In this paper, a novel distance-based density
estimation method is proposed, which considers the overall
density function in the goodness-of-fit. In detail, the parameters
of Gaussian mixture densities are estimated from samples,
based on the distance of the cumulative distributions over
the entire state space. Due to the ambiguous definition of the
standard multivariate cumulative distribution, the Localized
Cumulative Distribution and a modified Cramér-von Mises
distance measure are employed. A further contribution is the
derivation of a simple-to-implement optimization procedure
for the optimization problem. The proposed approach’s good
performance in estimating arbitrary Gaussian mixture densities
is shown in an experimental comparison to the Expectation
Maximization algorithm for Gaussian mixture densities.

In this paper, a new class of nonlinear Bayesian
estimators based on a special space partitioning structure, generalized
Octrees, is presented. This structure minimizes memory and calculation
overhead. It is used as a container framework for a set of node functions
that approximate a density piecewise. All necessary operations are derived
in a very general way in order to allow for a great variety of Bayesian
estimators. The presented estimators are especially well suited for
multi-modal nonlinear estimation problems. The running time performance
of the resulting estimators is first analyzed theoretically and then backed
by means of simulations. All operations have a linear running time in
the number of tree nodes.

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 Cramér-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.