Efficient estimation for non-linear and non-Gaussian state space models

A probabilistic approach for state space models with underlying Markov chains x_t and observation sequences y_t is explored. Firstly, we establish a recursive formula for calculating the Cramer-Rao (CR) lower bound for a general state space model with a transit pdf p (x_t|x_t-1) and conditional p (y_t |x_t). Secondly, we apply the CR bound to several models, including FM demodulation models and outlier noise models. Sometimes the simple Kalman filter (KF) can achieve the efficiency even for non-Gaussian cases. Sometimes extended Kalman filter (EKF) based methods, like phase locked loops (PLLs), can achieve the efficiency for non-linear models. To study the performance of non-linear filters like PLLs, an algebraic approach is given for calculating the stationary distribution of these filters when it exists. However, there are also cases that both KF and EKF are far away from efficiency. Thirdly, two techniques are suggested when conventional filtering methods are inefficient. One is based on Gaussian approximations for p(x_t|x_t-1) and p(y_t|x_t) by Taylor expansion or maximum entropy method. Then an identity for Gaussian density products can be used to derive non-linear filters. The other is based on so-called partial conditional expectations (PCE) yˆ_t,n=E(x_t|y_t, y_t-1, …, y_t-n+1) which can be viewed as a nonlinear transform of observations. Then optimal linear filters can be derived for tracking x_t based on yˆ_t,n. Simulation results show that under some circumstances these two approaches really can achieve the efficiency