[Paper Review] Optimally (Distributional-)Robust Kalman Filtering

Peter Ruckdeschel|arXiv (Cornell University)|Apr 20, 2010

Target Tracking and Data Fusion in Sensor Networks33 references3 citations

TL;DR

This paper presents optimally robust Kalman filtering under distributional uncertainty, formulating minimax MSE problems for both system-endogenous and observation-exogenous outliers. It derives closed-form saddle-point solutions that balance tracking and outlier attenuation, with a computationally simple rLS filter emerging when the ideal conditional mean is linear in the innovation—offering a surprising characterization of this linearity condition.

ABSTRACT

We present optimality results for robust Kalman filtering where robustness is understood in a distributional sense, i.e.; we enlarge the distribution assumptions made in the ideal model by suitable neighborhoods. This allows for outliers which in our context may be system-endogenous or -exogenous, which induces the somewhat conflicting goals of tracking and attenuation. The corresponding minimax MSE-problems are solved for both types of outliers separately, resulting in closed-form saddle-points which consist of an optimally-robust procedure and a corresponding least favorable outlier situation. The results are valid in a surprisingly general setup of state space models, which is not limited to a Euclidean or time-discrete framework. The solution however involves computation of conditional means in the ideal model, which may pose computational problems. In the particular situation that the ideal conditional mean is linear in the observation innovation, we come up with a straight-forward Huberization, the rLS filter, which is very easy to compute. For this linearity we obtain an again surprising characterization.

Motivation & Objective

• To develop a minimax robust Kalman filtering framework that accounts for both system-endogenous and observation-exogenous outliers under distributional uncertainty.
• To solve the corresponding minimax mean squared error (MSE) problems in a general state space model setup, extending beyond Euclidean and time-discrete frameworks.
• To identify conditions under which the optimal robust filter simplifies to a Huber-type estimator, enabling efficient computation.
• To provide a characterization of the linearity condition under which the rLS filter becomes optimal, offering a novel test for this property.

Proposed method

• Formulates distributional neighborhoods around the ideal Gaussian state space model to model contamination from outliers.
• Solves minimax MSE problems via saddle-point analysis, yielding robust filters and corresponding least favorable distributions.
• Introduces the rLS filter as a direct Huberization of the Kalman filter when the ideal conditional mean is linear in the innovation.
• Derives conditions under which the rLS filter achieves optimality, using a surprising characterization of linearity in the innovation space.
• Applies the theory to general state space models, including continuous-time and non-linear settings via linearization.
• Uses regular conditional densities and known measures to generalize beyond standard linear Gaussian models.

Experimental results

Research questions

• RQ1 Under what conditions does the minimax robust Kalman filter achieve optimal trade-off between tracking accuracy and outlier resistance in the presence of distributional contamination?
• RQ2 When does the optimal robust filter reduce to a simple Huber-type estimator (rLS filter), and what is the structural condition enabling this simplification?
• RQ3 How can the linearity of the ideal conditional mean in the observation innovation be tested, and what is the statistical basis for such a test?
• RQ4 What is the relationship between SO-neighborhoods and eSO-neighborhoods in the context of robust filtering, and how does this affect the robustness of the resulting filter?
• RQ5 How does the proposed framework extend to non-Euclidean, continuous-time, and non-linear state space models?

Key findings

• The optimal robust Kalman filter is derived as a saddle-point solution to a minimax MSE problem under distributional uncertainty, balancing tracking and outlier attenuation.
• For both system-endogenous and observation-exogenous outliers, the minimax solution yields closed-form expressions involving conditional means under the ideal model.
• When the ideal conditional mean is linear in the innovation, the optimal filter reduces to the rLS filter, a simple Huberization that is computationally efficient.
• A surprising characterization of linearity in the innovation space is derived, which enables a novel optimal test for this condition.
• The rLS filter achieves the minimax MSE bound on the corresponding eSO-neighborhood, establishing its optimality under the linearity condition.
• The framework generalizes to continuous-time, non-linear, and non-Euclidean state space models through appropriate linearization and density-based formulations.