Embargo
Non-symmetric over-time pooling using pseudo-grouping functions for convolutional neural networks
(Elsevier, 2024-07-01) 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
Admissible orders on fuzzy numbers
(IEEE, 2022) Zumelzu, Nicolás; Bedregal, Benjamin; Mansilla, Edmundo; Bustince Sola, Humberto; Díaz, Roberto; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas
From the more than two hundred partial orders for fuzzy numbers proposed in the literature, only a few are total. In this paper, we introduce the notion of admissible order for fuzzy numbers equipped with a partial order, i.e. a total order which refines the partial order. In particular, it is given special attention to the partial order proposed by Klir and Yuan in 1995. Moreover, we propose a method to construct admissible orders on fuzzy numbers in terms of linear orders defined for intervals considering a strictly increasing upper dense sequence, proving that this order is admissible for a given partial order. Finally, we use admissible orders to ranking the path costs in fuzzy weighted graphs. IEEE
Open Access
d-CC integrals: generalizing CC-integrals by restricted dissimilarity functions with applications to fuzzy-rule based systems
(Springer, 2023) Sartori, Joelson; Asmus, Tiago da Cruz; Santos, Helida; Borges, Eduardo N.; Bustince Sola, Humberto; Dimuro, Graçaliz Pereira; Lucca, Giancarlo; Bedregal, Benjamin; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC
The discrete Choquet Integral (CI) and its generalizations have been successfully applied in many different fields, with particularly good results when considered in Fuzzy Rule-Based Classification Systems (FRBCSs). One of those functions is the CC-integral, where the product operations in the expanded form of the CI are generalized by copulas. Recently, some new Choquet-like operators were developed by generalizing the difference operation by a Restricted Dissimilarity Function (RDF) in either the usual or the expanded form of the original CI, also providing good results in practical applications. So, motivated by such developments, in this paper we propose the generalization of the CC-integral by means of RDFs, resulting in a function that we call d-CC-integral. We study some relevant properties of this new definition, focusing on its monotonicity-like behavior. Then, we proceed to apply d-CC-integrals in a classification problem, comparing different d-CC-integrals between them. The classification acuity of the best d-CC-integral surpasses the one achieved by the best CC-integral and is statistically equivalent to the state-of-the-art in FRBCSs.
Open Access
A proposal for tuning the α parameter in CαC-integrals for application in fuzzy rule-based classification systems
(Springer, 2020) Lucca, Giancarlo; Sanz Delgado, José Antonio; Pereira Dimuro, Graçaliz; Bedregal, Benjamin; Bustince Sola, Humberto; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas
In this paper, we consider the concept of extended Choquet integral generalized by a copula, called CC-integral. In particular, we adopt a CC-integral that uses a copula defined by a parameter α, which behavior was tested in a previous work using different fixed values. In this contribution, we propose an extension of this method by learning the best value for the parameter α using a genetic algorithm. This new proposal is applied in the fuzzy reasoning method of fuzzy rule-based classification systems in such a way that, for each class, the most suitable value of the parameter α is obtained, which can lead to an improvement on the system's performance. In the experimental study, we test the performance of 4 different so called CαC-integrals, comparing the results obtained when using fixed values for the parameter α against the results provided by our new evolutionary approach. From the obtained results, it is possible to conclude that the genetic learning of the parameter α is statistically superior than the fixed one for two copulas. Moreover, in general, the accuracy achieved in test is superior than that of the fixed approach in all functions. We also compare the quality of this approach with related approaches, showing that the methodology proposed in this work provides competitive results. Therefore, we demonstrate that CαC-integrals with α learned genetically can be considered as a good alternative to be used in fuzzy rule-based classification systems.
Open Access
Improving the performance of fuzzy rule-based classification systems based on a nonaveraging generalization of CC-integrals named C-F1F2-integrals
(IEEE, 2019) Lucca, Giancarlo; Pereira Dimuro, Graçaliz; Fernández Fernández, Francisco Javier; Bustince Sola, Humberto; Bedregal, Benjamin; Sanz Delgado, José Antonio; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas
A key component of fuzzy rule-based classification systems (FRBCS) is the fuzzy reasoning method (FRM) since it infers the class predicted for new examples. A crucial stage in any FRM is the way in which the information given by the fired rules during the inference process is aggregated. A widely used FRM is the winning rule, which applies the maximum to accomplish this aggregation. The maximum is an averaging operator, which means that its result is within the range delimited by the minimum and the maximum of the aggregated values. Recently, new averaging operators based on generalizations of the Choquet integral have been proposed to perform this aggregation process. However, the most accurate FRBCSs use the FRM known as additive combination that considers the normalized sum as the aggregation operator, which is nonaveraging. For this reason, this paper is aimed at introducing a new nonaveraging operator named C-F1F2-integral, which is a generalization of the Choquet-like Copula-based integral (CC-integral). C-F1F2-integrals present the desired properties of an aggregation-like operator since they satisfy appropriate boundary conditions and have some kind of increasingness property. We show that C-F1F2 -integrals, when used to cope with classification problems, enhance the results of the previous averaging generalizations of the Choquet integral and provide competitive results (even better) when compared with state-of-the-art FRBCSs.
