secondly, the optimal state sequence is computed based on the equivalent first-order HMM by using our proposed algorithm. Finally, the optimal state sequence of the second-order HMM is inferred from the optimal state sequence from the equivalent first-order HMM.

The following is the observational sequence used for illustrating our algorithm:

We applied our EOFTA algorithm for computing the optimal state sequence based on the state entropy. The computed value of state entropy is shown in Figure

The evolution of the trellis structure for a transformation of a second-order into an equivalent first-order HMM with the observation sequence

The total entropy after each time step for the transformed model, that is, the second-order transformed into the equivalent first-order HMM is displayed at the bottom of Figure

We have introduced a novel algorithm for computing the optimal state sequence for HHMM that requires

The authors declare that there are no conflicts of interest regarding the publication of this paper.