Non-linear Filtering for State Space Models - High-Dimensional Applications and Theoretical Results
- Author(s): Lei, Jing
- Advisor(s): Bickel, Peter J.
- et al.
State space models are powerful modeling tools for stochastic dynamical systems and have been an important research area in the statistics community in the last several decades. This thesis makes contributions to the filtering problem, a key inference problem in general state space models. Our work in this area is motivated by both high-dimensional, nonlinear applications such as numerical weather forecasting and fundamental theoretical problems such as the convergence of filters.
First we study the ensemble Kalman filters (EnKF), a popular class of filtering methods in geophysics because they are easy to implement in large systems. However, their behavior in non-Gaussian situations is only partially understood. We compare two common versions of EnKF's under non-Gaussianity from a robustness perspective. The results support previous empirical studies on the same issue and provide additional insight in choosing a free parameter in the EnKF algorithms.
Second, we consider the filtering problem in high dimensional situations such as numerical weather forecasting. We review the EnKF from a statistical perspective and analyze its sources of bias. Then we propose a new method to reduces the bias, namely the non-linear ensemble adjustment filter (NLEAF). The one-step consistency of the NLEAF is studied and the performance is examined through simulations in two common testbeds in the weather forecasting literature. Finally we look at the theoretical properties of another popular class of filtering methods, the sequential Monte Carlo (SMC) filter. The convergence of SMC filters has been a challenging problem in both probability and statistics. The previous results either depend on strong mixing conditions which only hold in compact spaces or provide no rates of convergence or are under weak notions of distance, limiting the application of their practical use. We provide checkable sufficient conditions under which explicit rates of convergence of the SMC filter can be derived. The conditions essentially requires the regularity of the tail behavior of the process and they are general enough to include a wide class of autoregressive models as well as Gaussian linear models.