Recursive filtering with non-Gaussian noises

The Kalman filter is an optimal recursive filter, although its optimality can only be claimed under the Gaussian noise environment. In this paper, we consider the problem of recursive filtering with non-Gaussian noises. One of the most promising schemes, which was proposed by Masreliez (1972, 1975), uses the nonlinear score function as the correction term in the state estimate. Unfortunately, the score function cannot be easily implemented except for simple cases. In this paper, a new method for efficient evaluation of the score function is developed. The method employs an adaptive normal expansion to expand the score function followed by truncation of the higher order terms. Consequently, the score function can be approximated by a few central moments. The normal expansion is made adaptive by using the concept of conjugate recentering and the saddle point method. It is shown that the approximation is satisfactory, and the method is simple and practically feasible. Experimental results are reported to demonstrate the effectiveness of the new algorithm.