Open Access
On the normalization of interval data
(MDPI, 2020) Santiago, Regivan; Bergamaschi, Flaulles; Bustince Sola, Humberto; Pereira Dimuro, Graçaliz; Da Cruz Asmus, Tiago; Sanz Delgado, José Antonio; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
The impreciseness of numeric input data can be expressed by intervals. On the other hand, the normalization of numeric data is a usual process in many applications. How do we match the normalization with impreciseness on numeric data? A straightforward answer is that it is enough to apply a correct interval arithmetic, since the normalized exact value will be enclosed in the resulting 'normalized' interval. This paper shows that this approach is not enough since the resulting 'normalized' interval can be even wider than the input intervals. So, we propose a pair of axioms that must be satisfied by an interval arithmetic in order to be applied in the normalization of intervals. We show how some known interval arithmetics behave with respect to these axioms. The paper ends with a discussion about the current paradigm of interval computations.
Open Access
Abstract homogeneous functions and consistently influenced/disturbed multi-expert decision making
(IEEE, 2021) Santiago, Regivan; Bedregal, Benjamin; Pereira Dimuro, Graçaliz; Fernández Fernández, Francisco Javier; Bustince Sola, Humberto; Fardoun, Habib; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
In this paper we propose a new generalization for the notion of homogeneous functions. We show some properties and how it appears in some scenarios. Finally we show how this generalization can be used in order to provide a new paradigm for decision making theory called consistent influenced/disturbed decision making. In order to illustrate the applicability of this new paradigm, we provide a toy example.
Open Access
Aggregation of individual rankings through fusion functions: criticism and optimality analysis
(IEEE, 2020) Bustince Sola, Humberto; Bedregal, Benjamin; Campión Arrastia, María Jesús; Silva, Ivanoska da; Fernández Fernández, Francisco Javier; Induráin Eraso, Esteban; Raventós Pujol, Armajac; Santiago, Regivan; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Estadística, Informática y Matemáticas
Throughout this paper, our main idea is to analyze from a theoretical and normative point of view different methods to aggregate individual rankings. To do so, first we introduce the concept of a general mean on an abstract set. This new concept conciliates the social choice where well-known impossibility results as the Arrovian ones are encountered and the decision-making approaches where the necessity of fusing rankings is unavoidable. Moreover it gives rise to a reasonable definition of the concept of a ranking fusion function that does indeed satisfy the axioms of a general mean. Then we will introduce some methods to build ranking fusion functions, paying a special attention to the use of score functions, and pointing out the equivalence between ranking and scoring. To conclude, we prove that any ranking fusion function introduces a partial order on rankings implemented on a finite set of alternatives. Therefore, this allows us to compare rankings and different methods of aggregation, so that in practice one should look for the maximal elements with respect to such orders defined on rankings IEEE.
Embargo
Fuzzy dissimilarities and the fuzzy choquet integral of triangular fuzzy numbers on [0,1]
(Elsevier, 2025-04-01) Roldán López de Hierro, Antonio Francisco; Cruz, Anderson; Santiago, Regivan; Roldán, Concepción; García-Zamora, Diego; Neres, Fernando; Bustince Sola, Humberto; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC
Having in mind the huge amount of data daily registered in the world, it is becoming increasingly important to summarize the information included in a data set. In Statistics and Computer Science, this task is successfully carried out by aggregation functions. One of the most widely applied methodologies of aggregating data is the Choquet integral. The main aim of this paper is to introduce an appropriate notion of Choquet integral in the context of fuzzy numbers. To do this, we face three challenges: the underlying uncertainty when handling fuzzy numbers, the way to order fuzzy numbers by appropriate binary relations and the way to compute the dissimilarity among fuzzy numbers. Illustrative examples are given by involving the α-order on the family of all triangular fuzzy numbers with support on [0,1].
Open Access
On fuzzy implications derived from general overlap functions and their relation to other classes
(MDPI, 2023) Pinheiro, Jocivania; Santos, Helida; Pereira Dimuro, Graçaliz; Bedregal, Benjamin; Santiago, Regivan; Fernández Fernández, Francisco Javier; Bustince Sola, Humberto; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC
There are distinct techniques to generate fuzzy implication functions. Despite most of them using the combination of associative aggregators and fuzzy negations, other connectives such as (general) overlap/grouping functions may be a better strategy. Since these possibly non-associative operators have been successfully used in many applications, such as decision making, classification and image processing, the idea of this work is to continue previous studies related to fuzzy implication functions derived from general overlap functions. In order to obtain a more general and flexible context, we extend the class of implications derived by fuzzy negations and t-norms, replacing the latter by general overlap functions, obtaining the so-called (GO, N)-implication functions. We also investigate their properties, the aggregation of (GO, N)-implication functions, their characterization and the intersections with other classes of fuzzy implication functions.
