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 C- R bound to several models, including FM demodulation models and outlier noise models. Sometimes simple Kalman Filter (KF) can attend the efficiency even for non-Gaussian cases. Sometimes Extended Kalman Filter (EKF) based methods, like Phase Locked Loop (PLL), can achieve the efficiency for non-linear models. To study the performance of non-linear filters like PLL, 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 navel 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) (y) over cap(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 (y) over cap(t,n). Simulation results show that under some circumstances these two approaches really can achieve the efficiency.