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  • 1.
    Roth, Michael
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Kalman Filters for Nonlinear Systems and Heavy-Tailed Noise2013Licentiate thesis, monograph (Other academic)
    Abstract [en]

    This thesis is on filtering in state space models. First, we examine approximate Kalman filters for nonlinear systems, where the optimal Bayesian filtering recursions cannot be solved exactly. These algorithms rely on the computation of certain expected values. Second, the problem of filtering in linear systems that are subject to heavy-tailed process and measurement noise is addressed.

    Expected values of nonlinearly transformed random vectors are an essential ingredient in any Kalman filter for nonlinear systems, because of the required joint mean vector and joint covariance of the predicted state and measurement. The problem of computing expected values, however, goes beyond the filtering context. Insights into the underlying integrals and useful simplification schemes are given for elliptically contoured distributions, which include the Gaussian and Student’s t distribution. Furthermore, a number of computation schemes are discussed. The focus is on methods that allow for simple implementation and that have an assessable computational cost. Covered are basic Monte Carlo integration, deterministic integration rules and the unscented transformation, and schemes that rely on approximation of involved nonlinearities via Taylor polynomials or interpolation. All methods come with realistic accuracy statements, and are compared on two instructive examples.

    Heavy-tailed process and measurement noise in state space models can be accounted for by utilizing Student’s t distribution. Based on the expressions forconditioning and marginalization of t random variables, a compact filtering  algorithm for linear systems is derived. The algorithm exhibits some similarities with the Kalman filter, but involves nonlinear processing of the measurements in form of a squared residual in one update equation. The derived filter is compared to state-of-the-art filtering algorithms on a challenging target tracking example, and outperforms all but one optimal filter that knows the exact instances at which outliers occur.

    The presented material is embedded into a coherent thesis, with a concise introduction to the Bayesian filtering and state estimation problems; an extensive survey of available filtering algorithms that includes the Kalman filter, Kalman filters for nonlinear systems, and the particle filter; and an appendix that provides the required probability theory basis.

  • 2.
    Roth, Michael
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    On the Multivariate t Distribution2012Report (Other academic)
    Abstract [en]

    This technical report summarizes a number of results for the multivariate t distribution which can exhibit heavier tails than the Gaussian distribution. It is shown how t random variables can be generated, the probability density function (pdf) is derived, and marginal and conditional densities of partitioned t random vectors are presented. Moreover, a brief comparison with the multivariate Gaussian distribution is provided. The derivations of several results are given in an extensive appendix.

  • 3.
    Roth, Michael
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Fritsche, Carsten
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Hendeby, Gustaf
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering. Swedish Defence Research Agency (FOI), Linköping, Sweden.
    Gustafsson, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    The Ensemble Kalman Filter and its Relations to Other Nonlinear Filters2015In: Proceedings of the 2015 European Signal Processing Conference (EUSIPCO 2015), Institute of Electrical and Electronics Engineers (IEEE), 2015, p. 1236-1240Conference paper (Refereed)
    Abstract [en]

    The Ensemble Kalman filter (EnKF) is a standard algorithm in oceanography and meteorology, where it has got thousands of citations. It is in these communities appreciated since it scales much better with state dimension n than the standard Kalman filter (KF). In short, the EnKF propagates ensembles with N state realizations instead of mean values and covariance matrices and thereby avoids the computational and storage burden of working on n×n matrices. Perhaps surprising, very little attention has been devoted to the EnKF in the signal processing community. In an attempt to change this, we present the EnKF in a Kalman filtering context. Furthermore, its application to nonlinear problems is compared to sigma point Kalman filters and the particle filter, so as to reveal new insights and improvements for high-dimensional filtering algorithms in general. A simulation example shows the EnKF performance in a space debris tracking application.

  • 4.
    Roth, Michael
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Gustafsson, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    An Efficient Implementation of the Second Order Extended Kalman Filter2011In: Proceedings of the 14th International Conference on Information Fusion (FUSION), 2011, IEEE , 2011Conference paper (Refereed)
    Abstract [en]

    The second order extended Kalman filter (EKF2) is based on a second order Taylor expansion of a nonlinear system, in contrast to the more common (first order) extended Kalman filter (EKF1). Despite a solid theoretical ground for its approximation, it is seldom used in applications, where the EKF and the unscented Kalman filter (UKF) are the standard algorithms. One reason for this might be the requirement for analytical Jacobian and Hessian of the system equations, and the high complexity that scales with the state order $n_x$ as $n_x^5$. We propose a numerical algorithm which is based on an extended set of sigma points (compared to the UKF) that needs neither Jacobian nor Hessian (or numerical approximations of these). Further, it scales as $n_x^4$, which is an order of magnitude better than the EKF2 algorithm presented in literature.

