UC Riverside

In reality many time series are non-linear and non-gaussian. They show the characters like flat stretches, bursts of activity and outliers. (Bivariate) Mixture Transition Distribution model are introduced to study these time series data. EM algorithm is used for point estimations of parameters. However as is known, for many mixture models, the likelihoods couldn't be maximized since they will go to infinity. Number of mixtures should be prefixed in this way but in many realities it is unknown.

In our research, Bayesian methods are used to solve these problems. When the posterior is got, EM algorithm is used to maximize the posterior. Under some conditions these estimates are proved to be consistent. The second method is using MCMC to sample from the posterior and now number of mixtures itself can be treated as a random variable. Two methods for MCMC sampling are used. The first is Birth-Death process: if a birth happens, a new mixture component is added; if a death happens, an existed mixture will disappear. The second is Dirichlet process mixtures where we choose Dirichlet process priors for the parameters. When using MCMC, not only point estimations but also interval estimations can be constructed. For all these methods we do simulations to compare Bayesian methods with non-Bayesian methods and to show the excellence of Bayesian methods.