An application of a semihidden Markov model in wireless communication systems
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Abstract
Stochastic processes are approved presentation of real systems which its development in space or time can be supposed as random. A semihidden Markov model as a type of stochastic processes is a modification of hidden Markov models with states that are no longer totally unobservable and are less hidden. This mathematical model is employed for modeling data sequences with long runs, memory and statistical inertia. In this article, we investigate the theory of the semihidden Markov model along with its parameter estimation and order estimation methods. Moreover, the proposed model is applied to model the error traces generated by the wireless channels. A new Markovbased trace analysis algorithm is suggested to divide a nonstationary network error trace into stationary parts. By means of the best semihidden Markov model and fitting probability distribution, we would be able to model these parts accurately. Calculating the information measure criteria and the autocorrelation function by running the modified Baum–Welch algorithm several times help us to find the optimal order of the semihidden Markov model.
Keywords
Stochastic processes Semihidden Markov model Order estimation Wireless communicationMathematics Subject Classification
60J20 60K15 90B18Introduction
A stochastic process is a probability model employed for describing the evolution in time of a random event. The hidden Markov models (HMMs) make a profitable and flexible class of stochastic processes which have been used satisfactorily in a broad range of applied problems. The hidden Markov model (HMM) is an extension of a Markov chain whose states are hidden. It is a doubly stochastic model \(\lbrace (H_k,O_k) \rbrace \), where \(\lbrace H_k \rbrace \) denotes the hidden state sequence which is a finitestate Markov chain. Given \(\lbrace H_k \rbrace \), the \(O_k\) is conditionally independent and the conditional distribution of it relies on \(\lbrace H_k \rbrace \) only through \(H_n\). The (HMMs) have many applications in different fields such as speech recognition [1], hand gesture recognition [2], source coding [3], seismic hazard assessment [4], traffic prediction [5],wireless network [6, 7, 8], protein structure prediction [9] and finance [10]. The semihidden Markov models (SHMMs) are stochastic models related to HMMs. They are discussed in [11, 12], recently. A principal characteristic of these models is the involvement of statistical inertia which admits the generation, and analysis of observation symbol contains frequent runs. The SHMMs cause a substantial reduction in the model parameter set. Therefore in most cases, these models are computationally more efficient models compared to HMMs in most cases. As long stretches of errorfree transmission exist in wireless channels, the corresponding runlength vector is greatly shorter in length than the original binary data. Hence, the simulation runtime decreases considerably.
In this paper, the definition and the modified Baum–Welch algorithm for the parameter estimation of the SHMM are given first and next, the order estimation benchmarks of these models are discussed. Next, we present an application of the SHMM in wireless communication for modeling the error sequence generated in the CDMA system. Finally, conclusions are given.
The semihidden Markov model
The SHMMs are the advanced types of stochastic models connected to HMMs. They can model the behavior of symbolic sequences with inertia, memory and long runs. If we know the state of changing points, given these sequences it is probable to realize the sequence of states. Hence, the semihidden Markov model (SHMM) is not absolutely hidden. The SHMMs work by altering among distinct states and generating symbols of the alphabet in the same way as the HMMs. The input sequence is in terms of a runlength vector, and therefore, the length of the input sequence is extremely reduced, especially when the sequences involve long stretches of identical symbols.
Algorithm:
 1.The Akaike information criterion (AIC) was proposed by Akaike (1973) [13]. It is defined as$$\begin{aligned} \mathrm{AIC}=\,2\mathrm{log}(L)+2k \end{aligned}$$(18)
 2.The Bayesian information criterion (BIC) was introduced by Schwarz (1978) [14]. It is computed as$$\begin{aligned} \mathrm{BIC}=\,2\mathrm{log}(L)+k\mathrm{log}n \end{aligned}$$(19)
 3.The Hannan–Quinn information criterion (HQC) is calculated as$$\begin{aligned} \mathrm{HQC}=\,2\mathrm{log}(L)+2k \mathrm{log}(\mathrm{log}(n)) \end{aligned}$$(20)
An application of the SHMM in wireless communication
Modeling wireless communication errors is substantial for simulationbased performance assessment of network protocols or for utilizing information about these error characteristics within a protocol. Discrete channel models (DCMs) were employed in wireless systems such as codedivision multiple access (CDMA) [15], orthogonal frequency division modulation (OFDM) [16] and global system for mobile communication (GSM) [17]. HMMs are dominant tools with high accuracy which employed as the discrete channel model (DCM) for modeling stochastic processes. These models are applied for precise simulation of errors in wireless systems [18, 19, 20, 21, 22]. In this section, the SHMM is used for modeling the errors of the CDMA as the DCM. The order estimation is profitable for interpreting the model. Moreover, it is vital to ensure stability. The estimation of the order of the HMM was investigated in [23]. They discussed the optimal order estimation of the SHMM for the error sequences generated by the CDMA.
The CDMA specification
Analysis of the CDMA system with the SHMM

