The Monte Carlo Kalman Filter
In the sigma point Kalman filters, after an affine transformation of the state vector, the nonlinear functions were expanded in a polynomial reducing the Gaussian-weighted integrals to a set of moment equations that were solved for a set of weights and sigma points. Although these sigma point methods have been referred to in the literature as an approximation to the density function, no approximations for the Gaussian density have been applied in our derivations.
However, a method is presented here where the Gaussian density is approximated by a set of Monte Carlo samples from that density. Remember from (5.51) that
Suppose that we can generate a set of Monte Carlos sample from . Now we express as the discrete approximation to the density given by
where represents the delta-Dirac mass located at .
Substituting (12.2) into the state ...