Density Approximation Based on Dirac Mixtures with Regard to Nonlinear Estimation and Filtering

Abstract

Abstract-A deterministic procedure for optimal approximation of arbitrary probability density functions by means of Dirac mixtures with equal weights is proposed. The optimality of this approximation is guaranteed by minimizing the distance of the approximation from the true density. For this purpose a distance measure is required, which is in general not well defined for Dirac mixtures. Hence, a key contribution is to compare the corresponding cumulative distribution functions.This paper concentrates on the simple and intuitive integral quadratic distance measure. For the special case of a Dirac mixture with equally weighted components, closedform solutions for special types of densities like uniform and Gaussian densities are obtained. Closed-form solution of the given optimization problem is not possible in general. Hence, another key contribution is an efficient solution procedure for arbitrary true densities based on a homotopy continuation approach.In contrast to standard Monte Carlo techniques like particle filters that are based on random sampling, the proposed approach is deterministic and ensures an optimal approximation with respect to a given distance measure. In addition, the number of required components (particles) can easily be deduced by application of the proposed distance measure. The resulting approximations can be used as basis for recursive nonlinear filtering mechanism alternative to Monte Carlo methods.

Optimal parametric density estimation by minimizing an analytic distance measure

“…The other type consists of so called optimal samples. They are not drawn randomly but calculated as an optimal approximation of the given probability density function using the algorithm given in [7].…”

“…Since we want to find a minimum with regard to the inverse problem described in [7], we have to consider the partial derivative of (1) with respect toη. By setting the derivative to zero we obtain the following system of (non-linear) equations…”

“…Hence, the estimation problem is considered as an optimization problem. Furthermore, we require that this minimum also minimizes the dual problem given in [7], namely finding Dirac mixture parameters for approximating a continuous density function.…”

“…We have to choose a pair of parameter vectors κ Start and η Start that minimizes the distance between F (x, κ Start ) andF (x, η Start ). For that purpose, we apply the algorithm presented in [7] in order to calculate η Start for a given κ Start . The best results for the subsequent progression are obtained by choosing f (x, κ Start ) as a density function that covers the range of the samplesx i uniformly.…”

“…The method that will be presented here is inspired by previous work we have done on the dual problem. In [7], [8] we have derived a method for approximating continuous probability density functions by means of so called Dirac mixture functions. These Dirac mixtures can be interpreted as an analytical parametric representation of discrete samples.…”