Simulation-based sequential Bayesian filtering with Rao-Blackwellization applied to nonlinear dynamic state space models

Stochastic models are used to describe many real world random processes which necessitate the extraction of hidden (unobserved) states (signals) from noisy observable (measured) outputs. We consider a class of nonlinear dynamic state space models which contain conditionally linear and unknown static parameters. For tracking the a posteriori distribution of the hidden states of this type of models, one can apply particle filtering, which is an increasingly popular method in many fields of science and engineering. It is based on the Bayesian methodology and approximations of the distributions of interest with random measures composed of samples (particles) from the space of the states and weights associated to the particles. Particle filtering performs tracking of the desired distributions as new observations are made by modifying the random measure, that is, the particles and the weights. We address the application of particle filtering with the use of the Rao-Blackwellization principle. Rao-Blackwellization reduces the variance of estimators, and it is based on the Rao-Blackwell theorem. In the context of particle filtering, Rao-Blackwellization allows for integration of the conditionally linear unknowns thereby decreasing the dimension of the sampling space of the particles. One novelty in this dissertation is the implementation of Rao-Blackwellization by employing the implied integration method. Another novelty is the use of the approach on the standard stochastic volatility model and regime-switching stochastic volatility model. The latter model generalizes the former by allowing changes of the model parameters at unknown instants of time. All the models are nonlinear and contain conditionally linear parameters. Simulated datasets are used to compare the performances of our algorithms with popular ones based on the Liu and West method. Both the classical and auxiliary particle filtering algorithms are applied. We demonstrate that our particle filtering algorithms outperform the ones based on the Liu and West method.