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Minimum Error Entropy Kalman Filter

Badong Chen, Lujuan Dang, Yuantao Gu, Nanning Zheng, Jose C. Prıncipe

arXiv:1904.06617v2eess.SYcs.IT

TL;DR

Standard Kalman filters can degrade under non-Gaussian noise, while MCC-based methods may not handle multimodal noise well. The paper develops recursive MEE-KF and nonlinear MEE-EKF algorithms, with experiments reporting strong accuracy and robustness.

  • Problem

    MMSE-based Kalman filters can degrade under non-Gaussian noises, while MCC may not perform well for noises from multimodal distributions.

  • Method

    The paper develops recursive MEE-KF and MEE-EKF algorithms using MEE, propagation equations, and fixed-point posterior updates.

  • Results

    MEE-KF/MEE-EKF exhibits the best performance among compared Kalman fusion algorithms, while MEE-EKF provides much better infected-density prediction than EKF and MCEKF.

  • Takeaways & Limitations

    With an appropriate kernel size, MEE-KF can outperform KF and MCKF under complicated non-Gaussian noises, and MEE-EKF improves nonlinear estimation performance.

  • Takeaways & Limitations

    The convergence theorem requires the kernel size to satisfy σ ≥ max{σ1, σ2}.

Abstract

from arXiv · show

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 the present paper we develop a new Kalman-type filter, called minimum error entropy Kalman filter (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 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 minimum error entropy extended Kalman filter (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.

1 Introduction

Kalman filters commonly rely on Gaussian assumptions and MMSE, which can degrade under complicated non-Gaussian noise. The paper introduces MEE-KF and MEE-EKF to address multimodal noise and nonlinear estimation.

  • Kalman filters were originally developed for linear state-space models with Gaussian assumptions, with extensions including EKF, UKF, and CKF.
  • MMSE-based Kalman filters can suffer performance degradation when estimation errors arise from non-Gaussian noises.
  • MCC-based Kalman filters improve robustness against impulsive noises because correntropy is insensitive to large errors.
  • MCC may perform poorly for more complicated non-Gaussian noises, including noises from multimodal distributions.
  • The paper develops MEE-KF and MEE-EKF, using MEE optimization with recursive propagation and fixed-point posterior updates.

2 Background

The background describes the standard KF for linear dynamic systems and introduces MEE as an entropy-based alternative to MMSE and MCC. MEE estimation uses error information and can be approximated from samples.

  • 2.1 Kalman Filter: The KF estimates a dynamic system’s state through a state-space model with process and measurement equations.
  • 2.1 Kalman Filter: KF recursion consists of prediction, which computes prior state and covariance estimates, followed by updating posterior estimates and covariance.
  • 2.2 Minimum Error Entropy Criterion: MEE minimizes information contained in the error by measuring error information through Rényi entropy.
  • 2.2 Minimum Error Entropy Criterion: The error probability density can be estimated using Parzen’s window approach with error samples.
  • 2.2 Minimum Error Entropy Criterion: For second-order entropy, minimizing error entropy is equivalent to maximizing the estimated information potential.

3 Minimum Error Entropy Kalman Filter

The MEE-KF constructs an augmented model combining state prediction and measurement information, then obtains posterior estimates by maximizing an MEE-based cost function. A fixed-point iteration and covariance update yield Algorithm 1.

  • 3.1 Augmented Model: The state prediction error is combined with the measurement equation to form an augmented model.
  • 3.1 Augmented Model: The augmented noise vector contains state and measurement errors, with a positive-definite covariance assumption.
  • 3.1 Augmented Model: Cholesky decomposition and a transformation produce the augmented variables used in the MEE-KF derivation.
  • 3.2 Derivation of MEE-KF: The MEE-KF estimates the posterior state by maximizing an MEE-based cost function and setting its gradient to zero.
  • 3.2 Derivation of MEE-KF: The posterior state is solved through fixed-point iteration involving block matrices that separate state and measurement components.
  • 3.2 Derivation of MEE-KF: The posterior covariance is updated after the state derivation, and the complete procedure is summarized as Algorithm 1.

4 Computational Complexity

This section compares the computational complexity of KF, MCKF, and MEE-KF using floating-point operations. MEE-KF adds error-entropy computation and is slightly more complex than MCKF, but their orders of magnitude are not significantly different.

