Dirac Mixture Research Articles

A New Linear Regression Kalman Filter with Symmetric Samples

Nonlinear filtering is of great significance in industries. In this work, we develop a new linear regression Kalman filter for discrete nonlinear filtering problems. Under the framework of linear regression Kalman filter, the key step is minimizing the Kullback–Leibler divergence between standard normal distribution and its Dirac mixture approximation formed by symmetric samples so that we can obtain a set of samples which can capture the information of reference density. The samples representing the conditional densities evolve in a deterministic way, and therefore we need less samples compared with particle filter, as there is less variance in our method. The numerical results show that the new algorithm is more efficient compared with the widely used extended Kalman filter, unscented Kalman filter and particle filter.

A Discrete Quaternion Particle Filter Based on Deterministic Sampling for IMU Attitude Estimation

A discrete quaternion filtering algorithm following high-density uncertainty matching-based deterministic sampling scheme is proposed for IMU attitude estimation. Existing quaternion filters rely on the specific distributions directly or the on-tangent-plane Gaussian distribution indirectly to model the underlying uncertainty. The key idea of this paper is to apply a Dirac mixture at few deterministic weighted samples for non-parametric interpreting the high-density uncertainty of unit quaternion manifold. In contrast to standard Monte Carlo techniques based on random sampling, the proposed approach is deterministic. Furthermore, the deterministic quaternion particles are obtained by projecting on-sphere uniform points to the unit quaternion manifold via an exponential map. The on-manifold quaternions are then exploited in a discrete Bayesian filtering procedure. We compare the new filter with the genetic algorithm embedded quaternion particle filter (GA-QPF) and the unscented quaternion estimator (USQUE) for accuracy and computation cost by using simulated and practical IMU data. The results illustrate that the proposed estimator possesses lower computational cost than the GA-QPF while providing a comparable accuracy, higher accuracy than the USQUE, as well as more effective against large initialization error.

Sequential ensemble transform for Bayesian inverse problems

We present the Sequential Ensemble Transform (SET) method, an approach for generating approximate samples from a Bayesian posterior distribution. The method explores the posterior distribution by solving a sequence of discrete optimal transport problems to produce a series of transport plans which map prior samples to posterior samples. We prove that the sequence of Dirac mixture distributions produced by the SET method converges weakly to the true posterior as the sample size approaches infinity. Furthermore, our numerical results indicate that, when compared to standard Sequential Monte Carlo (SMC) methods, the SET approach is more robust to the choice of Markov mutation kernels and requires less computational efforts to reach a similar accuracy when used to explore complex posterior distributions. Finally, we describe adaptive schemes that allow to completely automate the use of the SET method.

Optimal Reduction of Dirac Mixture Densities on the 2-Sphere

This paper is concerned with optimal approximation of a given Dirac mixture density on the S2 manifold, i.e., a set of weighted samples located on the unit sphere, by an equally weighted Dirac mixture with a reduced number of components. The sample locations of the approximating density are calculated by minimizing a smooth global distance measure, a generalization of the well-known Cramér-von Mises Distance. First, the Localized Cumulative Distribution (LCD) together with the von Mises–Fisher kernel provides a continuous characterization of Dirac mixtures on the S2 manifold. Second, the L2 norm of the difference of two LCDs is a unique and symmetric distance between the corresponding Dirac mixtures. Thereby we integrate over all possible kernel sizes instead of choosing one specific kernel size. The resulting approximation method facilitates various efficient nonlinear sample-based state estimation methods.

Moment Ideals of Local Dirac Mixtures

In this paper we study ideals arising from moments of local Dirac measures and their mixtures. We provide generators for the case of first order local Diracs and explain how to obtain the moment ideal of the Pareto distribution from them. We then use elimination theory and Prony's method for parameter estimation of finite mixtures. Our results are showcased with applications in signal processing and statistics. We highlight the natural connections to algebraic statistics, combinatorics and applications in analysis throughout the paper.