Open Access
The law of O-conditionality for fuzzy implications constructed from overlap and grouping functions
(Elsevier, 2019) Pereira Dimuro, Graçaliz; Bedregal, Benjamin; Fernández Fernández, Francisco Javier; Sesma Sara, Mikel; Pintor Borobia, Jesús María; Bustince Sola, Humberto; Estatistika, Informatika eta Matematika; Ingeniaritza; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas; Ingeniería
Overlap and grouping functions are special kinds of non necessarily associative aggregation operators proposed for many applications, mainly when the associativity property is not strongly required. The classes of overlap and grouping functions are richer than the classes of t-norms and t-conorms, respectively, concerning some properties like idempotency, homogeneity, and, mainly, the self-closedness feature with respect to the convex sum and the aggregation by generalized composition of overlap/grouping functions. In previous works, we introduced some classes of fuzzy implications derived by overlap and/or grouping functions, namely, the residual implications R-0-implications, the strong implications (G, N)-implications and the Quantum Logic implications QL-implications, for overlap functions O, grouping functions G and fuzzy negations N. Such implications do not necessarily satisfy certain properties, but only weaker versions of these properties, e.g., the exchange principle. However, in general, such properties are not demanded for many applications. In this paper, we analyze the so-called law of O-Conditionality, O(x, 1(x, y)) <= y, for any fuzzy implication I and overlap function O, and, in particular, for Ro-implications, (G, N)-implications, QL-implications and D-implications derived from tuples (O, G, N), the latter also introduced in this paper. We also study the conditional antecedent boundary condition for such fuzzy implications, since we prove that this property, associated to the left ordering property, is important for the analysis of the O-Conditionality. We show that the use of overlap functions to implement de generalized Modus Ponens, as the scheme enabled by the law of O-Conditionality, provides more generality than the laws of T-conditionality and U-conditionality, for t-norms T and uninorms U, respectively.
Open Access
Application of two different methods for extending lattice-valued restricted equivalence functions used for constructing similarity measures on L-fuzzy sets
(Elsevier, 2018) Palmeira, Eduardo S.; Bedregal, Benjamin; Bustince Sola, Humberto; Paternain Dallo, Daniel; Miguel Turullols, Laura de; Automatika eta Konputazioa; Institute of Smart Cities - ISC; Automática y Computación; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
Based on previous investigations, we have proposed two different methods to extend lattice-valued fuzzy connectives (t-norms, t-conorms, negations and implications) and other related operators, considering a generalized notion of sublattices. Taking into account the results obtained and seeking to analyze the behavior of both extension methods in face of fuzzy operators related to image processing, we have applied these methods so as to extend restricted equivalence functions, restricted dissimilarity functions and Ee,N-normal functions. We also generalize the concepts of similarity measure, distance measure and entropy measure for L-fuzzy sets constructing them via restricted equivalence functions, restricted dissimilarity functions and Ee,N-normal functions
Open Access
A historical account of types of fuzzy sets and their relationships
(IEEE, 2016) Bustince Sola, Humberto; Barrenechea Tartas, Edurne; Pagola Barrio, Miguel; Fernández Fernández, Francisco Javier; Xu, Zeshui; Bedregal, Benjamin; Montero, Javier; Hagras, Hani; Herrera, Francisco; Baets, Bernard de; Automatika eta Konputazioa; Institute of Smart Cities - ISC; Automática y Computación
In this work we review the definition and basic properties of the different types of fuzzy sets that have appeared up to now in the literature. We also analyze the relationships between them and enumerate some of the applications in which they have been used.
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
Towards interval uncertainty propagation control in bivariate aggregation processes and the introduction of width-limited interval-valued overlap functions
(Elsevier, 2021) Da Cruz Asmus, Tiago; Pereira Dimuro, Graçaliz; Bedregal, Benjamin; Sanz Delgado, José Antonio; 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
Overlap functions are a class of aggregation functions that measure the overlapping degree between two values. They have been successfully applied as a fuzzy conjunction operation in several problems in which associativity is not required, such as image processing and classification. Interval-valued overlap functions were defined as an extension to express the overlapping of interval-valued data, and they have been usually applied when there is uncertainty regarding the assignment of membership degrees, as in interval-valued fuzzy rule-based classification systems. In this context, the choice of a total order for intervals can be significant, which motivated the recent developments on interval-valued aggregation functions and interval-valued overlap functions that are increasing to a given admissible order, that is, a total order that refines the usual partial order for intervals. Also, width preservation has been considered on these recent works, in an intent to avoid the uncertainty increase and guarantee the information quality, but no deeper study was made regarding the relation between the widths of the input intervals and the output interval, when applying interval-valued functions, or how one can control such uncertainty propagation based on this relation. Thus, in this paper we: (i) introduce and develop the concepts of width-limited interval-valued functions and width limiting functions, presenting a theoretical approach to analyze the relation between the widths of the input and output intervals of bivariate interval-valued functions, with special attention to interval-valued aggregation functions; (ii) introduce the concept of (a,b)-ultramodular aggregation functions, a less restrictive extension of one-dimension convexity for bivariate aggregation functions, which have an important predictable behaviour with respect to the width when extended to the interval-valued context; (iii) define width-limited interval-valued overlap functions, taking into account a function that controls the width of the output interval and a new notion of increasingness with respect to a pair of partial orders (≤1,≤2); (iv) present and compare three construction methods for these width-limited interval-valued overlap functions, considering a pair of orders (≤1,≤2), which may be admissible or not, showcasing the adaptability of our developments.