A Generalization of the Choquet Integral Defined in Terms of the Möbius Transform

Abstract

In this article, we propose a generalization of the Choquet integral, starting from its definition in terms of the Mobius transform. We modify the product on $\mathbb {R}$ considered in the Lovasz extension form of the Choquet integral into a function $F$ , and we discuss the properties of this new functional. For a fixed $n$ , a complete description of all $F$ yielding an $n$ -ary aggregation function with a fixed diagonal section, independent of the considered fuzzy measure, is given, and several particular examples are presented. Finally, all functions $F$ yielding an aggregation function, independent of the number $n$ of inputs and of the considered fuzzy measure, are characterized, and related aggregation functions are shown to be just the Choquet integrals over the distorted inputs.