Dirac Mixture Density Approximation Based on Minimization of the Weighted Cramér-von Mises Distance

This paper proposes a systematic procedure for approximating arbitrary probability density functions by means of Dirac mixtures. For that purpose, a distance measure is required, which is in general not well defined for Dirac mixture densities. Hence, a distance measure comparing the corresponding cumulative distribution functions is employed. Here, we focus on the weighted Cramer-von Mises distance, a weighted integral quadratic distance measure, which is simple and intuitive. Since a closed-form solution of the given optimization problem is not possible in general, an efficient solution procedure based on a homotopy continuation approach is proposed. Compared to a standard particle approximation, the proposed procedure ensures an optimal approximation with respect to a given distance measure. Although useful in their own respect, the results also provide the basis for a recursive nonlinear filtering mechanism as an alternative to the popular particle filters