Optimal Reduction of Multivariate Dirac Mixture Densities

TL;DR

This paper introduces a method for optimally approximating multivariate Dirac mixture densities by reducing components, using a generalized distance measure based on a novel localized cumulative distribution, enabling efficient nonlinear estimation.

Contribution

It presents a new approach to approximate multivariate Dirac mixtures by minimizing a generalized distance using the Localized Cumulative Distribution, extending univariate methods to multivariate cases.

Findings

Provides a novel approximation method for multivariate Dirac mixtures.

Enables efficient nonlinear state and parameter estimation.

Generalizes the Cramér-von Mises Distance to multivariate distributions.

Abstract

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 Cram\'{e}r-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.