Open Access
Computing with uncertainty truth degrees: a convolution-based degrees
(2017) Miguel Turullols, Laura de; Bustince Sola, Humberto; Baets, Bernard de; Induráin Eraso, Esteban; Automática y Computación; Automatika eta Konputazioa
La teoría de los conjuntos difusos puede contemplarse como un conjunto de herramientas matemáticas excepcionalmente adaptadas para trabajar con información incompleta, falta de nitidez e incertidumbre no aleatoria. De hecho, como herramienta en ingeniería, para traducir el lenguaje natural humano impreciso en un objeto matemático, los conjuntos difusos juegan un papel decisivo para superar la brecha entre el hombre y los ordenadores. Sin embargo, es ampliamente conocido que la asignación de un valor preciso como pertenencia no es una tarea sencilla. En la literatura, se han propuesto y estudiado varias generalizaciones de los conjuntos difusos para resolver esta dificultad. Más aún, estas generalizaciones han demostrado ser una herramienta útil, al mejorar los resultados en diferentes aplicaciones. Las generalizaciones difieren de los conjuntos difusos en el objeto matemático que se utiliza para modelar la imprecisión y/o incertidumbre. Especifícamente, los conjuntos difusos toman elementos en el intervalo unidad [0, 1] mientras que las generalizaciones toman objetos matemáticos más complejos como intervalos (conjuntos difusos intervalo-valorados), subconjuntos del intervalo unidad (conjuntos difusos "conjunto-valorados") o funciones (conjuntos difusos tipo-2), entre otros. No obstante, el uso de las generalizaciones de los conjuntos difusos tiene un gran inconveniente. Antes de aplicar las generalizaciones de los conjuntos difusos es necesario adaptar ad hoc cada noción teórica al correspondiente objeto matemático que modela la incertidumbre en la aplicación, es decir, es necesario redefinir cada noción teórica reemplazando el intervalo unidad [0, 1] por objetos matemáticos más complejos. En la historia de los conjuntos difusos quedó claro relativamente pronto que la relación natural entre la teoría de conjuntos y la lógica clásica podía ser imitada generando una relación entre la teoría de los conjuntos difusos y la lógica multi-valuada. Hoy en día esta lógica multivaluada recibe el nombre de lógica difusa. Del mismo modo, cada generalización de los conjuntos difusos genera un nuevo sistema lógico. Todos estos sistemas lógicos coinciden en que intentan modelar incertidumbre, pero difieren en el objeto matemático que representa esta incertidumbre. Es fácil comprobar que el mismo problema entre conjuntos difusos y sus generalizaciones puede encontrarse en los distintos sistemas lógicos, es decir, aunque todos ellos son similares, cada noción teórica tiene que ser redefinida para cada lógica. Este problema, junto con el gran número de lógicas que modelan incertidumbre, nos ha llevado a estudiar si es o no posible encontrar un sistema que englobe estas lógicas y nos ha motivado a proponer un sistema lógico que permita modelar la incertidumbre de manera más flexible. Centrándonos especialmente en sistemas lógicos provenientes de la lógica difusa, en esta tesis doctoral proponemos un nuevo sistema lógico que recupera varias de las lógicas de la literatura. La principales ventajas de nuestra propuesta son: evitará la excesiva repetición de las nociones teóricas; permitirá adaptar la aplicación a la generalización de los conjuntos difusos más adecuada de una manera mucho más sencilla. En esta tesis doctoral presentamos la semántica del modelo lógico propuesto junto con un estudio en profundidad de la operación de convolución que se utiliza para definir las conectivas disyunción y conjunción del sistema.
Open Access
The interval-valued Choquet integral based on admissible permutations
(IEEE, 2018) Paternain Dallo, Daniel; Miguel Turullols, Laura de; Ochoa Lezaun, Gustavo; Lizasoain Iriso, María Inmaculada; Mesiar, Radko; Bustince Sola, Humberto; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
Aggregation or fusion of interval data is not a trivial task, since the necessity of arranging data arises in many aggregation functions, such as OWA operators or the Choquet integral. Some arranging procedures have been given to solve this problem, but they need certain parameters to be set. In order to solve this problem, in this work we propose the concept of an admissible permutation of intervals. Based on this concept, which avoids any parameter selection, we propose a new approach for the interval-valued Choquet integral that takes into account every possible permutation fitting to the considered ordinal structure of data. Finally, a consensus among all the permutations is constructed.
Open Access
Construction of admissible linear orders for interval-valued Atanassov intuitionistic fuzzy sets with an application to decision making
(Elsevier, 2015) Miguel Turullols, Laura de; Bustince Sola, Humberto; Fernández Fernández, Francisco Javier; Induráin Eraso, Esteban; Kolesárová, Anna; Mesiar, Radko; Matemáticas; Matematika; Automática y Computación; Automatika eta Konputazioa; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
In this work we introduce a method for constructing linear orders between pairs of intervals by using aggregation functions. We adapt this method to the case of interval-valued Atanassov intuitionistic fuzzy sets and we apply these sets and the considered orders to a decision making problem.
Open Access
Pointwise aggregation of maps: its structural functional equation and some applications to social choice theory
(Elsevier, 2017) Miguel Turullols, Laura de; Campión Arrastia, María Jesús; Candeal, Juan Carlos; Induráin Eraso, Esteban; Paternain Dallo, Daniel; Automática y Computación; Matemáticas; Automatika eta Konputazioa; Matematika; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
We study a structural functional equation that is directly related to the pointwise aggregation of a finite number of maps from a given nonempty set into another. First we establish links between pointwise aggregation and invariance properties. Then, paying attention to the particular case of aggregation operators of a finite number of real-valued functions, we characterize several special kinds of aggregation operators as strictly monotone modifications of projections. As a case study, we introduce a first approach of type-2fuzzy sets via fusion operators. We develop some applications and possible uses related to the analysis of properties of social evaluation functionals in social choice, showing that those functionals can actually be described by using methods that derive from this setting.
Open Access
Hyperspectral imaging using notions from type-2 fuzzy sets
(Springer, 2019) López Maestresalas, Ainara; Miguel Turullols, Laura de; López Molina, Carlos; Arazuri Garín, Silvia; Bustince Sola, Humberto; Jarén Ceballos, Carmen; Ingeniería; Ingeniaritza; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
Fuzzy set theory has developed a prolific armamentarium of mathematical tools for each of the topics that has fallen within its scope. One of such topics is data comparison, for which a range of operators has been presented in the past. These operators can be used within the fuzzy set theory, but can also be ported to other scenarios in which data are provided in various representations. In this work, we elaborate on notions for type-2 fuzzy sets, specifically for the comparison of type-2 fuzzy membership degrees, to create function comparison operators. We further apply these operators to hyperspectral imaging, in which pixelwise data are provided as functions over a certain energy spectra. The performance of the functional comparison operators is put to the test in the context of in-laboratory hyperspectral image segmentation.
Open Access
Convolution lattices
(Elsevier, 2018) Miguel Turullols, Laura de; Bustince Sola, Humberto; Baets, Bernard de; Automatika eta Konputazioa; Institute of Smart Cities - ISC; Automática y Computación
We propose two convolution operations on the set of functions between two bounded lattices and investigate the algebraic structure they constitute, in particular the lattice laws they satisfy. Each of these laws requires the restriction to a specific subset of functions, such as normal, idempotent or convex functions. Combining all individual results, we identify the maximal subsets of functions resulting in a bounded lattice, and show this result to be equivalent to the distributivity of the lattice acting as domain of the functions. Furthermore, these lattices turn out to be distributive as well. Additionally, we show that for the larger subset of idempotent functions, although not satisfying the absorption laws, the convolution operations satisfy the Birkhoff equation.
Open Access
Type-2 fuzzy entropy-sets
(IEEE, 2017) Miguel Turullols, Laura de; Santos, Helida; Sesma Sara, Mikel; Bedregal, Benjamin; Jurío Munárriz, Aránzazu; Bustince Sola, Humberto; Automática y Computación; Automatika eta Konputazioa; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
The final goal of this study is to adapt the concept of fuzzy entropy of De Luca and Termini to deal with Type-2 Fuzzy Sets. We denote this concept Type-2 Fuzzy Entropy-Set. However, the construction of the notion of entropy measure on an infinite set, such us [0, 1], is not effortless. For this reason, we first introduce the concept of quasi-entropy of a Fuzzy Set on the universe [0, 1]. Furthermore, whenever the membership function of the considered Fuzzy Set in the universe [0, 1] is continuous, we prove that the quasi-entropy of that set is a fuzzy entropy in the sense of De Luca y Termini. Finally, we present an illustrative example where we use Type-2 Fuzzy Entropy-Sets instead of fuzzy entropies in a classical fuzzy algorithm.
Open Access
A framework for radial data comparison and its application to fingerprint analysis
(Elsevier, 2016) Marco Detchart, Cedric; Cerrón González, Juan; Miguel Turullols, Laura de; López Molina, Carlos; Bustince Sola, Humberto; Galar Idoate, Mikel; Automatika eta Konputazioa; Institute of Smart Cities - ISC; Automática y Computación; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
This work tackles the comparison of radial data, and proposes comparison measures that are further applied to fingerprint analysis. First, we study the similarity of scalar and non-scalar radial data, elaborated on previous works in fuzzy set theory. This study leads to the concepts of restricted radial equivalence function and Radial Similarity Measure, which model the perceived similarity between scalar and vectorial pieces of radial data, respectively. Second, the utility of these functions is tested in the context of fingerprint analysis, and more specifically, in the singular point detection. With this aim, a novel Template-based Singular Point Detection method is proposed, which takes advantage of these functions. Finally, their suitability is tested in different fingerprint databases. Different Similarity Measures are considered to show the flexibility offered by these measures and the behaviour of the new method is compared with well-known singular point detection methods.
Open Access
Strong negations and restricted equivalence functions revisited: an analytical and topological approach
(Elsevier, 2021) Bustince Sola, Humberto; Campión Arrastia, María Jesús; Miguel Turullols, Laura de; Induráin Eraso, Esteban; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
Throughout this paper, our main idea is to analyze the concepts of a strong negation and a restricted equivalence function, that appear in a natural way when dealing with theory and applications of fuzzy sets and fuzzy logic. Here we will use an analytical and topological approach, showing how to construct them in an easy way. In particular, we will also analyze some classical functional equation related to those key concepts.
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.