Nonlinear Fusion of Multi-Dimensional Densities in Joint State Space

Nonlinear fusion of multi-dimensional densities is an important application in Bayesian state estimation. In the approach proposed here, a joint density over all considered densities is build, which is then approximated by means of a Dirac mixture density by partitioning the joint state space into regions that are represented by single Dirac components. This approximation procedure depends on the nonlinear fusion model and only areas relevant to this model are considered. The processing in joint state space has advantages, especially when fusing Dirac mixture densities. Within this approach, degeneration can be avoided and even densities without mutual support can be combined. Thus, this approach gives an alternative to multiplication of Dirac mixtures with a likelihood, as used in the particle filter. Furthermore, a nonlinear Bayesian estimator with filter and prediction step can be formulated, which is able to cope with both discrete and continuous densities

Dirac Mixture Trees for Fast Suboptimal Multi-Dimensional Density Approximation

We consider the problem of approximating an arbitrary multi-dimensional probability density function by means of a Dirac mixture density. Instead of an optimal solution based on minimizing a global distance measure between the true density and its approximation, a fast suboptimal anytime procedure is proposed, which is based on sequentially partitioning the state space and component placement by local optimization. The proposed procedure adaptively covers the entire state space with a gradually increasing resolution. It can be efficiently implemented by means of a pre-allocated tree structure in a straightforward manner. The resulting computational complexity is linear in the number of components and linear in the number of dimensions. This allows a large number of components to be handled, which is especially useful in high-dimensional state spaces

Localized Cumulative Distributions and a Multivariate Generalization of the Cramér-von Mises Distance

This paper is concerned with distances for comparing multivariate random vectors with a special focus on the case that at least one of the random vectors is of discrete type, i.e., assumes values from a discrete set only. The first contribution is a new type of characterization of multivariate random quantities, the so called Localized Cumulative Distribution (LCD) that, in contrast to the conventional definition of a cumulative distribution, is unique and symmetric. Based on the LCDs of the random vectors under consideration, the second contribution is the definition of generalized distance measures that are suitable for the multivariate case. These distances are used for both analysis and synthesis purposes. Analysis is concerned with assessing whether a given sample stems from a given continuous distribution. Synthesis is concerned with both density estimation, i.e., calculating a suitable continuous approximation of a given sample, and density discretization, i.e., approximation of a given continuous random vector by a discrete one

Detection of a z=0.0515, 0.0522 absorption system in the QSO S4 0248+430 due to an intervening galaxy

In some of the few cases where the line of sight to a Quasi-Stellar Object (QSO) passes near a galaxy, the galaxy redshift is almost identical to an absorption redshift in the spectrum of the QSO. Although these relatively low redshift QSO-galaxy pairs may not be typical of the majority of the narrow heavy-element QSO absorption systems, they provide a direct measure of column densities in the outer parts of galaxies and some limits on the relative abundances of the gas. Observations are presented here of the QSO S4 0248+430 and a nearby anonymous galaxy (Kuhr 1977). The 14 second separation of the line of sight to the QSO (z sub e = 1.316) and the z=0.052 spiral galaxy, (a projected separation of 20 kpc ((h sub o = 50, q sub o = 0)), makes this a particularly suitable pair for probing the extent and content of gas in the galaxy. Low resolution (6A full width half maximum), long slit charge coupled device (CCD) spectra show strong CA II H and K lines in absorption at the redshift of the galaxy (Junkkarinen 1987). Higher resolution spectra showing both Ca II H and K and Na I D1 and D2 in absorption and direct images are reported here

Approximate Nonlinear Bayesian Estimation Based on Lower and Upper Densities

Recursive calculation of the probability density function characterizing the state estimate of a nonlinear stochastic dynamic system in general cannot be performed exactly, since the type of the density changes with every processing step and the complexity increases. Hence, an approximation of the true density is required. Instead of using a single complicated approximating density, this paper is concerned with bounding the true density from below and from above by means of two simple densities. This provides a kind of guaranteed estimator with respect to the underlying true density, which requires a mechanism for ordering densities. Here, a partial ordering with respect to the cumulative distributions is employed. Based on this partial ordering, a modified Bayesian filter step is proposed, which recursively propagates lower and upper density bounds. A specific implementation for piecewise linear densities with finite support is used for demonstrating the performance of the new approach in simulations

