[Paper Review] Maximum Correntropy Kalman Filter

Badong Chen, Xi Liu+2 more|arXiv (Cornell University)|Sep 15, 2015

Advanced Adaptive Filtering Techniques23 references3 citations

TL;DR

This paper proposes the Maximum Correntropy Kalman Filter (MCKF), a robust alternative to the traditional Kalman filter that replaces the minimum mean square error (MMSE) criterion with the maximum correntropy criterion (MCC) to improve performance under non-Gaussian, impulsive noise. By using a fixed-point algorithm for posterior state estimation and ensuring convergence under a sufficient condition, MCKF achieves superior robustness in heavy-tailed noise environments while retaining the state propagation framework of standard KF.

ABSTRACT

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.

Motivation & Objective

• To address the degradation of traditional Kalman filter performance under non-Gaussian, impulsive noise conditions.
• To develop a robust filtering framework that maintains accuracy in heavy-tailed noise environments where MMSE-based KF fails.
• To replace the MMSE criterion with the maximum correntropy criterion (MCC) for improved robustness.
• To ensure convergence of the proposed fixed-point update algorithm through a sufficient condition.
• To validate the effectiveness and robustness of the MCKF through illustrative examples and comparative analysis.

Proposed method

• Adopt the maximum correntropy criterion (MCC) as the optimality criterion instead of the traditional MMSE.
• Use standard Kalman filter state and covariance propagation equations for prior estimation.
• Develop a novel fixed-point algorithm to compute posterior state and covariance estimates under MCC.
• Establish a sufficient condition for the convergence of the fixed-point algorithm.
• Utilize a kernel-based similarity measure (correntropy) to enhance robustness against outliers and impulsive noise.
• Maintain the recursive structure of the Kalman filter while replacing the error metric with correntropy-based optimization.

Experimental results

Research questions

• RQ1 Can the maximum correntropy criterion improve Kalman filter robustness in the presence of impulsive noise?
• RQ2 How does the performance of the MCKF compare to the traditional KF under non-Gaussian noise conditions?
• RQ3 What conditions ensure the convergence of the fixed-point algorithm used in the MCKF update step?
• RQ4 Does the MCKF maintain computational efficiency while improving robustness?
• RQ5 Can the MCKF effectively handle heavy-tailed noise distributions where standard KF fails?

Key findings

• The MCKF significantly outperforms the traditional KF in environments with heavy-tailed impulsive noise, demonstrating superior estimation accuracy.
• The proposed fixed-point algorithm converges under a derived sufficient condition, ensuring numerical stability.
• The use of correntropy as a similarity measure effectively suppresses the impact of outliers and non-Gaussian disturbances.
• Simulation results show that MCKF maintains low estimation error even when the noise distribution deviates from Gaussianity.
• The method preserves the recursive and computationally efficient structure of the standard Kalman filter while enhancing robustness.
• Illustrative examples confirm the effectiveness of MCKF in practical scenarios involving impulsive noise.