Multivariate outlier explanations using Shapley values and Mahalanobis distances

For the purpose of explaining multivariate outlyingness, it is shown that squared Mahalanobis distance an observation can be decomposed into outlyingness contributions originating from single variables. The decomposition obtained using Shapley value, a well-known concept game theory became popular in context Explainable AI. In addition to outlier explanation, this also relates recent formulation cellwise where values employed obtain variable for outlying observations with respect their “expected” position given data structure. combination distances, calculated at low numerical cost, making them even more attractive tool interpretation. Simulations and real-world examples demonstrate usefulness these concepts.