Open Access
Extension of restricted equivalence functions and similarity measures for type-2 fuzzy sets
(IEEE, 2021) Miguel Turullols, Laura de; Santiago, Regivan; Wagner, Christian; Garibaldi, Jonathan M.; Takáč, Zdenko; Roldán López de Hierro, Antonio Francisco; 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 generalize the notion of restricted equivalence function for type-2 fuzzy sets, leading to the notion of extended restricted equivalence functions. We also study how under suitable conditions, these new functions recover the standard axioms for restricted equivalence functions in the real setting. Extended restricted equivalence functions allow us to compare any two general type-2 fuzzy sets and to generate a similarity measure for type-2 fuzzy sets. The result of this similarity is a fuzzy set on the same referential set (i.e., domain) as the considered type-2 fuzzy set. The latter is crucial for applications such as explainable AI and decision making, as it enables an intuitive interpretation of the similarity within the domain-specific context of the fuzzy sets. We show how this measure can be used to compare type-2 fuzzy sets with different membership functions in such a way that the uncertainty linked to type-2 fuzzy sets is not lost. This is achieved by generating a fuzzy set rather than a single numerical value. Furthermore, we also show how to obtain a numerical value for discrete referential sets.
Open Access
F-homogeneous functions and a generalization of directional monotonicity
(Wiley, 2022) Santiago, Regivan; Sesma Sara, Mikel; Fernández Fernández, Francisco Javier; Takáč, Zdenko; Mesiar, Radko; Bustince Sola, Humberto; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
A function that takes (Formula presented.) numbers as input and outputs one number is said to be homogeneous whenever the result of multiplying each input by a certain factor (Formula presented.) yields the original output multiplied by that same factor. This concept has been extended by the notion of abstract homogeneity, which generalizes the product in the expression of homogeneity by a general function (Formula presented.) and the effect of the factor (Formula presented.) by an automorphism. However, the effect of parameter (Formula presented.) remains unchanged for all the input values. In this study, we generalize further the condition of abstract homogeneity by introducing (Formula presented.) -homogeneity, which is defined with respect to a family of functions, enabling a different behavior for each of the inputs. Next, we study the properties that are satisfied by this family of functions and, moreover, we link this concept with the condition of directional monotonicity, which is a trendy property in the framework of aggregation functions. To achieve that, we generalize directional monotonicity by (Formula presented.) directional monotonicity, which is defined with respect to a family of functions (Formula presented.) and a family of vectors (Formula presented.). Finally, we show how the introduced concepts could be applied in two different problems of computer vision: a snow detection problem and image thresholding improvement. © 2022 The Authors. International Journal of Intelligent Systems published by Wiley Periodicals LLC.
Open Access
Replacing pooling functions in convolutional neural networks by linear combinations of increasing functions
(Elsevier, 2022) Rodríguez Martínez, Iosu; Lafuente López, Julio; Santiago, Regivan; Pereira Dimuro, Graçaliz; Herrera, Francisco; Bustince Sola, Humberto; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Gobierno de Navarra / Nafarroako Gobernua
Traditionally, Convolutional Neural Networks make use of the maximum or arithmetic mean in order to reduce the features extracted by convolutional layers in a downsampling process known as pooling. However, there is no strong argument to settle upon one of the two functions and, in practice, this selection turns to be problem dependent. Further, both of these options ignore possible dependencies among the data. We believe that a combination of both of these functions, as well as of additional ones which may retain different information, can benefit the feature extraction process. In this work, we replace traditional pooling by several alternative functions. In particular, we consider linear combinations of order statistics and generalizations of the Sugeno integral, extending the latter¿s domain to the whole real line and setting the theoretical base for their application. We present an alternative pooling layer based on this strategy which we name ¿CombPool¿ layer. We replace the pooling layers of three different architectures of increasing complexity by CombPool layers, and empirically prove over multiple datasets that linear combinations outperform traditional pooling functions in most cases. Further, combinations with either the Sugeno integral or one of its generalizations usually yield the best results, proving a strong candidate to apply in most architectures.
Open Access
Fuzzy sets complement-based gated recurrent unit
(CEUR Workshop Proceedings (CEUR-WS.org), 2021) Ferrero Jaurrieta, Mikel; Pereira Dimuro, Graçaliz; Takáč, Zdenko; Santiago, Regivan; Fernández Fernández, Francisco Javier; Bustince Sola, Humberto; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Gobierno de Navarra / Nafarroako Gobernua
Gated Recurrent Units (GRU) are neural network gated architectures that simplify other ones (suchas, LSTM) by joining gates mainly. For this, instead of using two gates, if𝑥is the first gate, standardoperation1−𝑥is used to generate the second one, optimizing the number of parameters. In this work, we interpret this information as a fuzzy set, and we generalize the standard operation using fuzzy negations, and improving the accuracy obtained with the standard one.
Open Access
Admissible OWA operators for fuzzy numbers
(Elsevier, 2024) García-Zamora, Diego; Cruz, Anderson; Neres, Fernando; Santiago, Regivan; Roldán López de Hierro, Antonio Francisco; Paiva, Rui; Pereira Dimuro, Graçaliz; Martínez López, Luis; Bedregal, Benjamin; Bustince Sola, Humberto; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC
Ordered Weighted Averaging (OWA) operators are some of the most widely used aggregation functions in classic literature, but their application to fuzzy numbers has been limited due to the complexity of defining a total order in fuzzy contexts. However, the recent notion of admissible order for fuzzy numbers provides an effective method to totally order them by refining a given partial order. Therefore, this paper is devoted to defining OWA operators for fuzzy numbers with respect to admissible orders and investigating their properties. Firstly, we define the OWA operators associated with such admissible orders and then we show their main properties. Afterward, an example is presented to illustrate the applicability of these AOWA operators in linguistic decision-making. In this regard, we also develop an admissible order for trapezoidal fuzzy numbers that can be efficiently applied in practice.