On a comparison between Mahalanobis distance and Choquet integral: The Choquet–Mahalanobis operator

Mahalanobis distance can be used in problems where variables are not independent. The presence of the covariance matrix in this expression permits us to represent the dependence between the variables. Fuzzy measures and Choquet integrals have a similar purpose. In this paper we compare these two expressions. To do so in the proper setting, we introduce a Choquet integral based distance. Then, we consider probability-density functions based on these two distances. In particular, we review the Gaussian distribution, which is based on the Mahalanobis distance and introduce another distribution based on the Choquet distance. Then, we introduce an operator that generalizes the Choquet integral and the Mahalanobis distance. It is the Choquet–Mahalanobis integral. Some propositions are also proven establishing equivalences and links between the Choquet–Mahalanobis integral, the Choquet integral, and the Mahalanobis distance.