Hybrid Gauss-Hermite filter

We present an algorithm for sequential Bayesian estimation consisting of a hybrid method that combines a particle-based representation of the prior state uncertainty with an efficient grid-based method to estimate the posterior probability density. The proposed filter uses a Monte-Carlo empirical measure of the prior probability density to induce a probability mass density that approximates the posterior probability density. Such an approximation enables accurate numerical integration, by means of the Gauss-Hermite quadrature, to compute the state estimates and error covariance. It is evident that the filter is prone to estimation errors dominated by the same approximation errors as those found in conventional particle filters, but it is well suited to generally solve nonlinear non-Gaussian filtering problems without the well-known particle degeneracy problem. Simulation results demonstrate the versatility of the filter for practical problems, showing performance similar to particle filters with optimal proposal density, for nonlinear non-Gaussian dynamic state-space models, with the advantage that the degeneracy problem is absent.