Variational Robust Kalman Filters: A Unified Framework

TL;DR

The paper addresses the challenge of achieving robustness and adaptivity in Kalman filtering under time-varying noise and outliers. It introduces a unified variational robust Kalman filter (STKF) built on a Student's t loss and solved via fixed-point iterations, enabling efficient joint handling of uncertain covariance and outliers. A formal link between robust filtering and variational/adaptive filtering is established, showing STKF is equivalent to a VBKF with fixed priors, and two robust-adaptive extensions (STKF-AR1/AR2) are developed with switching rules. Simulations demonstrate recovery of standard KF and superior performance in complex noise environments, validating the practical utility of the framework.

Abstract

Robustness and adaptivity are two competing objectives in Kalman filters (KF). Robustness involves temporarily inflating prior estimates of noise covariances, while adaptivity updates prior beliefs using real-time information. In practical applications, both process and measurement noise can be influenced by outliers, be time-varying, or both. Existing works may not effectively address the above complex noise scenarios, as there is an intrinsic incompatibility between robust filters and adaptive filters. In this work, we propose a unified variational robust Kalman filter, built on a Student's t-distribution induced loss function and variational inference, and solved through fixed-point iteration in a computationally efficient manner. We demonstrate that robustness can be understood as a prerequisite for adaptivity, making it possible to merge the above two competing goals into a single framework through switching rules. Additionally, our proposed filter can recover conventional KF, robust KF, and adaptive KF by adjusting parameters, and can suppress both the imperfect process and measurement noise, enabling it to perform superiorly in complex noise environments. Simulations verify the effectiveness of the proposed method.

Figures (9)

• Figure 1: The visualization of $\mathcal{L}_{st}$ and $\mathcal{L}_{gau}$ as well as their influence functions and induced PDFs. (a) The loss function of $\mathcal{L}_{st}$ and $\mathcal{L}_{gau}$. (b) The influence function of $\mathcal{L}_{st}$ and $\mathcal{L}_{gau}$. (c) The mapped Student's t distribution and Gaussian distribution. (d) The PDF of latent variable $\lambda$.
• Figure 2: Some noise scenarios considered in this work (but not limited to these examples). The first, second, and third column corresponds to Scenario 1, 2, and 3. The data with an absolute value bigger than 20 are visualized as $\pm 20$. (a) Case 1: $w_k \sim \mathcal{N}(0,1)$, $v_k \sim 0.99\mathcal{N}(0,1)+0.01\mathcal{N}(0,400)$. (b) Case 2: $w_k \sim \mathcal{N}(0,1)$, $v_k \sim \mathcal{N}(0,R_{k,t})$ where $R_{k,t}=(1+2|\sin(0.1\pi t)|)^{2}$. (c) Case 3: $w_k \sim \mathcal{N}(0,1)$, $v_k \sim 0.99\mathcal{N}(0,R_{k,t})+0.01\mathcal{N}(0,400)$. (d) Case 4: $w_k \sim 0.99\mathcal{N}(0,1)+0.01\mathcal{N}(0,400)$, $v_k \sim \mathcal{N}(0,1)$. (e) Case 5: $w_k \sim 0.99\mathcal{N}(0,R_{k,t})+0.01\mathcal{N}(0,400)$, $v_k \sim \mathcal{N}(0,1)$. (f) Case 6: $w_k \sim 0.99\mathcal{N}(0,1)+0.01\mathcal{N}(0,400)$, $v_k \sim \mathcal{N}(0,R_{k,t})$.
• Figure 3: Error performances of VBKF-fixed, STKF, and KF.
• Figure 4: Average RMSE (ARMSE) with different $\nu_3$ in STKF.
• Figure 5: The measurement noise covariance (or variance) tracking performance of VBKF and STKF-AR1.

Theorems & Definitions (11)

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