Outlier-robust Kalman Filtering through Generalised Bayes (2405.05646v2)

Abstract: We derive a novel, provably robust, and closed-form Bayesian update rule for online filtering in state-space models in the presence of outliers and misspecified measurement models. Our method combines generalised Bayesian inference with filtering methods such as the extended and ensemble Kalman filter. We use the former to show robustness and the latter to ensure computational efficiency in the case of nonlinear models. Our method matches or outperforms other robust filtering methods (such as those based on variational Bayes) at a much lower computational cost. We show this empirically on a range of filtering problems with outlier measurements, such as object tracking, state estimation in high-dimensional chaotic systems, and online learning of neural networks.

• The paper introduces WoLF, a new method that integrates generalized Bayes into Kalman filtering to mitigate outlier effects and model inaccuracies.
• The approach maintains closed-form updates and computational efficiency while being applicable to both linear and non-linear state-space models.
• The method's flexibility in selecting weighting functions enables robust performance across diverse real-time applications such as navigation and forecasting.

Enhancing Robustness of Kalman Filters with Generalised Bayes

Introduction

Kalman Filters (KFs) are staple tools in signal processing, object tracking, and other areas dealing with dynamic systems and time series data. However, they can struggle with outliers or model misspecifications due to their base assumption on data normality, which isn't always a practical scenario in real-world applications. The paper introduces a novel variation of the Kalman Filter, which integrates concepts from generalized Bayesian inference to achieve robustness against outliers and model misspecification in both linear and non-linear state-space models.

What's New: The WoLF Approach

The proposed approach, dubbed the Weighted Observation Likelihood Filter (WoLF), adapts the standard Kalman Filter process by introducing a weight into the observation likelihood function, effectively dampening the influence of data points identified as potential outliers based on a specified weighting function. Unlike traditional robust filtering methods, WoLF maintains the computational efficiency of the standard Kalman Filter by preserving closed-form update equations.

• Integration with Existing Filters: WoLF can be implemented in conjunction with traditional and Enhanced Kalman Filters (EKF) as well as the Ensemble Kalman Filter (EnKF), showing its versatility across various Kalman Filtering techniques.
• Weighting Functions: The flexibility of the approach also extends to the choice of the weighting function, which can be tailored to specific applications or outlier characteristics, such as using inverse multi-quadratic or Mahalanobis-based functions.
• Computational Efficiency: Even with the introduction of weighting, the method does not introduce significant computational overhead, retaining a complexity comparable to that of the original filters.

Key Advantages

WoLF presents several compelling advantages over existing methods:

1. Closed-form Updates: Keeping updates closed-form where possible ensures lower computational overhead and easier implementation.
2. Robustness to Outliers: By dampening outlier influences through weighting, WoLF provides more reliable performance in the presence of atypical or erroneous data.
3. Theoretical Soundness: The paper provides proof of the approach's robustness in theoretical terms, ensuring that its applications are grounded in solid statistical principles.
4. Flexibility: Easy adaptation to different types of Kalman Filters (KF, EKF, EnKF) and different systems (linear, non-linear) showcases the method's versatility.

Practical Implications and Speculations

WoLF's ability to handle outliers effectively without a significant computational penalty can make it a favorable choice in real-time systems where speed and accuracy are critical, such as in autonomous vehicle navigation, real-time economic forecasting, or adaptive control systems in engineering. Its adaptability to both linear and non-linear systems, as well as its integration with ensemble methods, also positions it well for complex dynamic modeling tasks such as weather forecasting or large-scale environmental modeling.

Future Directions

While the current approach improves robustness against certain types of data issues, several potential areas could be further explored to enhance its capabilities:

• Handling Non-Gaussian State Processes: Extending robustness to non-Gaussian scenarios in state transitions could widen applicational scopes.
• Adaptive Weighting Functions: Developing methods to dynamically adjust weighting functions based on real-time data characteristics could improve filter performance and robustness.
• Integration with Learning Models: Exploring deeper integration with machine learning models, where parameters of the dynamic models are learned from data, could open new avenues in adaptive filtering.

In conclusion, the Weighted Observation Likelihood Filter (WoLF) offers a robust, flexible, and computationally efficient method for enhancing the performance of Kalman Filters in outlier-prone environments, promising substantial benefits across various dynamic data modeling fields.