### Browsing by Author "Bustince, H."

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Publication A class of aggregation functions encompassing two-dimensional OWA operators(Elsevier Science Inc, 2010) Bustince, H.; Calvo, T.; De Baets, B.; Fodor, J.; Mesiar, R.; Montero, Javier; Paternain, D.; Pradera, A.In this paper we prove that, under suitable conditions, Atanassov’s Ka operators, which act on intervals, provide the same numerical results as OWA operators of dimension two. On one hand, this allows us to recover OWA operators from Ka operators. On the other hand, by analyzing the properties of Atanassov’s operators, we can generalize them. In this way, we introduce a class of aggregation functions – the generalized Atanassov operators – that,in particular, include two-dimensional OWA operators. We investigate under which conditions these generalized Atanassov operators satisfy some properties usually required for aggregation functions, such as bisymmetry, strictness, monotonicity, etc. We also show that if we apply these aggregation functions to interval-valued fuzzy sets, we obtain an ordered family of fuzzy sets.Publication A generalization of the migrativity property of aggregation functions(Elsevier Science Inc, 2012) Bustince, H.; De Baets, B.; Fernande, J.; Mesiar, R.; Montero, JavierThis paper brings a generalization of the migrativity property of aggregation functions, suggested in earlier work of some of the present authors by imposing the a-migrativity property of Durante and Sarkoci for all values of a instead of a single one. Replacing the algebraic product by an arbitrary aggregation function B naturally leads to the properties of a–B-migrativity and B-migrativity. This generalization establishes a link between migrativity and a particular case of Aczel’s general associativity equation, already considered by Cutello and Montero as a recursive formula for aggregation. Following a basic investigation, emphasis is put on aggregation functions that can be represented in terms of an additive generator, more specifically, strict t-norms, strict t-conorms and representable uninorms.Publication A historical account of types of fuzzy sets and their relationships(Institute of Electrical and Electronics Engineers Inc., 2016) Bustince, H.; Barrenechea, Edurne; Pagola, M.; Fernández, J.; Xu, Z.; Bedregal, B.; Montero, Javier; Hagras, H.; Herrera, F.; De Baets, B.In this paper, 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.Publication A Survey of Atanassov’s Intuitionistic Fuzzy Relations.(Springer International Publishing, 2016) Bustince, H.; Barrenechea, Edurne; Pagola, M.; Fernández, J.; Orduna, R.; Montero, Javier; Calvo Sanchez, Tomasa; Torrens Sastre, JoanIn this chapter we review several properties of Atanassov’s intuitionistic fuzzy relations, recalling the main concepts related to Atanassov’s intuitionistic fuzzy relations and the main properties that can be demanded to such conepts. We also consider the use of Atanassov’s operators over such relations.Publication Atanassov’s Intuitionistic Fuzzy Sets as a Classification Model.(Springer, 2007-06-18) Montero, Javier; Gómez, Daniel; Bustince, H.In this paper we show that Atanassov's Intuitionistic Fuzzy sets can be viewed as a classification model, that can be generalized in order to take into account more classes than the three classes considered by Atanassov's (membership, non-membership and non-determinacy). This approach will imply, on one hand, to change the meaning of these classes, so each one will have a positive definition. On the other hand, this approach implies the possibility of a direct generalization for alternative logics and additional valuation states, being consistent with Atanassov's focuss. From this approach we shall stress the absence of any structure within those three valuation states in Atanassov's model. In particular, we consider this is the main cause of the dispute about Atanassov's model: acknowledging that the name intuitionistic is not appropriate, once we consider that a crisp direct graph is defined in the valuation space, formal differences with other three-state models will appear.Publication Construction of Capacities from Overlap Indexes(Springer, 2017) Sanz, José Antonio; Galar, Mikel; Mesiar, Radko; Bustince, H.; Fernandez, Javier; Montero, Javier; Torra, Vicenç; Dahlbom, Anders; Narukawa, YasuoIn many problems, it is crucial to find a relation between groups of data. Such relation can be expressed, for instance, in terms of an appropriate fuzzy measure or capacity([10, 21]) which