Sensor Importance towards Observability Degree via Shapley Values

TL;DR

The paper tackles the challenge of quantifying each sensor's contribution to the degree of observability in linear time-invariant systems using Shapley values. By mapping observability metrics, particularly the observability Gramian's trace and its minimum eigenvalue, to value functions, it enables fair attribution of sensor importance prior to filter design. It shows that trace-based Shapley values are additive and miss interaction effects, while minimum-eigenvalue-based values capture inter-sensor interactions and yield intuitive allocations that align with observability improvements. Through two simulation scenarios, the authors demonstrate how the minimum-eigenvalue approach provides more reliable and interpretable sensor contributions, with sums matching the overall observability measure. The work offers a concept-level tool for sensor selection, placement, and state-estimator design, leveraging Shapley values to quantify each sensor's fair impact on observability.

Abstract

Sensor selection is an often under-appreciated aspect of state estimator or Kalman filter design. The basic minimum requirement for the choice of a sensor set while designing Kalman filters is that all states are observable. In addition, the sensors should be chosen with a view towards estimating the states with a desired accuracy. Often observability is treated as true/false check during filter design. Beyond observability -- the observability degree -- which measures \emph{how observable} the states are, has been used as the metric of choice to for sensor selection or placement applications. The higher the degree of observability, the better the possibility of designing Kalman filters that achieve the desired state estimation accuracy and consistency requirements. When a wide variety of sensors are available, sometimes with cost and physical constraints involved, sensor selection plays a crucial role in filter design. In such situations it is important to know the expected contribution of each sensor towards observability degree. Shapley values, developed in cooperative game theory for fair allocation of the payout of a multi-player game to individual players, are widely used in machine learning to assess feature importance. This paper shows that Shapley values can indeed be leveraged to quantify the expected marginal contribution of each sensor in any given sensor set towards the observability degree. This quantification of the fair contribution of each sensor towards the observability degree can be leveraged by filter designers for sensor selection, placement and filter (state estimator) design.