Aggregation functions defined by Choquet-partitioned functions

Abstract

[EN]Bustince et al. recently introduced a versatile framework for constructing aggregation functions from inputdependent [0, 1]-valued vectors under appropriate conditions. Here we leverage their framework to propose and investigate a new and structurally different family of operators. Whereas in Choquet-inspired aggregation functions, the weights applied to a difference of successive values depend on the inputs for which they are not weighting, the weights that are applied in the new model remain constant across the inputs that belong to an element of a fixed partition (in the sense of basic set theory) of the set of inputs. Consequently, a key structural distinction is that, in contrast to the former model, the number of weights is inherently finite when the partition is simplicial (a technical concept that aligns with the spirit of Choquet integration). When the partition is the finest (therefore, it is infinite) we obtain the general framework that motivates our investigation, hence the new model generalizes Choquet integrals. We prove that it also encompasses a type of aggregation operators based on weighting vectors that is more general than Ordered Weighted Averaging operators and Induced Ordered Weighted Averaging operators. Other mathematical proofs establish when this new class of idempotent operators has properties such as shift-invariance and continuity. Our proofs identify natural conditions guaranteeing that the new operators are aggregation functions. Numerical examples involving bivariate inputs illustrate their geometric interpretation. This new family of operators highlights the foundational importance and flexibility of the general framework designed by Bustince et al.