Bayesian Analysis Of Hidden Markov Model And Its Application

Hidden Markov model(HMM) comprise a Markov chain and a sequence depending on the Markov chain. The Markov chain is not observable and is called the state. Now, this model has been widely used in speech recognition, image analysis, and the study of gene sequence. With the development of research, the model has been successfully applied to problems in more other fields.In this paper, we study two popular Hidden Markov models: Poisson-HMM and NormalHMM. To explore the application of Hidden Markov model, two models are established for earthquakes series and logarithmic return series of S&P500 respectively. To be concrete, the paper includes the following two parts:Firstly, we introduce Poisson-HMM and its application in analyzing the earthquakes series. In order to avoid the problem of label switching, we order the states according to the means of the state-dependent distributions and estimate the parameters via Markov Chain Monte Carlo(MCMC) method which is based on Gibbs sample. For the sake of selecting a correct model, parallel sampling of competing models is established. This approach produce posterior model probabilities and hence Bayes factor estimated. The experimental results show that this method is effective. And then Poisson-HMM is established to analyze earthquake series. The dataset includes 107 samples and the series is over dispersion and serially dependent. The maximum number of state is 4, and the posterior probability of three-state Poisson-HMM is 0.762.Secondly, we introduce Normal-HMM and its application in analyzing logarithmic return series. For the likelihood of hidden Markov model is invariant to permutations of the components labels and this property give rise the label switching phenomenon. Imposing an artificial identifiability constraint to the MCMC sample is the simplest approach to deal with label switching problem, but this approach is not always work. So, we describe label switching problem and methods of reorder the MCMC output in detail. And then we establish a Normal-HMM for the logarithmic return series of S&P 500 index, the dataset includes 1258 samples. The MCMC output for parameters occurred label switching phenomenon. And we reorder the MCMC output successfully. Empirical studies shown that Bayesian Normal-HMM based on MCMC method can fit the stock returns well.The innovative points of this paper are the application of Poisson-HMM in analyzing the earthquake series and the application of Normal-HMM in analyzing the logarithmic return series of S&P500 index. We also introduce the parallel sampling algorithm which can produce posterior model probabilities of competing models and choose a correct model. When analyze the logarithmic return series of S&P500 index, we solve the label switching problem successfully.