# Statistical Analysis of using the Shapley Value for Sensor Anomaly Localization with Accurate Classifiers

arXiv: https://arxiv.org/abs/2606.00867 · HTML: https://mlsanity.com/paper/2606.00867/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 42 references that are in the map.
- First posted on arXiv 30 May 2026.

## Builds on (most cited first, 10 of 10)

- [Deep Learning for Anomaly Detection: A Survey](https://mlsanity.com/paper/1901.03407/) (2019)
- [The many Shapley values for model explanation](https://mlsanity.com/paper/1908.08474/) (2019)
- [Explaining individual predictions when features are dependent: More accurate approximations to Shapley values](https://mlsanity.com/paper/1903.10464/) (2019)
- [Explaining Predictive Uncertainty with Information Theoretic Shapley Values](https://mlsanity.com/paper/2306.05724/) (2023)
- [Shapley Values of Reconstruction Errors of PCA for Explaining Anomaly Detection](https://mlsanity.com/paper/1909.03495/) (2019)
- [Using Kernel SHAP XAI Method to optimize the Network Anomaly Detection Model](https://mlsanity.com/paper/2308.00074/) (2023)
- [Multivariate Generalized Gaussian Distribution: Convexity and Graphical Models](https://mlsanity.com/paper/1304.3206/) (2013)
- [Harvesting Ambient RF for Presence Detection Through Deep Learning](https://mlsanity.com/paper/2002.05770/) (2020)
- [A Characteristic Function for Shapley-Value-Based Attribution of Anomaly Scores](https://mlsanity.com/paper/2004.04464/) (2020)
- [On Using the Shapley Value for Anomaly Localization: A Statistical Investigation](https://mlsanity.com/paper/2507.21023/) (2025)

## Abstract (arXiv)

Recent publications have suggested using the Shap- ley value for sensor anomaly/attack localization. We study the performance of such an approach by using mathematically de- fined optimum binary classifiers in the Shapley value calculation. To judge localization performance, we study the ability of the Shapley value of a given sensor observation to determine if that observation is anomalous. First, we prove that for cases with independent sensor observations, an optimized anomaly test using the Shapley value is equivalent to an optimized lower-complexity anomaly test using a single term in the Shapley value calculation, yielding the exact same probability of error. For some popular dependent observation cases involving two sensors, including correlated bivariate Gaussian/Laplacian probability density functions and constant/Gaussian at- tacks/anomalies, we prove that these two tests are fundamentally different, yielding different decision regions and error probabil- ities. Further, we prove that the Shapley value test is sometimes strictly inferior to the other (single term in Shapley calculation) test in certain statistically dependent bivariate Gaussian scenarios with large correlation magnitude and additive attacks/anomalies, while it is strictly superior in others, depending on the sign of the correlation. One can combine these two approaches to obtain a strictly better approach in these cases. These results, which provide the first theoretical statistical analysis of Shapley-based localization, seem very interesting based on the wide acceptance of the Shapley value by many researchers and should encourage further research on this topic. Numerical results are provided which illustrate our findings.