Outlier explanation using Shapley values and Mahalanobis distances

Abstract

Multivariate outlier detection poses a topic of unabated popularity in statistics and computer science. In contrast to some of the most common methods, our aim is not only to detect outlying observations but also to explain which coordinates cause the outlyingness. We propose to use Shapley values for this purpose since they have been successfully used in explainable Artificial Intelligence to interpret the outcome of a ”black-box” method in terms of feature contributions for individual observations. Our approach is based on a combination of Shapley values and squared Mahalanobis distances, and we obtain a decomposition of this distance measure into an outlyingness score for each variable, which can be interpreted as the average marginal contribution to the outlyingness of an observation. While the computational complexity of the Shapley value is a major drawback in the general case, we show that the problem can be reformulated and significantly simplified in our case. This allows for an easy and time-efficient computation also in higher dimensions. Further, we present an extension to the method which is based on assigning outlyingness scores to pairs of variables, allowing for the evaluation of interaction effects. We illustrate the performance of our procedures on simulated as well as on real-world examples.