Deterministic and Stochastic Particle Filters in State-Space Models

Bølviken, Erik; Storvik, Geir

doi:10.1007/978-1-4757-3437-9_5

Cited by 15 publications

(15 citation statements)

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“…The predictor solves the system of equations given by (8) forη. We emphasize here, that we solve the system in this step as a linear system of equations for a given γ.…”

Section: A Predictormentioning

confidence: 99%

“…The particles are viewed as a mixture of weighted Dirac delta components used to systematically approximate the density at hand. This is different from the deterministic type of particle filters in [8] as a distance measure is employed to transform the {schrempf|uwe.hanebeck}@ieee.org, brunn@ira.uka.de approximation problem into an optimization problem. However, in the case of Dirac mixtures, typical distance measures quantifying the distance between two densities are not well defined.…”

Section: Introductionmentioning

confidence: 99%

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Dirac Mixture Density Approximation Based on Minimization of the Weighted Cramér¿von Mises Distance

Schrempf

Brunn

Hanebeck

2006

2006 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems

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Abstract-This paper proposes a systematic procedure for approximating arbitrary probability density functions by means of Dirac mixtures. For that purpose, a distance measure is required, which is in general not well defined for Dirac mixture densities. Hence, a distance measure comparing the corresponding cumulative distribution functions is employed. Here, we focus on the weighted Cramér-von Mises distance, a weighted integral quadratic distance measure, which is simple and intuitive. Since a closed-form solution of the given optimization problem is not possible in general, an efficient solution procedure based on a homotopy continuation approach is proposed. Compared to a standard particle approximation, the proposed procedure ensures an optimal approximation with respect to a given distance measure. Although useful in their own respect, the results also provide the basis for a recursive nonlinear filtering mechanism as an alternative to the popular particle filters.

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“…The predictor solves the system of equations given by (8) forη. We emphasize here, that we solve the system in this step as a linear system of equations for a given γ.…”

Section: A Predictormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Dirac Mixture Density Approximation Based on Minimization of the Weighted Cramér¿von Mises Distance

Schrempf

Brunn

Hanebeck

2006

2006 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems

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“…More in spirit with the original UKF, the above integrals can also be approximated using points determined by Gaussian quadrature [1,12]. …”

Section: Measurement Updatep(xmentioning

confidence: 99%

Deterministic and Stochastic Gaussian Particle Smoothing

Zoeter

Ypma²,

Heskes

2006

2006 IEEE Nonlinear Statistical Signal Processing Workshop

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In this article we study inference problems in non-linear dynamical systems. In particular we are concerned with assumed density approaches to filtering and smoothing. In models with uncorrelated (but dependent) state and observation, the extended Kalman filter and the unscented Kalman filter break down. We show that the Gaussian particle filter and the one-step unscented Kalman filter make less assumptions and potentially form useful filters for this class of models. We construct a symmetric smoothing pass for both filters that does not require the dynamics to be invertible.We investigate the characteristics of the methods in an interesting problem from mathematical finance. Among others we find that smoothing helps, in particular for the deterministic one-step unscented Kalman filter.

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“…The given data points are interpreted as a mixture of Dirac delta components in order to systematically approximate an arbitrary density function. The proposed method differs from the deterministic type of particle filters in [8] as a distance measure is employed to find an optimal approximation of the true density. However, typical distance measures quantifying the distance between two densities are not well defined for the case of Dirac mixtures.…”

Section: Introductionmentioning

confidence: 99%

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

Schrempf

Brunn

Hanebeck

2006

Proceedings of the 45th IEEE Conference on Decision and Control

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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.

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Deterministic and Stochastic Particle Filters in State-Space Models

Cited by 15 publications

References 0 publications

Dirac Mixture Density Approximation Based on Minimization of the Weighted Cramér¿von Mises Distance

Dirac Mixture Density Approximation Based on Minimization of the Weighted Cramér¿von Mises Distance

Deterministic and Stochastic Gaussian Particle Smoothing

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

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