Optimal Reduction of Multivariate Dirac Mixture Densities
For researchers in nonlinear estimation, this provides a new approximation technique for Dirac mixtures, though it is an incremental extension of existing distance-based methods to the multivariate case.
This paper introduces a method for optimally approximating a multivariate Dirac mixture with a reduced number of equally weighted components by minimizing a generalized Cramér-von Mises distance via a novel Localized Cumulative Distribution, enabling efficient nonlinear state and parameter estimation.
This paper is concerned with the optimal approximation of a given multivariate Dirac mixture, i.e., a density comprising weighted Dirac distributions on a continuous domain, by an equally weighted Dirac mixture with a reduced number of components. The parameters of the approximating density are calculated by minimizing a smooth global distance measure, a generalization of the well-known Cramér-von Mises Distance to the multivariate case. This generalization is achieved by defining an alternative to the classical cumulative distribution, the Localized Cumulative Distribution (LCD), as a characterization of discrete random quantities (on continuous domains), which is unique and symmetric also in the multivariate case. The resulting approximation method provides the basis for various efficient nonlinear state and parameter estimation methods.