Innovations of Hidden Markov Models

Hidden Markov Models (HMM) are widely used to represent time series with regime switching phenomenon. The most popular inference method of HMM is the Baum-Welch (B-W) algorithm, which is a special case of the E-M algorithm. However, it can be too slow for many real-time applications and is prone to being trapped into local optima. Spectral learning of HMM (SHMM), which is based on the method of moments estimation, has been proposed in the literature to overcome these obstacles. Despite its promises, asymptotic theory for SHMM has been elusive, and the long-run performance of SHMM can degrade due to unchecked propagation of error. In this thesis, for spectral estimation, we studied the theoretical property of the approximation error, improved the spectral learning by adding a 'project-onto-simplex' regularization, and provided the online learning of SHMM. We compare the performance of the proposed method with the state-of-the-art methods on both simulated data and real world applications, and find that the new method not only retains the computational advantages of SHMM, but also provides more robust estimation and forecasting.

SHMM is dedicated to the endogenous evolution of HMM. Some HMM's evolution is impacted by cross predictors. We propose a variant of HMM called latent control HMM, which provides a binary latent variable controlling whether each cross predictor has impact on the transition or not. This model is inferred by standard Markov Chain Monte Carlo (MCMC) sampling, which provides the predictability and interpretability. We show the latent control HMM's performance on both simulated and real-world data sets.