In this research thesis, we implement Markov Chain Monte Carlo techniques and polynomial-chaos expansion based techniques for states and parameters estimation in hidden Markov models (HMM). Our goal is to estimate the probability density function (PDF) of the states and parameters given noisy observations of the output of the hidden Markov model. We consider three problems, namely, (i) determining the PDF of the states in a non-linear HMM using sequential MCMC techniques, (ii) determining the parameters of discretized linear, ordinary differential equations (ODE) given noisy observations of the solutions and
(iii) Determining the PDF of the solution of a linear ordinary differential equation when the parameters of the ODE are random variables. While these problems naturally arise in several areas in engineering, this thesis is motivated by potential applications in bio-mechanics. One of the interesting research questions that is being considered by some researchers is whether the formation of clots can be predicted by observing the mechanical properties of arteries, such as their stiff ness. In order for this approach to be successful, it is critical to estimate the stiffness of arteries based on noisy measurements of their mechanical response. The parameters of these models can then be used to differentiate diseased arteries from healthy ones or, the parameters can be used to predict the probability of formation of plaques. From experimental data, we would like to infer the posterior density of the states and parameters (such as stiffness), and classify it as being healthy or diseased. If it is accomplished, this will improve the state-of-the art in modeling mechanical properties of arteries, which could lead to better prediction, and diagnosis of coronary artery disease