reflects the way the different data are connected to each other [20]. In this chapter, taking into account this fact and following the developments in [8],we introduce a method to build capacities ([20, 21]) directly from the data (inputs) of a given problem. In order to do so, we make use of the notions of overlap function and overlap index ([5, 12, 13, 7, 4, 14, 16]) for constructing capacities which reflect how different data are related to each other. This paper is organized as follows: after providing some preliminaries, we analyse, in Section 3, some properties of overlap functions and indexes. In Sections 4 we discuss a method for constructing capacities from overlap functions and overlap indexes. Finally, we present some conclusions and references.Publication Contrast of a fuzzy relation.(Elsevier Science Inc, 2010) Bustince, H.; Barrenechea, E.; Fernández, J.; Pagola, M.; Montero, Javier; Guerra, C.In this paper we address a key problem in many fields: how a structured data set can be analyzed in order to take into account the neighborhood of each individual datum. We propose representing the dataset as a fuzzy relation, associating a membership degree with each element of the relation. We then introduce the concept of interval-contrast, a means of aggregating information contained in the immediate neighborhood of each element of the fuzzy relation. The interval-contrast measures the range of membership degrees present in each neighborhood. We use interval-contrasts to define the necessary properties of a contrast measure, construct several different local contrast and total contrast measures that satisfy these properties, and compare our expressions to other definitions of contrast appearing in the literature. Our theoretical results can be applied to several different fields. In an Appendix A, we apply our contrast expressions to photographic images.Publication From Trillas’ Negations and Antonyms to a Set Representation of Contradiction Within Bipolar and Other Extensions of Fuzzy Sets(Springer International Publishing, 2015) Bustince, H.; Barrenechea, Edurne; Burusco, Ana; Fernandez, Javier; Rodríguez, Juan Tinguaro; Montero, Javier; Pagola, Miguel; Gómez, D.; Seising, RudolfIn 1979, Enric Trillas started to interest in in fuzzy connectives. His first paper on this topic was ”Funciones de negacin en la teora de subconjuntos difusos” ([9]) (Negation functions in the theory of fuzzy subsets), which appeared in Spanish in the Stochastica journal. This work, focused on the characterization of strong negations, has been so relevant for the development of fuzzy theory that it was translated into English and widely cited in the last 35 yearsPublication Generation of Interval-Valued Fuzzy Negations from Trillas’ Theorem. The Case of Interval Type-2 Fuzzy Sets(Springer International Publishing, 2015) Bustince, H.; Barrenechea, E.; Fernández, J.; Pagola, M.; Montero, Javier; Magdalena, Luis; Verdegay, José Luis; Esteva, FrancescIn this work we introduce a method for building interval-valued negations using the characterization theorem for strong negations which was proposed by Trillas in 1979. We also show that interval type-2 fuzzy sets are a three dimensional representation of interval-valued fuzzy sets and we analyze the problems to build complementation for such interval type-2 fuzzy sets. We finally propose a method to construct this complementation.Publication Ignorance functions. An application to the calculation of the threshold in prostate ultrasound images.(Elsevier Science Bv, 2010) Bustince, H.; Pagola, M.; Fernandez, J.; Melo-Pinto, P.; Couto, P.; Tizhoosh, H.R.; Montero, JavierIn this paper, we define the concept of an ignorance function and use it to determine the best threshold with which to binarize an image. We introduce a method to construct such functions from t-norms and automorphisms. By means of these new measures, we represent the degree of ignorance of the expert when given one fuzzy set to represent the background and another to represent the object. From this ignorance degree, we assign interval-valued fuzzy sets to the image in such a way that the best threshold is given by the interval-valued fuzzy set with the lowest associated ignorance. We prove that the proposed method provides better thresholds than the fuzzy classical methods when applied to transrectal prostateultrasoundimages. The experimental results on ultrasound and natural images also allow us to determine the best choice of the function to represent the ignorance.Publication Laws for Conjunctions and Disjunctions in Interval Type 2 Fuzzy Sets(IEEE, 2008-06-01) Bustince, H.; Montero, Javier; Barrenechea, E.; Pagola, M.In this paper we study in depth certain properties of interval type 2 fuzzy sets. In particular we recall a method to construct different interval type 2 fuzzy connectives starting from an operator. We further study the law of contradiction and the law of excluded middle for these sets. Furthermore we analyze the properties: idempotency, absorption, and distributiveness.Publication Migrativity of aggregation functions.(Elsevier Science Bv, 2009) Bustince, H.; Montero, Javier; Mesiar, R.We introduce a slight modification of the definition of migrativity for aggregation functions that allows useful characterization of this property. Among other things, in this context we prove that there are no t-conorms, uninorms or nullnorms that satisfy migrativity (with the product being the only migrative t-norm, as already shown by other authors) and that the only migrative idempotent aggregation function is the geometric mean. The k-Lipschitz migrative aggregation functions are also characterized and the product is shown to be the only 1-Lipschitz migrative aggregation function. Similarly, it is the only associative migrative aggregation function possessing a neutral element. Finally, the associativity and bisymmetry of migrative aggregation functions are discussed.Publication n-Dimensional overlap functions(Elsevier, 2014) Gómez, Daniel; Rodríguez, Juan Tinguaro; Montero, Javier; Bustince, H.; Barrenechea, E.In this paper we introduce the definition of n-dimensional overlap functions and we justify the axiomatization proposed in its definition. Basically, these functions allow to measure the degree of overlapping of several classes in a given classification system and for any given object. We also show a construction method for this class of functions, studying its relationships with the properties of migrativity, homogeneity and Lipschitz continuity. Finally, we propose an example where the use of n-dimensional overlap functions provides better results than those obtained with the commonly used product t-norm.Publication On the relevance of some families of fuzzy Sets.(Elsevier Science BV, 2007) Montero, Javier; Gómez, D.; Bustince, H.In this paper we stress the relevance of a particular family of fuzzy sets, where each element can be viewed as the result of a classification problem. In particular, we assume that fuzzy sets are defined from a well-defined universe of objects into a valuation space where a particular graph is being defined, in such a way that each element of the considered universe has a degree of membership with respect to each state in the valuation space. The associated graph defines the structure of such a valuation space, where an ignorance state represents the beginning of a necessary learning procedure. Hence, every single state needs a positive definition, and possible queries are limited by such an associated graph. We then allocate this family of fuzzy sets with respect to other relevant families of fuzzy sets, and in particular with respect to Atanassov's intuitionistic fuzzy sets. We postulate that introducing this graph allows a natural explanation of the different visions underlying Atanassov's model and interval valued fuzzy sets, despite both models have been proven equivalent when such a structure in the valuation space is not assumed.Publication Overlap functions and homogeneity(Edicions UIB, 2009) Bustince, H.; Mesiar, R.; Montero, Javier; Fernandez, J. F.; González, M.; Mayor, G.; Suñer, J.; Torrens, J.In this work we study some properties of overlap functions from an analytical point of view. We study the relation between overlap functions and continuous t-norms. We focus on homogeneous overlap functions and we get a characterization of them. Finally we completely identify homogeneous migrative overlap functions.Publication Overlap functions.