Generalized C F 1 F 2-integrals

This paper introduces the theoretical framework for a generalization of C F 1 F 2-integrals, a family of Choquet-like integrals used successfully in the aggregation process of the fuzzy reasoning mechanisms of fuzzy rule based classification systems. The proposed generalization, called by g C F 1 F 2-integrals, is based on the so-called pseudo pre-aggregation function pairs ( F 1 , F 2 ), which are pairs of fusion functions satisfying a minimal set of requirements in order to guarantee that the g C F 1 F 2-integrals to be either an aggregation function or just an ordered directionally increasing function satisfying the appropriate boundary conditions. We propose a dimension reduction of the input space, in order to deal with repeated elements in the input, avoiding ambiguities in the definition of g C F 1 F 2-integrals. We study several properties of g C F 1 F 2-integrals, considering different constraints for the functions F 1 and F 2, and state under which conditions g C F 1 F 2-integrals present or not averaging behaviors. Several examples of g C F 1 F 2-integrals are presented, considering different pseudo pre-aggregation function pairs, defined on, e.g., t-norms, overlap functions, copulas that are neither t-norms nor overlap functions and other functions that are not even pre-aggregation functions.