Open Access
d-Choquet integrals: Choquet integrals based on dissimilarities
(Elsevier, 2020) Bustince Sola, Humberto; Mesiar, Radko; Fernández Fernández, Francisco Javier; Galar Idoate, Mikel; Paternain Dallo, Daniel; Altalhi, A. H.; Pereira Dimuro, Graçaliz; Bedregal, Benjamin; Takáč, Zdenko; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa, PJUPNA13
The paper introduces a new class of functions from [0,1]n to [0,n] called d-Choquet integrals. These functions are a generalization of the 'standard' Choquet integral obtained by replacing the difference in the definition of the usual Choquet integral by a dissimilarity function. In particular, the class of all d-Choquet integrals encompasses the class of all 'standard' Choquet integrals but the use of dissimilarities provides higher flexibility and generality. We show that some d-Choquet integrals are aggregation/pre-aggregation/averaging/functions and some of them are not. The conditions under which this happens are stated and other properties of the d-Choquet integrals are studied.
Open Access
Dissimilarity based choquet integrals
(Springer, 2020) Bustince Sola, Humberto; Mesiar, Radko; Fernández Fernández, Francisco Javier; Galar Idoate, Mikel; Paternain Dallo, Daniel; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
In this paper, in order to generalize the Choquet integral, we replace the difference between inputs in its definition by a restricted dissimilarity function and refer to the obtained function as d-Choquet integral. For some particular restricted dissimilarity function the corresponding d-Choquet integral with respect to a fuzzy measure is just the ‘standard’ Choquet integral with respect to the same fuzzy measure. Hence, the class of all d-Choquet integrals encompasses the class of all 'standard' Choquet integrals. This approach allows us to construct a wide class of new functions, d-Choquet integrals, that are possibly, unlike the 'standard' Choquet integral, outside of the scope of aggregation functions since the monotonicity is, for some restricted dissimilarity function, violated and also the range of such functions can be wider than [0, 1], in particular it can be [0, n].
Open Access
Generalized decomposition integral
(Elsevier, 2020) Horanská, Lubomíra; Bustince Sola, Humberto; Fernández Fernández, Francisco Javier; Mesiar, Radko; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas
In this paper we propose two different generalizations of the decomposition integral introduced by Even and Lehrer. We modify the product operator merging a given capacity and the decomposition coefficients by some more general functions F and G and compare properties of the obtained functionals with properties of the original decomposition integral. Generalized decomposition integrals corresponding to the particular decomposition systems, being generalizations of Shilkret, Choquet and concave integrals, are studied and exemplified.
Open Access
A generalization of the Choquet integral defined in terms of the Mobius transform
(IEEE, 2020) Fernández Fernández, Francisco Javier; Bustince Sola, Humberto; Horanská, Lubomíra; Mesiar, Radko; Stupñanová, Andrea; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
In this article, we propose a generalization of the Choquet integral, starting fromits definition in terms of the Mobius transform. We modify the product on 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.

Fernández Fernández, Francisco Javier

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