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

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.

Cyber Security of Sensor Systems for State Sequence Estimation: an AI Approach

Sensor systems are extremely popular today and vulnerable to sensor data attacks. Due to possible devastating consequences, counteracting sensor data attacks is an extremely important topic, which has not seen sufficient study. This paper develops the first methods that accurately identify/eliminate only the problematic attacked sensor data presented to a sequence estimation/regression algorithm under a powerful attack model constructed based on known/observed attacks. The approach does not assume a known form for the statistical model of the sensor data, allowing data-driven and machine learning sequence estimation/regression algorithms to be protected. A simple protection approach for attackers not endowed with knowledge of the details of our protection approach is first developed, followed by additional processing for attacks based on protection system knowledge. In the cases tested for which it was designed, experimental results show that the simple approach achieves performance indistinguishable, to two decimal places, from that for an approach which knows which sensors are attacked. For cases where the attacker has knowledge of the protection approach, experimental results indicate the additional processing can be configured so that the worst-case degradation under the additional processing and a large number of sensors attacked can be made significantly smaller than the worst-case degradation of the simple approach, and close to an approach which knows which sensors are attacked, for the same number of attacked sensors with just a slight degradation under no attacks. Mathematical descriptions of the worst-case attacks are used to demonstrate the additional processing will provide similar advantages for cases for which we do not have numerical results. All the data-driven processing used in our approaches employ only unattacked training data.