On parameter estimation of state space models and its applications

State space model is a class of models where the observations are driven by underlying stochastic processes. It is widely used in computer vision, economics and financial data analysis, engineering, environmental sciences and etc. My thesis mainly addresses the parameter estimation problem of state space model and the applications of it. This thesis starts with a brief introduction and the motivation for studying the problems in the first chapter. The second chapter follows the first one by covering the main tools used to study the topics in the thesis. The general framework of state space models and its related filtering methods, Kalman Filtering for linear Gaussian models and sequential Monte Carlo for other cases, are introduced. The information criteria, as a tool for model selection, are also covered in this chapter. The parameter estimation problem is mainly discussed in the third chapter. Two algorithms under the general framework of Stochastic Approximation methods are proposed. These two algorithms attain much faster convergence rate and less computational cost by variance reduction techniques which utilize the property of sequential Monte Carlo methods. Two numerical examples are examined to compare the performance. Another contribution of Chapter 3 is the application of sequantial Monte Carlo methods in modeling and predicting the bond yield curve with regime-switching Dynamic Nelson-Siegel model. The fourth chapter, which is a joint work with Hao Chang, develops a state space model with regime switching to detect periodically collapsing rational bubbles in stock price. The present-value stock-price model is expressed in a state space form and the bubble process is modeled as a conditional dynamic linear system. The asset-bubble system is estimated by a novel sequential Monte Carlo based method, Mixture Kalman Filter (MKF). The efficacy of the proposed method is examined by simulated observations and real stock index of the US market. Another application of state space model with regime switching is discussed in the fifth chapter, in which real-time Blood Glucose Monitoring problem is addressed using a conditional dynamic linear system modeling. A study with a biostatistical dataset, Star 1 dataset, has shown the advantage of the proposed novel estimation framework. In the sixth chapter, a nonparametric regression model, $ l1 $ trend filtering method is discussed. Two trend filtering models out of state space representation, both of which have similar property as $ l1 $ trend filtering, are proposed. With the implementation of sequential Monte Carlo methods as well as a greedy Viterbi algorithm, both trend filtering models can operate on-line rather than just on batch data. To better emphasize the two models' improvement in on-line trend filtering, a real world econometrics topic is introduced. The econometric example shows the competence of trend filtering as well as the efficiency of the proposed models.Ph.D.Includes bibliographical referencesby Liang Wan