Kernel based filter for nonlinear estimation

For highly nonlinear problems, the linear minimum mean-square error (LMMSE) estimation using a nonlinearly converted measurement can outperform the one using the original measurement. For a function space of measurement conversions, every function in the space can be represented as a linear combination of a basis of the space. Then the LMMSE estimator using a vector with its entries forming a basis of the space is optimal among all the estimators that are linear in the measurement conversions in the space. However, a basis of a space may consist of infinitely many functions and may be hard to determine. To solve this problem, this paper proposes a kernel based filter (KBF), which is equivalent to the LMMSE estimator using the conversion vector but is simpler, more convenient and feasible. Using deterministic sampling and the kernel trick, the KBF can be represented in kernel functions of the sampled and the real measurements provided the space is a reproducing kernel Hilbert space associated with the kernel functions. Because calculating the basis functions directly is avoided by using kernel functions, the computational complexity can be greatly reduced and it is unnecessary to get the basis explicitly. Simulation results show the effectiveness of the proposed filter.