A Concept of Approximated Densities for Efficient Nonlinear Estimation
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A Concept of Approximated Densities for Efficient Nonlinear Estimation Virginie F. Ruiz Department of Cybernetics, The University of Reading, Whiteknights, Reading RG6 6AY, UK Email: [email protected] Received 31 July 2001 and in revised form 11 July 2002 This paper presents the theoretical development of a nonlinear adaptive filter based on a concept of filtering by approximated densities (FAD). The most common procedures for nonlinear estimation apply the extended Kalman filter. As opposed to conventional techniques, the proposed recursive algorithm does not require any linearisation. The prediction uses a maximum entropy principle subject to constraints. Thus, the densities created are of an exponential type and depend on a finite number of parameters. The filtering yields recursive equations involving these parameters. The update applies the Bayes theorem. Through simulation on a generic exponential model, the proposed nonlinear filter is implemented and the results prove to be superior to that of the extended Kalman filter and a class of nonlinear filters based on partitioning algorithms. Keywords and phrases: nonlinear estimation, nonlinear adaptive filter, exponential distribution, maximum entropy, nonGaussian signal processing.
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INTRODUCTION
This paper describes a recursive algorithm based on a nonlinear approach to parameter estimation. The most common procedure applied for nonlinear estimation is the extended Kalman filter (EKF) [1, 2, 3]. It is known that the nonlinear estimator for nonlinear models does not provide a finite solution. By linearising the state equations about the conditional means xk|k−1 and xk|k , the EKF provides a solution in terms of minimal mean square error to the original nonlinear problem. A number of other procedures have also been applied including partitioning approaches, statistical linearisation, maximum a posteriori, least square criteria, and functional approximation of conditional state density [3, 4, 5, 6, 7]. Crucially, finite models or filters that completely define physical systems are rare [8, 9]. Nevertheless, the state distributions depend on a finite number of parameters, which are recursively computed as the observed data arrive [10, 11, 12]. The computation of a finite number of pertinent parameters is a common procedure for the definition of approximated probability distributions. As nonlinear state space models rarely yield explicit or analytic distributions from output measurements, the distribution normally has to be approximated [12]. This paper presents the theoretical development of a simple method of approximation [13]. In the following, the proposed nonlinear adaptive filter may be referred to as the FAD
filter. The approach uses a maximum entropy principle to approximate the filtering equations arising from a state model with nonlinear equations. The probability density functions (pdf) created by applying the entropy principle are of an exponential type and depend on a finite number of parameters. The nonlinear filtering leads to recursive equa
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