Generalized linear minimum mean-square error estimation

The linear minimum mean-square error (LMMSE) estimation plays an important role in nonlinear estimation. Generalized LMMSE (GLMMSE) estimation is proposed in this work. LMMSE estimation finds the best estimator in the set of all estimators that are linear in the data. We extend this candidate set in GLMMSE estimation by employing a vector-valued function of the data and hence find the best one among all estimators that are linear in this function, rather than the data itself. The estimation performance may be enhanced since linear functions may not be adequate to provide good accuracy for a highly nonlinear problem. Theoretically speaking, GLMMSE estimation should perform at least as well as LMMSE estimation if the moments involved can be evaluated exactly. Unfortunately, similar to LMMSE estimation, those moments are difficult to evaluate analytically in general. Many numerical approximations for LMMSE estimation are also applicable to GLMMSE estimation. Computation of GLMMSE estimation based on the Gaussian-Hermite quadrature is presented, and its superior performance, compared with the unscented filter and the Gaussian filter, is demonstrated by several numerical examples.