A state space model approach to functional time series and time series driven by differential equations
Creators: name: Wang, Jiabin
Publisher: name: No Publisher Supplied
Year: 2012

description: This dissertation studies the modeling of time series driven by unobservable processes using state space model. New models and methodologies are proposed and applied 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 model. An efficient filtering scheme is proposed to overcome the difficulty caused by the sparsity of the observed data. The methodology is illustrated and evaluated in the simulation studies and the analysis of  bartonella infection data set.  %When the coefficients in the converted difference equation only involve the observed time series, the system could be formed to a linear state space model and the optimal filtering scheme, Kalman Filter, could be easily applied to get the likelihood for inferring unknown parameters. This is demonstrated in the first application of modeling the HIV dynamic from limited clinical data. Simulation studies are conducted to compare the performance of the proposed model with some previous approaches and show superior performance. On the clinical data of two individual HIV infected patients treated with antiretroviral therapies, the proposed model is successful in estimating all HIV viral dynamic parameters without many constraints on the parameters. When the converted state space model takes a nonlinear form and the stochastic perturbation in the state process is large, an efficient filtering scheme, taken from the idea in cite{doucetPFSDE}, are proposed. It alleviated the large perturbation accumulated over the large time interval between two observations, by using a more informed propagation distribution. It utilizes the information from next observable data. Due to the effectiveness of the filtering scheme, a small filter size yields reasonable approximation of the loglikelihood and a multi-level grid search is applied to locate the MLE. The proposed methodology is applied in the calibration of the well-studied ecological SIR model. Simulation studies for both deterministic and stochastic SIR models are conducted and shown superior estimation accuracy than existing methods in the SIR model via state space model. It is also illustrated in the bartonella infection data set.    The second part of the thesis applied state space model approach on functional time series driven by its feature process , with illustration on two financial data sets. We first find the underlying feature process and build its transitional relationship, which provides the basis to build a SSM form. Then we infer the unknown parameters from likelihood calculated from the filtering scheme. The first application analyzes the U.S. treasury yield curve from January 1985 through June 2000 and proposed a two-regime AR model on its feature process: level, slope and curvature of the yield curve. The second application applies the framework on the daily return distributions of the 1000 largest capitalization stocks from 1991 to 2002. A novel skew-t distribution is used to fit the target distribution and to extract the parameters of the distribution as the feature process, which is further fitted by a vector moving average model. Compared to competing models, our model shows superior prediction performance in both applications.