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Maximum Correntropy Unscented Filter

Xi Liu, Badong Chen, Bin Xu, Zongze Wu, Paul Honeine

arXiv:1608.07526v1stat.ML

TL;DR

The paper addresses the UKF’s sensitivity to heavy-tailed non-Gaussian noise. It proposes the MCUF, combining UT-based prior estimation with MCC-based statistical linearization, and reports improved performance over existing filters under impulsive noise when the kernel bandwidth is properly chosen.

  • Problem

    The UKF may perform poorly under heavy-tailed non-Gaussian noise because it is based on the MMSE criterion.

  • Method

    MCUF uses the UT for prior state and covariance estimates and MCC-based statistical linearization, solved through fixed-point iteration, for posterior estimates.

  • Results

    With a proper kernel bandwidth, MCUF can outperform EKF, HEKF, UKF, and HUKF, particularly under impulsive noise.

  • Takeaways & Limitations

    MCUF provides a recursive nonlinear filtering approach whose performance depends on selecting an appropriate kernel bandwidth.

Abstract

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The unscented transformation (UT) is an efficient method to solve the state estimation problem for a non-linear dynamic system, utilizing 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 nonlinear 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 linearization regression based on the maximum correntropy criterion (MCC) is then used to obtain the posterior estimates of the state and covariance. The satisfying performance of the new algorithm is confirmed by two illustrative examples.

I. INTRODUCTION

The UKF provides a derivative-free nonlinear filtering alternative to the EKF, but its MMSE basis makes it sensitive to heavy-tailed non-Gaussian noise. This paper proposes the MCUF, combining unscented transformation with MCC-based statistical linearization for robust recursive estimation.

  • I. INTRODUCTION: The UKF avoids Jacobian calculations by propagating deterministically chosen sigma points, but can perform poorly under heavy-tailed non-Gaussian noise.Its sensitivity follows from reliance on the MMSE criterion.
  • I. INTRODUCTION: MCC-based adaptive filtering has shown strong performance in heavy-tailed non-Gaussian noise, motivating its use for nonlinear state estimation.Correntropy is presented as an information-theoretic robust cost for estimation.
  • I. INTRODUCTION: The MCUF combines UT-based prior state and covariance estimates with MCC-based statistical linearization for posterior estimation.The method retains a recursive structure suitable for online implementation.
  • I. INTRODUCTION: The paper evaluates the proposed filter through two illustrative examples and reports desirable performance.The paper organizes its development around MCC, MCUF derivation, examples, and conclusions.

II. MAXIMUM CORRENTROPY CRITERION

The maximum correntropy criterion uses correntropy as a similarity-based objective for adaptive estimation. With a Gaussian kernel, it incorporates second- and higher-order error moments, while finite samples provide its practical estimator.

  • II. MAXIMUM CORRENTROPY CRITERION: Correntropy is a generalized similarity measure between two random variables, computed using a shift-invariant Mercer kernel.The paper uses a Gaussian kernel unless stated otherwise.
  • II. MAXIMUM CORRENTROPY CRITERION: Finite data permit correntropy estimation through a sample-mean estimator when the joint distribution is unknown.The estimator uses samples drawn from the joint distribution.
  • II. MAXIMUM CORRENTROPY CRITERION: A Taylor expansion shows that correntropy is a weighted sum of all even-order moments of the error variable.The kernel bandwidth controls the weighting of second- and higher-order moments.
  • II. MAXIMUM CORRENTROPY CRITERION: MCC-based learning formulates parameter estimation as an optimization problem over a feasible parameter set.The optimal parameter vector is defined as the solution of this optimization.

III. MAXIMUM CORRENTROPY UNSCENTED FILTER

The MCUF combines MCC with statistical linear regression to form a nonlinear Kalman-type filter designed for non-Gaussian noise. It operates through time and measurement updates for nonlinear state-space systems.

  • III. MAXIMUM CORRENTROPY UNSCENTED FILTER: The MCUF combines MCC and statistical linear regression to handle nonlinear filtering under non-Gaussian noise.Correntropy incorporates second- and higher-order error moments.
  • III. MAXIMUM CORRENTROPY UNSCENTED FILTER: The filter addresses nonlinear systems with n-dimensional states and m-dimensional measurements governed by nonlinear system and measurement functions.The functions are assumed continuously differentiable, with process and measurement noises modeled as independent, zero-mean quantities.
  • III. MAXIMUM CORRENTROPY UNSCENTED FILTER: Like other Kalman-type filters, MCUF separates estimation into time-update and measurement-update steps.These two stages structure the recursive filtering procedure.

A. Time update

The MCUF time update uses sigma points generated from the previous state estimate and covariance, propagates them through the process model, and forms prior state and covariance estimates.

