On Using the Shapley Value for Anomaly Localization: A Statistical Investigation

ABSTRACT Recent publications have suggested using the Shapley value for anomaly localization for sensor data systems. We use a reasonable statistical model for the classifiers required to compute the Shapley value to provide repeatable and rigorous analysis in the anomaly localization application. Then we provide a proof that using a single fixed term in the Shapley value calculation achieves a lower complexity anomaly localization test, with the same probability of error, as a test using the Shapley value in cases with independent observation. While it is impossible to test all possible cases numerically, we found this to be true in all the cases we tested with independent observations. For some dependent observation cases with two sensors, where only the second sensor data is anomalous, we show numerically that the Shapley value test can falsely decide an anomaly occurs at the first (nonanomalous) sensor with a probability which approaches one for increasing anomaly magnitude. On the other hand, using a single fixed term in the Shapley value calculation in these cases gives a reasonably small probability of an anomaly occurring at the first (nonanomalous) sensor for any anomaly magnitude. These results are the first of this type we have seen, could encourage new algorithm development, and should encourage future research to more fully understand these observations. A better understanding of the Shapley value, given its popularity, seems an important topic which could lead to improvements in algorithms and real implementations in the future.