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2016 | Buch

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Fuzzy Logic and Information Fusion

To commemorate the 70th birthday of Professor Gaspar Mayor

herausgegeben von: Tomasa Calvo Sánchez, Joan Torrens Sastre

Verlag: Springer International Publishing

Buchreihe : Studies in Fuzziness and Soft Computing

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book offers a timely report on key theories and applications of soft-computing. Written in honour of Professor Gaspar Mayor on his 70th birthday, it primarily focuses on areas related to his research, including fuzzy binary operators, aggregation functions, multi-distances, and fuzzy consensus/decision models. It also discusses a number of interesting applications such as the implementation of fuzzy mathematical morphology based on Mayor-Torrens t-norms. Importantly, the different chapters, authored by leading experts, present novel results and offer new perspectives on different aspects of Mayor’s research. The book also includes an overview of evolutionary fuzzy systems, a topic that is not one of Mayor’s main areas of interest, and a final chapter written by the Spanish pioneer in fuzzy logic, Professor E. Trillas. Computer and decision scientists, knowledge engineers and mathematicians alike will find here an authoritative overview of key soft-computing concepts and techniques.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Gaspar Mayor: A Prolific Career on Fuzzy Sets and Aggregation Functions

Abstract

The LOBFI Research Group was founded many years ago by Professor Gaspar Mayor at the beginning of his research career at the University of the Balearic Islands. Since then, he has been the leader of the group and has devoted many of his efforts to getting LOBFI to a prestigious and recognized group in the fuzzy community. This chapter has been written by all the current members of the LOBFI group in gratitude to him, and it is devoted to recall his main scientific achievements in the field of fuzzy sets theory and aggregation fusion.

LOBFI Research Group

Chapter 2. Smooth Finite T-norms and Their Equational Axiomatization

Abstract

In this paper, as homage to Professor Gaspar Mayor in his 70 anniversary, we present a summary of results on BL-algebras and related structures that, using the one-to-one correspondence between divisible finite t-norms and finite BL-chains, allows us to provide an equational characterization of any divisible finite t-norm.

Francesc Esteva, Àngel García-Cerdaña, Lluís Godo

Chapter 3. Associative Copulas: A Survey

Abstract

Copulas—functions that join multivariate distribution functions to their one-dimensional margins—are special cases of binary 1-Lipschitz aggregation functions, commonly used in aggregation processes. Here we consider a significant class of copulas: Associative copulas. We explore briefly the subclass of Archimedean copulas, and some of the properties and applications of associative copulas, such as the simultaneous associativity, the Kendall distribution functions, topological aspects, etc. Finally, some open problems are posed.

Juan Fernández-Sánchez, José Juan Quesada-Molina, Manuel Úbeda-Flores

Chapter 4. Powers with Respect to t-Norms and t-Conorms and Aggregation Functions

Abstract

Aggregation functions A stable with respect to powers of t-norms and t-conorms (i.e.: satisfying \(A(x^{(r)},y^{(r)})=(A(x,y))^{(r)}\)) where \(x^{(r)}\) is the r-th power of \(x \in [0,1]\) with respect to a t-norm or t-conorm) are characterized. This result generalizes the characterization of power stable aggregation functions in [5].

D. Boixader, J. Recasens

Chapter 5. Modus Tollens on Fuzzy Implication Functions Derived from Uninorms

Abstract

The most used inference schemes in approximate reasoning are the so-called Modus Ponens for forward inferences, and Modus Tollens for backward inferences. In this way, finding new fuzzy implication functions satisfying these two properties has become an important topic for researchers. In the framework of fuzzy logic, they can be written as two inequalities involving fuzzy implication functions. In this paper, the property of Modus Tollens with respect to a continuous t-norm and a continuous fuzzy negation is studied for residual implication functions derived from uninorms, that is, for RU-implications. The corresponding inequality is solved in the cases of an RU-implication derived from a uninorm U in the class of \(\mathscr {U}_{\min }\), from an idempotent uninorm or from a representable uninorm.

M. Mas, J. Monreal, M. Monserrat, J. V. Riera, Joan Torrens Sastre

Chapter 6. A Survey of Atanassov’s Intuitionistic Fuzzy Relations

Abstract

In 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.

Humberto Bustince, Edurne Barrenechea, Miguel Pagola, Javier Fernandez, Raul Orduna, Javier Montero

Chapter 7. On Weighting Triangles Using Fuzzy Relations and Its Application to Aggregation Functions

Abstract

In this work, a new lattice L determined by the class of weighting triangles as a base of L-fuzzy subsets is proposed. Furthermore, extended orders and operators which are obtained by means of fuzzy binary relations \(F_{\triangle }\) associated to a weighting triangle are included. Moreover, some new expressions have been defined for Extended Ordered Weighted Averaging operators, and Extended Aggregation functions.

Tomasa Calvo Sánchez, Ramón Fuentes-González, Pilar Fuster-Parra

Chapter 8. New Advances in the Aggregation of Asymmetric Distances. The Bounded Case

Abstract

In 1981, J. Borsík and J. Doboš studied the problem of how to merge, by means of a function, a family of distances into a single one. To this end, they introduced the notion of distance aggregation function and gave a characterization of such functions. Later on, in 2010, the notion of distance aggregation function was extended to the framework of asymmetric distances by G. Mayor and O. Valero. Thus, asymmetric distance aggregation functions were introduced and a characterization of this new type of functions was also given. Concretely, the aforesaid characterization states that the functions which allow to merge a family of asymmetric distances into a single one are exactly those that are amenable, monotone and subadditive. In the present chapter we consider the problem of aggregating a family of bounded asymmetric distances. To this end, the notion of bounded asymmetric distance aggregation function is introduced and a full description of such functions is provided. The obtained results are illustrated by means of examples. Furthermore, the relationship between asymmetric aggregation functions and the bounded ones is discussed.