  • A. Time update: The time update generates 2n + 1 sigma points from the previous state estimate and covariance matrix.The points are constructed using a matrix square root and the scaling factor λ.
  • A. Time update: The sigma-point spread is controlled by α, while φ is often set to 3 − n and λ combines these scaling parameters.The parameter β incorporates prior distribution knowledge and equals 2 for a Gaussian distribution.
  • A. Time update: The sigma points are propagated through the nonlinear process equation to estimate the prior state mean and covariance.This applies the unscented transformation to the process model.
  • A. Time update: The state and covariance estimates use corresponding sigma-point weights in the unscented transformation.The weights are defined for the central point and the remaining 2n points.

B. Measurement update

The measurement update combines sigma-point propagation with MCC-based statistical linear regression to obtain robust posterior state and covariance estimates.

  • The algorithm propagates 2n + 1 sigma points through the process and measurement models before constructing the update quantities.
  • The MCUF forms a statistical linear regression model for the measurement update using prior state-error information and a measurement slope matrix.
  • The MCC update estimates the posterior state by maximizing correntropy over the regression residuals, yielding a fixed-point equation.
  • The fixed-point solution is iterated from a prior-state or least-squares initialization until the state estimate converges.
  • After convergence, MCUF updates the posterior covariance and repeats the time-update and measurement-update cycle.
  • The kernel bandwidth controls robustness and convergence: smaller values improve outlier robustness, but excessively small values can slow or prevent convergence.

IV. ILLUSTRATIVE EXAMPLES

The paper evaluates MCUF in two illustrative examples using benchmark measures computed across time steps and Monte Carlo runs.

  • Two illustrative examples are used to demonstrate the performance of the proposed MCUF algorithm.
  • The evaluation benchmarks aggregate estimation error over K time steps and M Monte Carlo runs.

A. Example 1

The univariate nonstationary growth model evaluates MCUF across Gaussian, impulsive measurement, and mixed-Gaussian process-and-measurement noise settings. UKF is best in Gaussian noise, while MCUF achieves the best performance under non-Gaussian noise with a proper kernel bandwidth.

  • In Gaussian noise, UKF achieves the smallest MSE among the filters, while MCUF performs better with a larger kernel bandwidth.MCUF also has relatively small average iteration numbers, especially for large kernel bandwidths.
  • With impulsive measurement noise, MCUF achieves the smallest MSE at σ = 2.0, while larger kernel bandwidths improve convergence speed.The fixed-point algorithm generally converges to the optimal solution in only a few iterations.
  • A smaller ε usually slightly lowers MSE but increases the iteration count required for convergence; the study uses ε = 10^-6 by default.
  • With a proper kernel bandwidth, MCUF achieves the best performance when both process and measurement noises are non-Gaussian.

B. Example 2

Example 2 evaluates EKF, HEKF, UKF, HUKF, and MCUF for vertically falling-body state estimation under Gaussian and impulsive non-Gaussian measurement noise. MCUF is competitive in Gaussian noise and most advantageous under the heavy-tailed disturbance.

  • In Gaussian noise, UKF performs best, while MCUF with a large kernel bandwidth outperforms HEKF and HUKF.The Gaussian-noise comparison uses MSEs for x1, x2, and x3 across the evaluated filters.
  • With σ = 2.0, MCUF achieves the smallest MSE among all algorithms under heavy-tailed non-Gaussian measurement noise.The robust filters outperform their non-robust counterparts, while MCUF performance depends on kernel bandwidth.

V. CONCLUSION

The paper proposes MCUF, combining UT-based prior estimation with MCC-based statistical linearization for posterior state and covariance estimates. Simulations show better performance than several existing filters, particularly under impulsive noise, when the kernel bandwidth is properly chosen.

  • The reported evaluations include MSEs and average fixed-point iteration numbers for non-Gaussian-noise experiments.These quantities are summarized in Tables VII and VIII.
  • MCUF combines UT-based prior state and covariance estimates with MCC-based statistical linearization for posterior estimates solved by fixed-point iteration.This design replaces first-order approximation with UT and statistical linear approximation while using MCC instead of MMSE.
  • With a proper kernel bandwidth, MCUF outperforms EKF, HEKF, UKF, and HUKF particularly under impulsive noises.The conclusion attributes this outcome to the reported simulation results.

APPENDIX A DERIVATION OF (33)

The appendix derives equation (33) through intermediate expressions and a matrix inversion lemma. The supplied derivation identifies the final combination of equations but does not expose the full intermediate algebra.

  • The derivation introduces an intermediate expression involving the predicted state, measurement, and measurement function.
  • The derivation simplifies notation by denoting Sp(k|k−1), Sr(k), Cx(k), and Cy(k) as Sp, Sr, Cx, and Cy.
  • Equation (33) follows by combining equations (32), (56), and (57) after applying the matrix inversion lemma.
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