Ab Initio Molecular Dynamics Simulations of the Influence of Lithium Bromide on the Structure of the Aqueous Solution-Air Interface

We present the results of ab initio molecular dynamics simulations of the solution-air interface of aqueous lithium bromide (LiBr). We find that, in agreement with the experimental data and previous simulation results with empirical polarizable force field models, Br- anions prefer to accumulate just below the first molecular water layer near the interface, whereas Li+ cations remain deeply buried several molecular layers from the interface, even at very high concentration. The separation of ions has a profound effect on the average orientation of water molecules in the vicinity of the interface. We also find that the hydration number of Li+ cations in the center of the slab Na-c,Na-Li+-H2O approximate to 4.7 +/- 0.3, regardless of the salt concentration. This estimate is consistent with the recent experimental neutron scattering data, confirming that results from nonpolarizable empirical models, which consistently predict tetrahedral coordination of Li+ to four solvent molecules, are incorrect. Consequently, disruption of the hydrogen bond network caused by Li+ may be overestimated in nonpolarizable empirical models. Overall, our results suggest that empirical models, in particular nonpolarizable models, may not capture all of the properties of the solution-air interface necessary to fully understand the interfacial chemistry.Peer reviewe

Density Trees for Efficient Nonlinear State Estimation

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

State Estimation with Sets of Densities considering Stochastic and Systematic Errors

In practical applications, state estimation requires the consideration of stochastic and systematic errors. If both error types are present, an exact probabilistic description of the state estimate is not possible, so that common Bayesian estimators have to be questioned. This paper introduces a theoretical concept, which allows for incorporating unknown but bounded errors into a Bayesian inference scheme by utilizing sets of densities. In order to derive a tractable estimator, the Kalman filter is applied to ellipsoidal sets of means, which are used to bound additive systematic errors. Also, an extension to nonlinear system and observation models with ellipsoidal error bounds is presented. The derived estimator is motivated by means of two example applications

Simultaneous State and Parameter Estimation of Distributed-Parameter Physical Systems based on Sliced Gaussian Mixture Filter

This paper presents a method for the simultaneous state and parameter estimation of finite-dimensional models of distributed systems monitored by a sensor network. In the first step, the distributed system is spatially and temporally decomposed leading to a linear finite-dimensional model in state space form. The main challenge is that the simultaneous state and parameter estimation of such systems leads to a high-dimensional nonlinear problem. Thanks to the linear substructure contained in the resulting finite-dimensional model, the development of an overall more efficient estimation process is possible. Therefore, in the second step, we propose the application of a novel density representation - sliced Gaussian mixture density - in order to decompose the estimation problem into a (conditionally) linear and a nonlinear problem. The systematic approximation procedure minimizing a certain distance measure allows the derivation of (close to) optimal and deterministic results. The proposed estimation process provides novel prospects in sensor network applications. The performance is demonstrated by means of simulation results

Dirac Mixture Approximation of Multivariate Gaussian Densities

For the optimal approximation of multivariate Gaussian densities by means of Dirac mixtures, i.e., by means of a sum of weighted Dirac distributions on a continuous domain, a novel systematic method is introduced. The parameters of this approximate density are calculated by minimizing a global distance measure, a generalization of the well-known Cram\\u27{e}r- von Mises distance to the multivariate case. This generalization is obtained by defining an alternative to the classical cumulative distribution, the Localized Cumulative Distribution (LCD). In contrast to the cumulative distribution, the LCD is unique and symmetric even in the multivariate case. The resulting deterministic approximation of Gaussian densities by means of discrete samples provides the basis for new types of Gaussian filters for estimating the state of nonlinear dynamic systems from noisy measurements