  • 5.
    Roth, Michael
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Gustafsson, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Orguner, Umut
    Dept. of Electrical and Electronics Engineering, Middle East Technical University Ankara, Turkey.
    On-road Trajectory Generation from GPS Data: A Particle Filtering/Smoothing Application2012In: 2012 15th International Conference on Information Fusion, IEEE , 2012, p. 779-786Conference paper (Refereed)
    Abstract [en]

    Many studies in target localization and tracking use GPS measurements as ground truth. These GPS locations might be in conflict with computed estimates in applications where road network information is available (and employed in the estimation procedure). This paper proposes to use particle methods to generate on-road trajectories that can be used as improved ground truth for such road constrained estimation schemes. A bootstrap particle filter and three different particle smoothers are utilized to obtain kinematic target state estimates. The particle smoothers require important adjustments for their implementation in the resulting hybrid state space. The performances of the presented methods are compared on challenging real data obtained from an urban area.

    Although particle filters and smoothers can be applied to general localization problems, with arbitrary sensors, we concentrate on GPS measurements, motivated by an application in cellular network systems.

  • 6.
    Roth, Michael
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Hendeby, Gustaf
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Gustafsson, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    EKF/UKF Maneuvering Target Tracking using Coordinated Turn Models with Polar/Cartesian Velocity2014In: 17th International Conference on Information Fusion (FUSION), 2014, Institute of Electrical and Electronics Engineers (IEEE), 2014, p. 1-8Conference paper (Refereed)
    Abstract [en]

    Nonlinear Kalman filter adaptations such as extended Kalman filters (EKF) or unscented Kalman filters (UKF) provide approximate solutions to state estimation problems in nonlinear models. The algorithms utilize mean values and covariance matrices to represent the probability densities in the otherwise intractable Bayesian filtering equations. As a consequence, their estimation performance can show significant dependence on the choice of state coordinates. The here considered problem of tracking maneuvering targets using coordinated turn (CT) models is one practically relevant example: The velocity in the target state can either be formulated in Cartesian or polar coordinates. We extend a previous study to a broader range of CT models that allow for changes in target speed and turn rate, and investigate UKF as well as EKF variants in terms of their performance and sensitivity to noise parameters. The results advocate for the use of polar CT models.

  • 7.
    Roth, Michael
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Hendeby, Gustaf
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Gustafsson, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, Faculty of Science & Engineering.
    Nonlinear Kalman Filters Explained: A Tutorial on Moment Computations and Sigma Point Methods2016In: Journal of Advances in Information Fusion, ISSN 1557-6418, Vol. 11, no 1, p. 47-70Article in journal (Refereed)
    Abstract [en]

    Nonlinear Kalman filters are algorithms that approximately solve the Bayesian filtering problem by employing the measurement update of the linear Kalman filter (KF). Numerous variants have been developed over the past decades, perhaps most importantly the popular sampling based sigma point Kalman filters.In order to make the vast literature accessible, we present nonlinear KF variants in a common framework that highlights the computation of mean values and covariance matrices as the main challenge. The way in which these moment integrals are approximated distinguishes, for example, the unscented KF from the divided difference KF.With the KF framework in mind, a moment computation problem is defined and analyzed. It is shown how structural properties can be exploited to simplify its solution. Established moment computation methods, and their basics and extensions, are discussed in an extensive survey. The focus is on the sampling based rules that are used in sigma point KF. More specifically, we present three categories of methods that use sigma-points 1) to represent a distribution (as in the UKF); 2) for numerical integration (as in Gauss-Hermite quadrature); 3) to approximate nonlinear functions (as in interpolation). Prospective benefits and downsides are listed for each of the categories and methods, including accuracy statements. Furthermore, the related KF publications are listed.The theoretical discussion is complemented with a comparative simulation study on instructive examples.

  • 8.
    Roth, Michael
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Ozkan, Emre
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Gustafsson, Fredrik
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    A student's t filter for heavy tailed process and measurement noise2013In: Proceedings of the 38th International Conference on Acoustics, Speech, and Signal Processing (ICASSP), IEEE conference proceedings, 2013, p. 5770-5774Conference paper (Refereed)
    Abstract [en]

    We consider the filtering problem in linear state space models with heavy tailed process and measurement noise. Our work is based on Student's t distribution, for which we give a number of useful results. The derived filtering algorithm is a generalization of the ubiquitous Kalman filter, and reduces to it as special case. Both Kalman filter and the new algorithm are compared on a challenging tracking example where a maneuvering target is observed in clutter.

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