Distributed track‐to‐track fusion for non‐linear systems with Gaussian mixture noise

The authors propose a distributed state estimation algorithm based on optimal track-to-track fusion for local posteriors in terms of Gaussian mixtures. The track-to-track fusion system is implemented with both parallel and sequential structure based on generalised covariance intersection rule. They obtain the optimal fusion coefficients in a computationally efficient manner via Monte Carlo importance sampling method. The Dirac mixture approximation method is proposed for the computation of arbitrary power of a Gaussian mixture density. The resulting Gaussian mixture fusion rule is analytical and applicable to the multi-sensor case. Numerical examples are presented to demonstrate the performance advantages of the proposed method in comparison with existing track-to-track fusion algorithms.

Dirac mixture distributions for the approximation of mixed effects models

Mixed effect modeling is widely used to study cell-to-cell and patient-to-patient variability. The population statistics of mixed effect models is usually approximated using Dirac mixture distributions obtained using Monte-Carlo, quasi Monte-Carlo, and sigma point methods. Here, we propose the use of a method based on the Cramér-von Mises Distance, which has been introduced in the context of filtering. We assess the accuracy of the different methods using several problems and provide the first scalability study for the Cramér-von Mises Distance method. Our results indicate that for a given number of points, the method based on the modified Cramér-von Mises Distance method tends to achieve a better approximation accuracy than Monte-Carlo and quasi Monte-Carlo methods. In contrast to sigma-point methods, the method based on the modified Cramér-von Mises Distance allows for a flexible number of points and a more accurate approximation for nonlinear problems.

Nonlinear Measurement Update for Recursive Filtering Based on the Gauss von Mises Distribution

In conventional Kalman-based state estimation algorithms, there is an assumption that the uncertainties in the system state and measurements are Gaussian distributed. However, this Gaussian assumption ignores the periodic nature of angular or orientation quantities. In this paper, the Gauss von Mises (GVM) distribution model defined on a cylindrical manifold is employed, the Dirac mixture approximation method is extended to deal with sampling with GVM, in order to perform recursive filtering, the GVM approximation to joint distribution is proposed, the formula to compute posterior distribution is derived. Finally, the measurement update algorithm is developed. Simulation results show that when the system state contains a circular variable, the proposed GVM filter can achieve more accurate estimates than the traditional extended Kalmanfilter(EKF), thereby providing a novel method to estimate system state specialized to GVM distribution.

Optimal Reduction of Multivariate Dirac Mixture Densities

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 a Dirac mixture with a reduced number of components. The parameters of the approximating density are calculated by numerically minimizing a smooth 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 smooth characterization of discrete random quantities (on continuous domains). The resulting approximation method provides the basis for various efficient nonlinear estimation and control methods.

Moment-based Dirac Mixture Approximation of Circular Densities

Given a circular probability density function, called the true probability density function, the goal is to find a Dirac mixture approximation based on some circular moments of the true density. When keeping the locations of the Dirac points fixed, but almost arbitrarily located, we are applying recent results on the circulant rational covariance extension problem to the problem of calculating the weights. For the case of simultaneously calculating optimal locations, additional constraints have to be deduced from the given density. For that purpose, a distance measure for the deviation of the Dirac mixture approximation from the true density is derived, which then is minimized while considering the moment conditions as constraints. The method is based on progressive numerical minimization, converges quickly and gives well- distributed Dirac mixtures that fulfill the constraints, i.e., have the desired circular moments.

EVALUATION OF HYBRID BAYESIAN NETWORKS USING ANALYTICAL DENSITY REPRESENTATIONS

In this article, a new mechanism is described for modeling and evaluating Hybrid Dynamic Bayesian networks. The approach uses Gaussian mixtures and Dirac mixtures as messages to calculate marginal densities. As these densities are approximated by means of Gaussian mixtures, any desired precision is possible.The presented approach removes the restrictions of sample based evaluation of Bayesian networks since it uses an analytical approximation scheme for probability densities which systematically minimizes the distance between the exact and the approximate density.