In this paper, a novel Gaussian–Student's t mixture (GSTM) distribution is proposed to model non-stationary heavy-tailed noises. The proposed GSTM distribution can be formulated as a hierarchical Gaussian form by introducing a Bernoulli random variable, based on which a new hierarchical linear Gaussian state-space model is constructed. A novel robust GSTM distribution based Kalman filter is proposed based on the constructed hierarchical linear Gaussian state-space model using the variational Bayesian approach. The Kalman filter and robust Student's t based Kalman filter (RSTKF) with fixed distribution parameters are two existing special cases of the proposed filter. The novel GSTM distributed Kalman filter has the important advantage over the RSTKF that the adaptation of the mixing parameter is much more straightforward than learning the degrees of freedom parameter. Simulation results illustrate that the proposed filter has better estimation accuracy than those of the Kalman filter and RSTKF for a linear state-space model with non-stationary heavy-tailed noises.

A Novel Robust Gaussian–Student's t Mixture Distribution Based Kalman Filter

Mixture correntropy for robust learning

In this paper, we investigate the state estimation problem of nonlinear systems with non-Gaussian measurement noise. Based on a newly defined cost function which is obtained by a combination of weighted least square (WLS) and maximum correntropy criterion (MCC), we derive our maximum correntropy unscented Kalman filter (MCUKF) and the corresponding maximum correntropy unscented information filter (MCUIF). Comparing with existing MCUKF, our MCUKF avoids the numerical problem occurred when the measurements contain large outliers, and can obtain similar or even better estimation results. When the kernel bandwidth goes infinity, we prove that our MCUKF and MCUIF will converge to UKF and UIF, respectively, while existing MCUIF will not in this case and it generally has poor estimation accuracy as well. Two typical nonlinear models are used to illustrate the advantages of our proposed algorithms.

Maximum correntropy unscented Kalman and information filters for non-Gaussian measurement noise

The unscented transformation UT is an efficient method to solve the state estimation problem for a non-linear dynamic system, utilising a derivative-free higher-order approximation by approximating a Gaussian distribution rather than approximating a non-linear function. Applying the UT to a Kalman filter type estimator leads to the well-known unscented Kalman filter UKF. Although the UKF works very well in Gaussian noises, its performance may deteriorate significantly when the noises are non-Gaussian, especially when the system is disturbed by some heavy-tailed impulsive noises. To improve the robustness of the UKF against impulsive noises, a new filter for non-linear systems is proposed in this work, namely the maximum correntropy unscented filter MCUF. In MCUF, the UT is applied to obtain the prior estimates of the state and covariance matrix, and a robust statistical linearisation regression based on the maximum correntropy criterion is then used to obtain the posterior estimates of the state and covariance matrix. The satisfying performance of the new algorithm is confirmed by two illustrative examples.

https://hal.archives-ouvertes.fr/hal-01965044/document

Maximum correntropy unscented filter

To date, most linear and nonlinear Kalman filters (KFs) have been developed under the Gaussian assumption and the well-known minimum mean square error (MMSE) criterion. In order to improve the robustness with respect to impulsive (or heavy-tailed) non-Gaussian noises, the maximum correntropy criterion (MCC) has recently been used to replace the MMSE criterion in developing several robust Kalman-type filters. To deal with more complicated non-Gaussian noises such as noises from multimodal distributions, in this article, we develop a new Kalman-type filter, called minimum error entropy KF (MEE-KF), by using the minimum error entropy (MEE) criterion instead of the MMSE or MCC. Similar to the MCC-based KFs, the proposed filter is also an online algorithm with the recursive process, in which the propagation equations are used to give prior estimates of the state and covariance matrix, and a fixed-point algorithm is used to update the posterior estimates. In addition, the MEE extended KF (MEE-EKF) is also developed for performance improvement in the nonlinear situations. The high accuracy and strong robustness of MEE-KF and MEE-EKF are confirmed by experimental results.

Minimum Error Entropy Kalman Filter

Maximum correntropy Kalman filter

Traditional Kalman filter (KF) is derived under the well-known minimum mean square error (MMSE) criterion, which is optimal under Gaussian assumption. However, when the signals are non-Gaussian, especially when the system is disturbed by some heavy-tailed impulsive noises, the performance of KF will deteriorate seriously. To improve the robustness of KF against impulsive noises, we propose in this work a new Kalman filter, called the maximum correntropy Kalman filter (MCKF), which adopts the robust maximum correntropy criterion (MCC) as the optimality criterion, instead of using the MMSE. Similar to the traditional KF, the state mean and covariance matrix propagation equations are used to give prior estimations of the state and covariance matrix in MCKF. A novel fixed-point algorithm is then used to update the posterior estimations. A sufficient condition that guarantees the convergence of the fixed-point algorithm is given. Illustration examples are presented to demonstrate the effectiveness and robustness of the new algorithm.

Xi Liu

Papers

Maximum correntropy Kalman filter

Maximum Correntropy Kalman Filter