State space models originate from control theories. A state space model usually consists of two sets of equations, the system equations and the observation equations. The system (dynamic) equations model the dynamics of state variables and the observation (output) equations model the observed state variables. For a linear Gaussian state space model, the well-known Kalman filtering approach provides optimal estimates for state variables based on the information from the two sources, the dynamic equations and the observations. However, for nonlinear and non-Gaussian state space models, it is quite challenging to estimate the state variables (including filtering, smoothing and forecasting) and model parameters. Thanks to the rapid development of computational power and Monte Carlo techniques in Bayesian computation, many new computation-based methods such as sequential Monte Carlo approaches, particle filtering and Gibbs sampler techniques have been developed for nonlinear and non-Gaussian state space models in the past decade. Various applications of state space models in econometrics, engineering, biomedical studies and other areas have attracted the attention of more statisticians. Recent developments of biotechnologies allow us to understand biomedical systems more fundamentally or even at a cellular level. There is a great need to study the biomedical systems using an elegant quantitative method. Our group is focusing on the development of state space models for HIV dynamics at both subject and population levels. Thus, the mixed-effects modeling idea and other techniques for longitudinal data analysis are introduced into state space models. In particular, we are developing statistical inference methods for parameter estimation (model identification) and state variable estimation. The research in this area reflects an important interplay among mathematical modeling, statistical inference, engineering techniques and biomedical sciences. Some other potential applications of state space modeling techniques include tumor cell kinetics in cancer research, hepatitis virus dynamics, genetic regulatory network modeling, and other biomedical process modeling.

State Space Models

.

Kalman filter and related filtering/smoothing techniques

.

Maximum likelihood estimation of parameters and inference

.

Monte Carlo approaches for state space models

.

Related books on Monte Carlo methods

Particle Filter