## The Unscented Kalman Filter for nonlinear estimation (2000)

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Citations: | 73 - 4 self |

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@INPROCEEDINGS{Wan00theunscented,

author = {Eric A. Wan and Rudolph Van Der Merwe},

title = {The Unscented Kalman Filter for nonlinear estimation},

booktitle = {},

year = {2000},

pages = {153--158}

}

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### Abstract

The Extended Kalman Filter (EKF) has become a standard technique used in a number of nonlinear estimation and machine learning applications. These include estimating the state of a nonlinear dynamic system, estimating parameters for nonlinear system identification (e.g., learning the weights of a neural network), and dual estimation (e.g., the Expectation Maximization (EM) algorithm) where both states and parameters are estimated simultaneously. This paper points out the flaws in using the EKF, and introduces an improvement, the Unscented Kalman Filter (UKF), proposed by Julier and Uhlman [5]. A central and vital operation performed in the Kalman Filter is the propagation of a Gaussian random variable (GRV) through the system dynamics. In the EKF, the state distribution is approximated

### Citations

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Citation Context ...Equation 7). As expressed earlier, a number of algorithmic approaches exist for this problem. We present results for the Dual UKF and Joint UKF. Development of a Unscented Smoother for an EM approach =-=[2]-=- was presented in [13]. As in the prior state-estimation example, we utilize a noisy time-series application modeled with neural networks for illustration of the approaches. In the the dual extended K... |

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Citation Context ...pplied directly as an efficient "second-order" technique for learning the parameters. In the linear case, the relationship between the Kalman Filter (KF) and Recursive Least Squares (RLS) is=-= given in [3]-=-. The use of the EKF for training neural networks has been developed by Singhal and Wu [9] and Puskorious and Feldkamp [8]. Dual Estimation A special case of machine learning arises when the input x k... |

429 | A new extension of the Kalman filter to nonlinear systems
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(Show Context)
Citation Context ...h states and parameters are estimated simultaneously. This paper points out the flaws in using the EKF, and introduces an improvement, the Unscented Kalman Filter (UKF), proposed by Julier and Uhlman =-=[5]-=-. A central and vital operation performed in the Kalman Filter is the propagation of a Gaussian random variable (GRV) through the system dynamics. In the EKF, the state distribution is approximated by... |

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Citation Context ...s for all nonlinearities. For non-Gaussian inputs, approximations are accurate to at least the second-order, with the accuracy of third and higher order moments determined by the choice of ands(See [=-=4]-=- for a detailed discussion of the UT). A simple example is shown in Figure 1 for a 2-dimensional system: the left plot shows the true mean and covariance propagation using Monte-Carlo sampling; the ce... |

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Citation Context ...he linear case, the relationship between the Kalman Filter (KF) and Recursive Least Squares (RLS) is given in [3]. The use of the EKF for training neural networks has been developed by Singhal and Wu =-=[9]-=- and Puskorious and Feldkamp [8]. Dual Estimation A special case of machine learning arises when the input x k is unobserved, and requires coupling both state-estimation and parameter estimation. For ... |

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Citation Context ... between the Kalman Filter (KF) and Recursive Least Squares (RLS) is given in [3]. The use of the EKF for training neural networks has been developed by Singhal and Wu [9] and Puskorious and Feldkamp =-=[8]-=-. Dual Estimation A special case of machine learning arises when the input x k is unobserved, and requires coupling both state-estimation and parameter estimation. For these dual estimation problems, ... |

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Citation Context .... The EKF and its Flaws Consider the basic state-space estimation framework as in Equations 1 and 2. Given the noisy observation y k , a recursive estimation for x k can be expressed in the form (see =-=[6-=-]), ^ xk = (prediction of xk ) +Kk [yk (prediction of yk )] (8) This recursion provides the optimal minimum mean-squared error (MMSE) estimate for x k assuming the prior estimate ^ x k 1 and current ... |

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Citation Context ...KF to a broader class of nonlinear estimation problems, including nonlinear system identification, training of neural networks, and dual estimation problems. Our preliminary results were presented in =-=[13]-=-. In this paper, the algorithms are further developed and illustrated with a number of additional examples. This work was sponsored by the NSF under grant grant IRI-9712346 1. Introduction The EKF has... |

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Citation Context ...ght-estimation are done with the UKF. Note that the state-transition is linear in the weight filter, so the nonlinearity is restricted to the measurement equation. In the joint extended Kalman filter =-=[7]-=-, the signal-state and weight vectors are concatenated into a single, joint state vector: [x T k w T k ] T . Estimation is done recursively by writing the state-space equations for the joint state as:... |

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Citation Context ...d in [13]. As in the prior state-estimation example, we utilize a noisy time-series application modeled with neural networks for illustration of the approaches. In the the dual extended Kalman filter =-=[11]-=-, a separate state-space representation is used for the signal and the weights. The state-space representation for the state x k is the same as in Equation 20. In the context of a time-series, the sta... |

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Citation Context ... i=0 W (m) i Y i (17) Py 2L X i=0 W (c) i fY i yg fY i yg T (18) Note that this method differs substantially from general "sampling " methods (e.g., Monte-Carlo methods such as particle f=-=ilters [1]-=-) which require orders of magnitude more sample points in an attempt to propagate an accurate (possibly nonGaussian) distribution of the state. The deceptively simple approach taken with the UT result... |

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