Isabel Aguiló, Tomasa Calvo Sánchez, Pilar Fuster-Parra, Javier Martín, Jaume Suñer, Oscar Valero

Chapter 9. Multidistances and Dispersion Measures

Abstract

In this paper, we provide a formal notion of absolute dispersion measure that is satisfied by some classical dispersion measures used in Statistics, such as the range, the variance, the mean deviation and the standard deviation, among others, and also by the absolute Gini index, used in Welfare Economics for measuring inequality. The notion of absolute dispersion measure shares some properties with the notion of multidistance introduced and analyzed by Martín and Mayor in several recent papers. We compare absolute dispersion measures and multidistances and we establish that these two notions are compatible by showing some functions that are simultaneously absolute dispersion measures and multidistances. We also establish that remainders obtained through the dual decomposition of exponential means, introduced by García-Lapresta and Marques Pereira, are absolute dispersion measures up to sign.

Miguel Martínez-Panero, José Luis García-Lapresta, Luis Carlos Meneses

Chapter 10. Soft Consensus Models in Group Decision Making

Abstract

In group decision making problems, when a consensual solution is required, a natural question is how to measure the closeness among experts’ opinions in order to obtain the consensus level. To do so, different approaches have been proposed. Following this research line, several authors have introduced hard consensus measures varying between 0 (no consensus or partial consensus) and 1 (full consensus or complete agreement). However, consensus as a full and unanimous agreement is far from being achieved in real situations. So, in practice, a more realistic approach is to use some softer consensus measures, which assess the consensus degree in a more flexible way reflecting better all possible partial agreements obtained through the process. The aim of this chapter is to identify and describe the different existing approaches to compute soft consensus measures in fuzzy group decision making problems. Additionally, we analyze the current models and new challenges on this field.

Ignacio Javier Perez, Francisco Javier Cabrerizo, Sergio Alonso, Francisco Chiclana, Enrique Herrera-Viedma

Chapter 11. Relation Between AHP and Operators Based on Different Scales

Abstract

Obtaining the value of the weights in any decision problem is of great importance, because it can change the course of action for the final decision. The value of these weights is approximate due to the vagueness and ambiguity of the data. Our study is based on the Analytic Hierarchy Process and its relation with the Prioritized Aggregation Operators. We propose their obtaining starting from a proportionality relationship, and we study the main properties of the prioritized operator with proportionality ratio and linear scale.

E. Cables, M. T. Lamata, J. L. Verdegay

Chapter 12. Evolutionary Fuzzy Systems: A Case Study in Imbalanced Classification

Abstract

The use of evolutionary algorithms for designing fuzzy systems provides them with learning and adaptation capabilities, resulting on what is known as Evolutionary Fuzzy Systems. These types of systems have been successfully applied in several areas of Data Mining, including standard classification, regression problems and frequent pattern mining. This is due to their ability to adapt their working procedure independently of the context we are addressing. Specifically, Evolutionary Fuzzy Systems have been lately applied to a new classification problem showing good and accurate results. We are referring to the problem of classification with imbalanced datasets, which is basically defined by an uneven distribution between the instances of the classes. In this work, we will first introduce some basic concepts on linguistic fuzzy rule based systems. Then, we will present a complete taxonomy for Evolutionary Fuzzy Systems. Then, we will review several significant proposals made in this research area that have been developed for addressing classification with imbalanced datasets. Finally, we will show a case study from which we will highlight the good behavior of Evolutionary Fuzzy Systems in this particular context.

A. Fernández, F. Herrera

Chapter 13. Mayor-Torrens t-norms in the Fuzzy Mathematical Morphology and Their Applications

Abstract

Fuzzy mathematical morphology has been extensively used in many different applications such as edge detection, noise reduction and shape and pattern recognition. The fundamentals of this morphology are based on an appropriate selection of the operators involved, namely the conjunction and implication. In this work we investigate the use of the Mayor-Torrens family of t-norms, from both theoretical and practical point of view. The results suggest that competitive results can be obtained by using the t-norms of this family.

P. Bibiloni, M. González-Hidalgo, S. Massanet, A. Mir, D. Ruiz-Aguilera

Chapter 14. A Short Dialogue Concerning “What Is” and “What Is Not” with Imprecise Words

Abstract

What follows is a virtual conversation between two imagined characters, Karl and Carla. They try to debate on how, in fuzzy set algebras, what is not covered under a linguistic label should be represented; that is, and mainly, on both the negation and the opposites of a predicate, and on which fuzzy and crisp expressions of not covered by can be obtained.

Enric Trillas

Titel: Fuzzy Logic and Information Fusion
herausgegeben von: Tomasa Calvo Sánchez
Joan Torrens Sastre
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-30421-2
Print ISBN: 978-3-319-30419-9
DOI: https://doi.org/10.1007/978-3-319-30421-2