Title: Theoretical Statistical Analysis of Shapley Value-Based Sensor Anomaly Localization Using Optimal Classifiers

Abstract

Recent literature has proposed the application of Shapley values for the localization of sensor anomalies and attacks. This study evaluates the performance of such methodologies by employing mathematically defined optimal binary classifiers within the Shapley value computation. To assess localization efficacy, we analyze the capacity of the Shapley value associated with a specific sensor observation to identify whether that observation is anomalous.

First, we demonstrate that for scenarios involving independent sensor observations, an optimized anomaly test based on the Shapley value is equivalent to a lower-complexity optimized test utilizing only a single term from the Shapley value calculation. Both approaches yield identical error probabilities. However, in cases involving dependent observations—specifically those involving two sensors with correlated bivariate Gaussian or Laplacian probability density functions, as well as constant or Gaussian attacks/anomalies—we prove that the two testing methods are fundamentally distinct. These methods result in different decision regions and error probabilities.

Furthermore, we establish that in certain statistically dependent bivariate Gaussian scenarios characterized by high correlation magnitudes and additive attacks/anomalies, the full Shapley value test is strictly inferior to the single-term test. Conversely, in other scenarios, the Shapley value test is strictly superior, depending on the sign of the correlation. By combining these two approaches, it is possible to derive a method that is strictly superior in these contexts.

These findings represent the first theoretical statistical analysis of Shapley-based localization. Given the widespread adoption of the Shapley value among researchers, these results are particularly significant and should stimulate further investigation into this topic. Numerical results are presented to illustrate and validate our theoretical findings.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC