Open Access
Pre-aggregation functions: construction and an application
(IEEE, 2015) Lucca, Giancarlo; Sanz Delgado, José Antonio; Pereira Dimuro, Graçaliz; Bedregal, Benjamin; Mesiar, Radko; Kolesárová, Anna; Bustince Sola, Humberto; Automática y Computación; Automatika eta Konputazioa
In this work we introduce the notion of preaggregation function. Such a function satisfies the same boundary conditions as an aggregation function, but, instead of requiring monotonicity, only monotonicity along some fixed direction (directional monotonicity) is required. We present some examples of such functions. We propose three different methods to build pre-aggregation functions. We experimentally show that in fuzzy rule-based classification systems, when we use one of these methods, namely, the one based on the use of the Choquet integral replacing the product by other aggregation functions, if we consider the minimum or the Hamacher product t-norms for such construction, we improve the results obtained when applying the fuzzy reasoning methods obtained using two classical averaging operators like the maximum and the Choquet integral.
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 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
Interval-valued Atanassov intuitionistic OWA aggregations using admissible linear orders and their application to decision making
(IEEE, 2016) Miguel Turullols, Laura de; Bustince Sola, Humberto; Pekala, Barbara; Bentkowska, Urszula; Silva, Ivanoska da; Bedregal, Benjamin; Mesiar, Radko; Ochoa Lezaun, Gustavo; Automatika eta Konputazioa; Matematika; Institute of Smart Cities - ISC; Automática y Computación; Matemáticas
Based on the definition of admissible order for interval-valued Atanassov intuitionistic fuzzy sets, we study OWA operators in these sets distinguishing between the weights associated to the membership and those associated to the nonmembership degree which may differ from the latter. We also study Choquet integrals for aggregating information which is represented using interval-valued Atanassov intuitionistic fuzzy sets. We conclude with two algorithms to choose the best alternative in a decision making problem when we use this kind of sets to represent information.
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
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.