Optimal Dirac Approximation by Exploiting Independencies

The sample-based recursive prediction of discrete-time nonlinear stochastic dynamic systems requires a regular reapproximation of the Dirac mixture densities characterizing the state estimate with an exponentially increasing number of components. For that purpose, a systematic approximation method is proposed that is deterministic and guaranteed to minimize a new type distance measure, the so called modified Cram\'{e}r-von Mises distance. A huge increase in approximation performance is achieved by exploiting structural independencies usually occurring between the random variables used as input to the system. The corresponding prediction step achieves optimal performance when no further assumptions can be made about the system function. In addition, the proposed approach shows a much better convergence compared to the prediction step of the particle filter and by far fewer Dirac components are required for achieving a given approximation quality. As a result, the new approximation method opens the way for the development of new fully deterministic and optimal stochastic state estimators for nonlinear dynamic systems.