\(\{L_n n\ge {0}\}\): The lossy state length process, where \(L_n\) shows the length of the nth state.

\(\{G_n n\ge {0}\}\): The errorfree state length process, where \(G_n\) shows the number of elements in the nth errorfree state.
MSE ACFs of different SHMMs for the lossy trace
Model  MSE ACF 

2state  0.000989 
3state  0.001202 
4state  0.001132 
5state  0.001283 
6state  0.001269 
AIC, BIC and HQC values of different SHMMs for the lossy trace
Model  k  Loglikelihood  AIC  BIC  HQC 

2state SHMM  4  305.3651  618.7302  624.0041229  614.8977144 
3state SHMM  9  305.3651  628.7302  640.5965263  620.1071073 
4state SHMM  16  305.1949  642.3898  663.4854912  627.0598575 
5state SHMM  25  305.1932  660.3864  693.3484175  636.4333648 
6state SHMM  36  305.1948  682.3896  729.8549052  647.8972293 
 1.
Choose the number of lossy and errorfree frames (N) to generate in the artificial trace.
 2.
Identify the length of an errorfree state \((g_{len})\) from the \(G_n\) using the inverse CDF method.
 3.
Generate a sequence of zeros \((g_{len})\) to make an errorfree burst.
 4.
Identify the length of lossy state \((l_{len})\) from the \(L_n\) using the inverse CDF method.
 5.
Generate a sequence of \((l_{len})\) burst which is either lossy or errorfree frames based on 2state SHMM.
 6.
Compound the two sequences to the artificial trace.
 7.
Stop if all N frames have been generated, else return to step 2. The original and artificial error traces are compared according to the ACF in Fig. 6. It is evident that the two plots are matched. Moreover, from Fig. 7, it can be concluded that the distributions of errorfree intervals for these two sequences are in the same manner which validates the accuracy of our mathematical model.
Conclusion
Analyzing the network protocol and performance depends on the methods of modeling and simulating channel conditions. Wireless channels usually face bursty errors. In this paper, we demonstrated that the SHMM as a discrete channel model can accurately model the errors of a CDMA system. The simulation contained the effects of multipath, additive white Gaussian noise and multiple access interference to generate the error sequence. The original error trace exhibited a nonstationary performance. Therefore, we divided the data into two lossy and errorfree traces to obtain stationary behavior. The SHMM was used to model the lossy trace. The AIC, BIC, HQC and sample autocorrelation criteria were employed to find the best model as a 2state SHMM. The best fitting of a runlength of the lossy and errorfree trace was the twoparameter Gamma distribution and the generalized Pareto distribution, respectively. An artificial binary error trace was generated by integrating the 2state (SHMM) and generalized Pareto distribution according to the algorithm we explained. The original error trace matched closely with the artificial one according to the sample autocorrelation function. All in all, the semihidden Markov model is a reliable stochastic model for modeling symbolic sequences with long runs and statistical inertia. It has become a precise mathematical feature to model the error traces generated by wireless channels.
Notes
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