  • 4 Computational Complexity: The section evaluates KF, MCKF, and MEE-KF computational complexity in terms of floating-point operations.The comparison is organized around the algorithms’ update equations and tabulated operation counts.
  • 4 Computational Complexity: MEE-KF’s fixed-point iteration number T is generally relatively small in the simulations.T denotes the number of fixed-point iterations used by the algorithm.
  • 4 Computational Complexity: MEE-KF has additional computational burden from error-entropy functions and slightly higher complexity than MCKF.The extra burden is attributed to the error-entropy functions used in the MEE-KF updates.
  • 4 Computational Complexity: MEE-KF, MCKF, and KF have no significant difference in computational complexity in the sense of order of magnitude.This conclusion follows the operation-count comparison reported for the three filters.

5 Convergence Issue

The convergence analysis gives a sufficient condition for the fixed-point iterations in MEE-KF. Under a sufficiently large kernel size and a bounded initial state, the iterations converge to a unique fixed point.

  • 5 Convergence Issue: The section establishes a sufficient condition ensuring convergence of the MEE-KF fixed-point iterations.The proof is stated to be similar to prior work and is not provided in this section.
  • 5 Convergence Issue: If σ ≥ max{σ1, σ2}, the kernel-size condition required by Theorem 1 is satisfied.σ1 and σ2 are defined through norm conditions involving g(x(k)) and its Jacobian.
  • 5 Convergence Issue: Under the theorem’s conditions, the fixed-point algorithm converges to a unique fixed point within ||x(k)||1 ≤ β.The initial state must also satisfy ||x(k)0||1 ≤ β.

6 Minimum Error Entropy Extended Kalman Filter

The MEE-EKF extends the MEE-KF framework to nonlinear systems by linearizing the state and measurement functions and constructing an augmented measurement model. The resulting filter follows from the MEE-KF derivation.

  • 6 Minimum Error Entropy Extended Kalman Filter: For nonlinear systems, EKF approximates the state-transition and measurement functions with first-order Taylor expansions.The expansions are taken at the prior state estimate and the current predicted state estimate, respectively.
  • 6 Minimum Error Entropy Extended Kalman Filter: The Jacobian matrices F(k −1) and H(k) represent the derivatives of the state-transition and measurement functions.These Jacobians provide the linearized model used by EKF.
  • 6 Minimum Error Entropy Extended Kalman Filter: Substituting the linearized functions yields the EKF prediction and update equations, including the EKF gain.The nonlinear measurement equation is then approximated for the MEE-EKF construction.
  • 6 Minimum Error Entropy Extended Kalman Filter: The MEE-EKF is obtained by replacing the nonlinear measurement equation with an augmented model and applying a derivation similar to MEE-KF.The resulting procedure is provided as Algorithm 2.

7 Experimental Results

The experiments evaluate MEE-KF and MEE-EKF across multiple scenarios, including land vehicle navigation, autonomous-driving tracking, and infectious-disease epidemic prediction. Linear and nonlinear comparisons use corresponding conventional and maximum-correntropy Kalman filters, with MSE averaged over 100 Monte Carlo runs.

  • 7 Experimental Results: The proposed MEE-KF and MEE-EKF are evaluated in different scenarios through three examples.The supplied passage identifies land vehicle navigation, autonomous-driving tracking, and infectious-disease epidemic prediction among the applications.
  • 7 Experimental Results: Linear-model results compare MEE-KF with KF and MCKF, while nonlinear-model results compare MEE-EKF with EKF and MCEKF.The comparison uses corresponding conventional and maximum-correntropy filters for linear and nonlinear settings.
  • 7 Experimental Results: Performance is evaluated using mean square error (MSE), with estimates compared against the true state over N samples.The MSE is defined from the state-estimation error.
  • 7 Experimental Results: The simulations compute MSE by averaging results over 100 independent Monte Carlo runs.This averaging procedure is used for the reported simulation evaluation.

7.1 Land Vehicle Navigation

The land vehicle navigation experiment evaluates MEE-KF against KF and MCKF under different noise conditions and kernel sizes. MEE-KF remains more stable for small kernels and achieves desirable performance around kernel size 1.5, with slightly higher computational burden than MCKF.

