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
New classes of the moderate deviation functions
(Springer Nature, 2021) Špirková, Jana; Bustince Sola, Humberto; Fernández Fernández, Francisco Javier; Sesma Sara, Mikel; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
At present, in the field of aggregation of various input values, attention is focused on the construction of aggregation functions using other functions that can affect the resulting aggregated value. This resulting value should characterize the properties of the individual input values as accurately as possible. Attention is also paid to aggregation using the so-called moderate deviation function. Using this function in aggregation ensures that all properties of aggregation functions are preserved. This work offers constructions of the moderate deviation functions using negations and automorphisms on the symmetric interval [−1, 1] and a general closed interval [a, b] ⊂ [−∞, ∞].
Open Access
A framework for generalized monotonicity of fusion functions
(Elsevier, 2023) Sesma Sara, Mikel; Šeliga, Adam; Boczek, Michał; Jin, LeSheng; Kaluszka, Marek; Kalina, Martin; 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
The relaxation of the property of monotonicity is a trend in the theory of aggregation and fusion functions and several generalized forms of monotonicity have been introduced, most of which are based on the notion of directional monotonicity. In this paper, we propose a general framework for generalized monotonicity that encompasses the different forms of monotonicity that we can find in the literature. Additionally, we introduce various new forms of monotonicity that are not based on directional monotonicity. Specifically, we introduce dilative monotonicity, which requires that the function increases when the inputs have increased by a common factor, and a general form of monotonicity that is dependent on a function T and a subset of the domain Z. This two new generalized monotonicities are the basis to propose a set of different forms of monotonicity. We study the particularities of each of the new proposals and their links to the previous relaxed forms of monotonicity. We conclude that the introduction of dilative monotonicity complements the conditions of weak monotonicity for fusion functions and that (T,Z)-monotonicity yields a condition that is slightly stronger than weak monotonicity. Finally, we present an application of the introduced notions of monotonicity in sentiment analysis.
Open Access
Interval subsethood measures with respect to uncertainty for the interval-valued fuzzy setting
(Atlantis Press, 2020) Pekala, Barbara; Bentkowska, Urszula; Sesma Sara, Mikel; Fernández Fernández, Francisco Javier; Lafuente López, Julio; Altalhi, A. H.; Knap, Maksymilian; Bustince Sola, Humberto; Pintor Borobia, Jesús María; Estatistika, Informatika eta Matematika; Ingeniaritza; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas; Ingeniería
In this paper, the problem of measuring the degree of subsethood in the interval-valued fuzzy setting is addressed. Taking into account the widths of the intervals, two types of interval subsethood measures are proposed. Additionally, their relation and main properties are studied. These developments are made both with respect to the regular partial order of intervals and with respect to admissible orders. Finally, some construction methods of the introduced interval subsethood measures with the use interval-valued aggregation functions are examined.
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
Directions of directional, ordered directional and strengthened ordered directional increasingness of linear and ordered linear fusion operators
(IEEE, 2019) Sesma Sara, Mikel; Marco Detchart, Cedric; 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; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
In this work we discuss the forms of monotonicity that have been recently introduced to relax the monotonicity condition in the definition of aggregation functions. We focus on directional, ordered directional and strengthened ordered directional monotonicity, study their main properties and provide some results about their links and relations among them. We also present two families of functions, the so-called linear fusion functions and ordered linear fusion functions and we study the set of directions for which these types of functions are directionally, ordered directionally and strengthened ordered directionally increasing. In particular, OWA operators are an example of ordered linear fusion functions.
Open Access
Description and properties of curve-based monotone functions
(Springer, 2019) Sesma Sara, Mikel; Miguel Turullols, Laura de; Roldán López de Hierro, Antonio Francisco; Špirková, Jana; Mesiar, Radko; Bustince Sola, Humberto; Institute of Smart Cities - ISC
Curve-based monotonicity is one of the lately introduced relaxations of monotonicity. As directional monotonicity regards monotonicity along fixed rays, which are given by real vectors, curve-based monotonicity studies the increase of functions with respect to a general curve. In this work we study some theoretical properties of this type of monotonicity and we relate this concept with previous relaxations of monotonicity.