Abstract

This dissertation studies the modeling of time series driven by unobservable pro-cesses using state space model. New models and methodologies are proposed and ap-plied on a variety of real life examples arising from finance and biology. More specifically, we mainly consider two types of time series: partially observed dynamic systems driven by differential equations and functional time series driven by its feature process. The first type of time series data is generated by a hidden dynamic process controlled by some underlying differential equation with a set of unknown parameters. We propose a state space approach to fit these models with observation data, which is only available at sparsely separated time points as well as with measurement error, and estimate the corresponding parameters. More specifically, we approximate the target nonlinear deterministic/stochastic differential equations by difference equations and convert the dynamic into a state space model(SSM), which is further calibrated by the likelihood calculated from the filtering scheme. The first application converts the HIV dynamic into a linear SSM and estimates all HIV viral dynamic parameters successfully without many constraints. The second application focus on the well-studied ecological SIR