Open Access
Weak and directional monotonicity of functions on Riesz spaces to fuse uncertain data
(Elsevier B.V., 2019) Sesma Sara, Mikel; Mesiar, Radko; Bustince Sola, Humberto; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
In the theory of aggregation, there is a trend towards the relaxation of the axiom of monotonicity and also towards the extension of the definition to other domains besides real numbers. In this work, we join both approaches by defining the concept of directional monotonicity for functions that take values in Riesz spaces. Additionally, we adapt this notion in order to work in certain convex sublattices of a Riesz space, which makes it possible to define the concept of directional monotonicity for functions whose purpose is to fuse uncertain data coming from type-2 fuzzy sets, fuzzy multisets, n-dimensional fuzzy sets, Atanassov intuitionistic fuzzy sets and interval-valued fuzzy sets, among others. Focusing on the latter, we characterize directional monotonicity of interval-valued representable functions in terms of standard directional monotonicity.
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
Pre-aggregation functions: construction and an application
(IEEE, 2015) Lucca, Giancarlo; Sanz Delgado, José Antonio; Pereira Dimuro, Graçaliz; Bedregal, Benjamin; Mesiar, Radko; Kolesárová, Anna; Bustince Sola, Humberto; Automática y Computación; Automatika eta Konputazioa
In this work we introduce the notion of preaggregation function. Such a function satisfies the same boundary conditions as an aggregation function, but, instead of requiring monotonicity, only monotonicity along some fixed direction (directional monotonicity) is required. We present some examples of such functions. We propose three different methods to build pre-aggregation functions. We experimentally show that in fuzzy rule-based classification systems, when we use one of these methods, namely, the one based on the use of the Choquet integral replacing the product by other aggregation functions, if we consider the minimum or the Hamacher product t-norms for such construction, we improve the results obtained when applying the fuzzy reasoning methods obtained using two classical averaging operators like the maximum and the Choquet integral.
Open Access
Mixture functions and their monotonicity
(Elsevier, 2019) Špirková, Jana; Beliakov, Gleb; Bustince Sola, Humberto; Fernández Fernández, Francisco Javier; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas
We consider mixture functions, which are a type of weighted averages for which the corresponding weights are calculated by means of appropriate continuous functions of their inputs. In general, these mixture function need not be monotone increasing. For this reason we study su cient conditions to ensure standard, weak and directional monotonicity for speci c types of weighting functions. We also analyze directional monotonicity when di erentiability is assumed.
Open Access
Strengthened ordered directional and other generalizations of monotonicity for aggregation functions
(Springer, 2018) Sesma Sara, Mikel; Miguel Turullols, Laura de; Lafuente López, Julio; Barrenechea Tartas, Edurne; Mesiar, Radko; Bustince Sola, Humberto; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
A tendency in the theory of aggregation functions is the generalization of the monotonicity condition. In this work, we examine the latest developments in terms of different generalizations. In particular, we discuss strengthened ordered directional monotonicity, its relation to other types of monotonicity, such as directional and ordered directional monotonicity and the main properties of the class of functions that are strengthened ordered directionally monotone. We also study some construction methods for such functions and provide a characterization of usual monotonicity in terms of these notions of monotonicity.
Open Access
Directional monotonicity of multidimensional fusion functions with respect to admissible orders
(Elsevier, 2023-03-09) Sesma Sara, Mikel; Bustince Sola, Humberto; Mesiar, Radko; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa, PJUPNA25-2022
The notion of directional monotonicity emerged as a relaxation of the monotonicity condition of aggregation functions. As the extension of aggregation functions to fuse more complex information than numeric data, directional monotonicity was extended to the framework of multidimensional data, with respect to the product order, which is a partial order. In this work, we present the notion of admissible order for multidimensional data and we define the concept of directional monotonicity for multidimensional fusion functions with respect to an admissible order. Moreover, we study the main properties of directionally monotone functions in this new context. We conclude that, while some of the properties are still valid (e.g. the set of directions of increasingness is still closed under convex combinations), some of the main ones no longer hold (e.g. there does not exist a finite set of directions that characterize standard monotonicity in terms of directional monotonicity).
Open Access
Curve-based monotonicity: a generalization of directional monotonicity
(Taylor & Francis, 2019) Roldán López de Hierro, Antonio Francisco; Sesma Sara, Mikel; Špirková, Jana; Lafuente López, Julio; Pradera, Ana; Mesiar, Radko; Bustince Sola, Humberto; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas
In this work we propose a generalization of the notion of directional monotonicity. Instead of considering increasingness or decreasingness along rays, we allow more general paths defined by curves in the n-dimensional space. These considerations lead us to the notion of α-monotonicity, where α is the corresponding curve. We study several theoretical properties of α-monotonicity and relate it to other notions of monotonicity, such as weak monotonicity and directional monotonicity.
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
Strengthened ordered directionally monotone functions. Links between the different notions of monotonicity
(Elsevier, 2019) Sesma Sara, Mikel; Lafuente López, Julio; Roldán López de Hierro, Antonio Francisco; Mesiar, Radko; Bustince Sola, Humberto; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas
In this work, we propose a new notion of monotonicity: strengthened ordered directional monotonicity. This generalization of monotonicity is based on directional monotonicity and ordered directional monotonicity, two recent weaker forms of monotonicity. We discuss the relation between those different notions of monotonicity from a theoretical point of view. Additionally, along with the introduction of two families of functions and a study of their connection to the considered monotonicity notions, we define an operation between functions that generalizes the Choquet integral and the Lukasiewicz implication.
Open Access
Ordered directional monotonicity in the construction of edge detectors
(Elsevier, 2021) Marco Detchart, Cedric; Bustince Sola, Humberto; Fernández Fernández, Francisco Javier; Mesiar, Radko; Lafuente López, Julio; Barrenechea Tartas, Edurne; Pintor Borobia, Jesús María; Estatistika, Informatika eta Matematika; Ingeniaritza; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas; Ingeniería
In this paper we provide a specific construction method of ordered directionally monotone functions. We show that the functions obtained with this construction method can be used to build edge detectors for grayscale images. We compare the results of these detectors to those obtained with some other ones that are widely used in the literature. Finally, we show how a consensus edge detector can be built improving the results obtained both by our proposal and by those in the literature when applied individually.