Open Access
On a total order on the set of Z-numbers based on discrete fuzzy numbers
(Springer, 2024-07-02) Mir Fuentes, Arnau; Miguel Turullols, Laura de; Massanet, Sebastia; Mir Torres, Arnau; Riera, Juan Vicente; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
Z-numbers were introduced by Zadeh in 2011 as a pair of fuzzy numbers (A, B), where A is interpreted as a fuzzy restriction on the values of a variable, while B is interpreted as a measure of certainty or sureness of A. From the initial proposal, several other approaches have been introduced in order to reduce the computational cost of the involved operations. One of such approaches is called discrete Z-numbers where A and B are modelled as discrete fuzzy numbers. In this paper, the construction of total orders on the set of discrete Z-numbers is investigated for the first time. Specifically, the total order is designed for discrete Z-numbers where the second component has membership values belonging to a finite and prefixed set of values. The method relies on solid and coherent linguistic criteria and several linguistic properties are analyzed. The order involves the transformation of the first components of the discrete Z-numbers by using the credibility of the second components in the sense that a lower credibility enlarges in a greater extent the uncertainty of the first component. Then a total order on the set of discrete fuzzy numbers is applied. Finally, a practical example on how to order discrete Z-numbers is presented and a comparison with other ranking methods is performed from which the strengths of our method are stressed.
Open Access
On the notion of fuzzy dispersion measure and its application to triangular fuzzy numbers
(Elsevier, 2023) Roldán López de Hierro, Antonio Francisco; Bustince Sola, Humberto; Rueda, María del Mar; Roldán, Concepción; Miguel Turullols, Laura de; Guerra Errea, Carlos; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
In this paper, based on the analysis of the most widely used dispersion measure in the real context (namely, the variance), we introduce the notion of fuzzy dispersion measure associated to a finite set of data given by fuzzy numbers. This measure is implemented as a fuzzy number, so there is no loss of information caused by any defuzzification. The proposed concept satisfies the usual properties in a genuinely fuzzy sense and it avoids limitations in terms of its geometric shape or its analytical properties: under this conception, it could have a piece of its support in the negative part of the real line. This novel notion can be interpreted as a way of fusing the information included in a fuzzy data set in order to make a decision based on its dispersion. To illustrate the main characteristics of this approach, we present an example of a fuzzy dispersion measure that allows to conclude that this new way to deal this problem is coherent, at least, from the point of view of human intuition.
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.
Embargo
Non-symmetric over-time pooling using pseudo-grouping functions for convolutional neural networks
(Elsevier, 2024) Ferrero Jaurrieta, Mikel; Paiva, Rui; Cruz, Anderson; Bedregal, Benjamin; Miguel Turullols, Laura de; Takáč, Zdenko; López Molina, Carlos; Bustince Sola, Humberto; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC
Convolutional Neural Networks (CNNs) are a family of networks that have become state-of-the-art in several fields of artificial intelligence due to their ability to extract spatial features. In the context of natural language processing, they can be used to build text classification models based on textual features between words. These networks fuse local features to generate global features in their over-time pooling layers. These layers have been traditionally built using the maximum function or other symmetric functions such as the arithmetic mean. It is important to note that the order of input local features is significant (i.e. the symmetry is not an inherent characteristic of the model). While this characteristic is appropriate for image-oriented CNNs, where symmetry might make the network robust to image rigid transformations, it seems counter-productive for text processing, where the order of the words is certainly important. Our proposal is, hence, to use non-symmetric pooling operators to replace the maximum or average functions. Specifically, we propose to perform over-time pooling using pseudo-grouping functions, a family of non-symmetric aggregation operators that generalize the maximum function. We present a construction method for pseudo-grouping functions and apply different examples of this family to over-time pooling layers in text-oriented CNNs. Our proposal is tested on seven different models and six different datasets in the context of engineering applications, e.g. text classification. The results show an overall improvement of the models when using non-symmetric pseudo-grouping functions over the traditional pooling function.
Open Access
Neuro-inspired edge feature fusion using Choquet integrals
(Elsevier, 2021) Marco Detchart, Cedric; Lucca, Giancarlo; López Molina, Carlos; Miguel Turullols, Laura de; Pereira Dimuro, Graçaliz; Bustince Sola, Humberto; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
It is known that the human visual system performs a hierarchical information process in which early vision cues (or primitives) are fused in the visual cortex to compose complex shapes and descriptors. While different aspects of the process have been extensively studied, such as lens adaptation or feature detection, some other aspects, such as feature fusion, have been mostly left aside. In this work, we elaborate on the fusion of early vision primitives using generalizations of the Choquet integral, and novel aggregation operators that have been extensively studied in recent years. We propose to use generalizations of the Choquet integral to sensibly fuse elementary edge cues, in an attempt to model the behaviour of neurons in the early visual cortex. Our proposal leads to a fully-framed edge detection algorithm whose performance is put to the test in state-of-the-art edge detection datasets.
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
Distance transformations based on ordered weighted averaging operators
(University of Hawaii Press, 2021) López Molina, Carlos; Miguel Turullols, Laura de; Iglesias Rey, Sara; Bustince Sola, Humberto; Baets, Bernard de; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika
Binary image comparison has been a study subject for a long time, often rendering in context-specific solutions that depend upon the type of visual contents in the binary images. Distance transformations have been a recurrent tool in many of such solutions. The literature contains works on the generation and definition of distance transformations, but also on how to make a sensible use of their results. In this work, we attempt to solve one of the most critical problems in the application of distance transformations to real problems: their oversensitivity to certain spurious pixels which, even if having a minimal visual impact in the binary images to be compared, may have a severe impact on their distance transforms. With this aim, we combine distance transformations with Ordered Weighted Averaging (OWA) operators, a well-known information fusion tool from Fuzzy Set Theory.