Kalman Filtering in Non-Gaussian Model Errors: A New Perspective a
It is well-known that the optimality of the Kalman filter relies on the Gaussian distribution of the process and observation model errors, which in many situations is well justified [1, 2, 3]. However, its optimality is useless in applications that the distribution assumptions of the model errors do not hold in practice. Even minor deviation from the assumed (or nominal) distribution may cause the Kalman filter’s performance to drastically degrade or to completely breakdown. In particular, when dealing with perceptually important signals such as speech, images, medical, campaign and ocean engineering, the measurements have confirmed the presence of non-Gaussian impulsive (heavy tailed) or Laplace noises . Therefore, the classical Kalman filter which is derived under nominal Gaussian probability model is biased or even breaks down in such situations. This article presents a simple modification approach to overcome this limitation of the Kalman filter. We show that in the smoothing Wiener filter, the estimated state is chosen to minimize the ℓ2-norm of the variation of the system model error. The ℓ2-norm, however, corresponds to Gaussian priors. It confirms the optimality of Wiener filter and Kalman filter in linear Gaussian systems and explains why they suffer from the sensitivity to Laplase or impulsive error statistics. To address this limitation, we propose a variation on smoothing Wiener filter which substitutes a sum of absolute values (i.e., ℓ1-norm) for the sum of squares used in ℓ2 smoothing Wiener filter to penalize variations in the system model error. The proposed ℓ1 smoothing Wiener filter is suitable for analyzing systems with impulsive or Laplace model errors. The idea of ℓ1 smoothing Wiener filter is then used to recast and correct the system (or process) model equation. The modified model puts a Laplace or a sparse distribution on the system model error to enforce the sparsity on the states of the system. Based on the fact that the formulations of the optimum filter by Wiener and Kalman are equivalent in discrete time steady state , the Kalman filter can be used to estimate the desired states using the modified model.
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