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