(Elsevier, 2010) Bustince, H.; Fernandez, J.; Mesiar, R.; Montero, Javier; Orduna, RIn this paper we address a key issue in scenario classification, where classifying concepts show a natural overlapping. In fact, overlapping needs to be evaluated whenever classes are not crisp, in order to be able to check if a certain classification structure fits reality and still can be useful for our declared decision making purposes. In this paper we address an object recognition problem, where the best classification with respect to background is the one with less overlapping between the class object and the class background. In particular, in this paper we present the basic properties that must be fulfilled by overlap functions, associated to the degree of overlapping between two classes. In order to define these overlap functions we take as reference properties like migrativity, homogeneity of order 1 and homogeneity of order 2. Hence we define overlap functions, proposing a construction method and analyzing the conditions ensuring that t-norms are overlap functions. In addition, we present a characterization of migrative and strict overlap functions by means of automorphisms.Publication Overlap index, overlap functions and migrativity(European Society of Fuzzy Logic and Technology, 2009-07-20) Bustince, H.; Fernandez, J.; Mesiar, R.; Montero, Javier; Orduna, R; Carvalho, Joao Paulo; Dubois, Didier; Kaymak, Uzay; Costa Sousa, Joao Miguel doIn this work we study overlap degrees expressed in terms of overlap functions. We present the basic properties that from our point of view must satisfy these overlap functions. We study a construction method, we analyze which t-norms are also overlap functions and we prove that if we apply particular aggregations to such functions we recover the overlap index between fuzzy sets as defined by Dubois, and the consistency index of Zadeh. We also consider some properties that can be required to overlap functions, as k-Lipschitzianity or migrativityPublication Paired structures and bipolar knowledge representation(University of Copenhagen, Department of Food and Resource Economics, 2014) Montero, Javier; Bustince, H.; Franco, C.; Rodríguez, Juan Tinguaro; Gómez, Daniel; Pagola, Miguel; Fernandez, Javier; Barrenechea, EdurneIn this strictly positional paper we propose a general approach to bipolar knowledge representation, where the meaning of concepts can be modelled by examining their decomposition into opposite and neutral categories. In particular, it is the semantic relationship between the opposite categories which suggests the emergence of a paired structure and its associated type of neutrality, being there three general types of neutral categories, namely indeterminacy, ambivalence and conflict. Hence, the key issue consists in identifying the semantic opposition characterizing the meaning of concepts and at the same time the type of neutrality rising in between opposites. Based on this first level of bipolar knowledge representation, paired structures in fact offer the means to characterize a specific bipolar valuation scale depending on the meaning of the concept that has to be verified. In this sense, a paired structure is a standard basic structure that allows learning and building different valuation scales, leading to linear or even more complex valuation scales.Publication Paired structures in knowledge representation.(Elservier, 2016) Montero, Javier; Bustince, H.; Franco, C.; Rodríguez, Juan Tinguaro; Gómez, Daniel; Pagola, M.; Fernández, J.; Barrenechea, E.In this position paper we propose a consistent and unifying view to all those basic knowledge representation models that are based on the existence of two somehow opposite fuzzy concepts. A number of these basic models can be found in fuzzy logic and multi-valued logic literature. Here it is claimed that it is the semantic relationship between two paired concepts what determines the emergence of different types of neutrality, namely indeterminacy, ambivalence and conflict, widely used under different frameworks (possibly under different names). It will be shown the potential relevance of paired structures, generated from two paired concepts together with their associated neutrality, all of them to be modeled as fuzzy sets. In this way, paired structures can be viewed as a standard basic model from which different models arise. This unifying view should therefore allow a deeper analysis of the relationships between several existing knowledge representation formalisms, providing a basis from which more expressive models can be later developed.Publication Semiautoduality in a restricted family of aggregation operators(Elsevier Science Bv, 2007) Bustince, H.; Montero, Javier; Pagola, M.In this paper we consider aggregation operators satisfying non-decreasingness and some specific boundary conditions. We then analyze some properties of such a family of aggregation operators, introducing the semiautoduality condition, which is weaker than the standard autoduality condition (i.e., the standard self De Morgan identity). Particular families of aggregation operators will appear depending on the context.