  • 7.1 Land Vehicle Navigation: The experiment evaluates MEE-KF against KF and MCKF for a linear land vehicle navigation problem under different noise conditions.The simulation uses 30000 samples to calculate MSE.
  • 7.1.2 Estimation results of different algorithms: When the kernel size is too small or too large, MCKF and MEE-KF performance becomes worse.For small kernel sizes, MCKF may diverge.
  • 7.1.2 Estimation results of different algorithms: MEE-KF is more stable than MCKF for small kernel sizes.The comparison is reported for the same measurement noise as Case (4).
  • 7.1.2 Estimation results of different algorithms: MEE-KF achieves desirable performance when the kernel size is around 1.5.
  • 7.1.2 Estimation results of different algorithms: MEE-KF achieves better performance than MCKF at the cost of slightly higher computational burden.Computing time is measured per iteration using MATLAB 2016b on an i5-4590 3.30 GHZ CPU.

7.2 Tracking of Autonomous Driving

The autonomous-driving experiment uses lidar and radar measurements to track vehicles across two trajectories with linear and nonlinear Kalman fusion algorithms. MEE-KF/MEE-EKF exhibits the best performance among the compared methods.

  • 7.2 Tracking of Autonomous Driving: Lidar provides position information, while radar provides better velocity accuracy in poor weather compared with lidar.The experiment uses Udacity self-driving-car data measured by lidar and radar sensors.
  • 7.2.1 Model: The linear lidar model uses KF, MCKF, and MEE-KF, whereas the nonlinear radar model uses EKF, MCEKF, and MEE-EKF.
  • 7.2.1 Model: The compared algorithms are evaluated on two trajectories using KF/EKF, MCKF/MCEKF, and MEE-KF/MEE-EKF.Figures 2 and 3 show tracking results for trajectories 1 and 2, respectively.
  • 7.2.2 Estimation results of different algorithms: MEE-KF/MEE-EKF exhibits the best performance among all compared Kalman fusion algorithms.The comparison uses MSE, with kernel sizes 20.0 for MEE-KF, 1.66 for MEE-EKF, 20.0 for MCC-KF, and 15.0 for MCC-EKF.

7.3 Prediction of Infectious Disease Epidemics

The epidemic experiment applies MEE-EKF to a nonlinear SIR-based model using weekly infected-population observations. MEE-EKF provides much better infected-density prediction than EKF and MCEKF.

  • 7.3 Prediction of Infectious Disease Epidemics: The experiment validates MEE-EKF on United States Influenza-Like-Illness data supplied by a disease control center.The data represent 52 weeks of infected-population density in 2010.
  • 7.3.1 Model: The epidemic state is represented by susceptible and infected densities, with recovered density determined by b(k)=1−s(k)−i(k).The state is x(k)=[s(k), i(k)]T, and the dynamics follow the SIR model.
  • 7.3.1 Model: Syndromic surveillance observes only the infected state i(k), so the measurement matrix is H=[0 1].
  • 7.3.2 Estimation results of different algorithms: MEE-EKF provides a much better prediction of infected-population density than EKF and MCEKF.The result is reported from Figure 4 and Table 7, using kernel sizes 20.0 for MEE-EKF and 16.0 for MCEKF.

8 Conclusion

The paper proposes MEE-KF and MEE-EKF for state estimation under complicated non-Gaussian noises. Simulations in navigation, autonomous-driving tracking, and epidemic prediction confirm their reported performance.

  • 8 Conclusion: MEE-KF replaces MMSE and MCC with the MEE criterion and uses fixed-point iteration.
  • 8 Conclusion: With an appropriate kernel size, MEE-KF can outperform KF and MCKF, especially under complicated non-Gaussian noises.
  • 8 Conclusion: MEE-EKF extends the MEE approach to nonlinear state estimation under non-Gaussian noises.
  • 8 Conclusion: The paper provides MEE-KF computational-complexity analysis and a sufficient convergence condition for its fixed-point iteration.
  • 8 Conclusion: Simulations cover land vehicle navigation, autonomous-driving tracking, and infectious-disease epidemic prediction.
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