Ramos-Guajardo A.B.,European Center for Soft Computing
Computational Statistics and Data Analysis | Year: 2012
The problem of testing equality of variances often arises when distributions of random variables are compared or linear models between them are considered. The usual tests for variances given normality of the underlying populations are highly non-robust to non-normality and are strongly dependent on the kurtosis. Some alternative formulations of Levene's test statistic for testing the homoscedasticity have been shown to be powerful and robust under non-normality. On the basis of Levene's classical procedure, a test for the equality of variances of k fuzzy-valued random elements is developed. Accordingly, consistent asymptotic and bootstrap tests are established and their empirical behaviour is analyzed by means of extensive simulation studies. In addition, the proposed test is compared with a Bartlett-type test. A case-study illustrating the applicability of the procedure is presented. © 2011 Elsevier B.V. All rights reserved.
Cordon O.,European Center for Soft Computing
International Journal of Approximate Reasoning | Year: 2011
The need for trading off interpretability and accuracy is intrinsic to the use of fuzzy systems. The obtaining of accurate but also human-comprehensible fuzzy systems played a key role in Zadeh and Mamdani's seminal ideas and system identification methodologies. Nevertheless, before the advent of soft computing, accuracy progressively became the main concern of fuzzy model builders, making the resulting fuzzy systems get closer to black-box models such as neural networks. Fortunately, the fuzzy modeling scientific community has come back to its origins by considering design techniques dealing with the interpretability-accuracy tradeoff. In particular, the use of genetic fuzzy systems has been widely extended thanks to their inherent flexibility and their capability to jointly consider different optimization criteria. The current contribution constitutes a review on the most representative genetic fuzzy systems relying on Mamdani-type fuzzy rule-based systems to obtain interpretable linguistic fuzzy models with a good accuracy. © 2011 Elsevier Inc. All rights reserved.
Alonso J.M.,European Center for Soft Computing |
Magdalena L.,European Center for Soft Computing
Information Sciences | Year: 2011
Interpretability is acknowledged as one of the most appreciated advantages of fuzzy systems in many applications, especially in those with high human interaction where it actually becomes a strong requirement. However, it is important to remark that there is a somehow misleading but widely extended belief, even in part of the fuzzy community, regarding fuzzy systems as interpretable no matter how they were designed. Of course, we are aware the use of fuzzy logic favors the interpretability of designed models. Thanks to their semantic expressivity, close to natural language, fuzzy variables and rules can be used to formalize linguistic propositions which are likely to be easily understandood by human beings. Obviously, this fact makes easier the knowledge extraction and representation tasks carried out when modeling real-world complex systems. Notwithstanding, fuzzy logic is not enough by itself to guarantee the interpretability of the final model. As it is thoroughly illustrated in this special issue, achieving interpretable fuzzy systems is a matter of careful design because fuzzy systems cannot be deemed as interpretable per se. Thus, several constraints have to be imposed along the whole design process with the aim of producing really interpretable fuzzy systems, in the sense that every element of the whole system may be checked and understood by a human being. Otherwise, fuzzy systems may even become black-boxes. © 2011 Elsevier Inc. All rights reserved.
Borgelt C.,European Center for Soft Computing
Journal of Computer and System Sciences | Year: 2010
When it comes to learning graphical models from data, approaches based on conditional independence tests are among the most popular methods. Since Bayesian networks dominate research in this field, these methods usually refer to directed graphs, and thus have to determine not only the set of edges, but also their direction. At least for a certain kind of possibilistic graphical models, however, undirected graphs are a much more natural basis. Hence, in this area, algorithms for learning undirected graphs are desirable, especially, since first learning a directed graph and then transforming it into an undirected one wastes resources and computation time. In this paper I present a general algorithm for learning undirected graphical models, which is strongly inspired by the well-known Cheng-Bell-Liu algorithm for learning Bayesian networks from data. Its main advantage is that it needs fewer conditional independence tests, while it achieves results of comparable quality. © 2009 Elsevier Inc. All rights reserved.
Magdalena L.,European Center for Soft Computing
International Journal of Computational Intelligence Systems | Year: 2010
The term Soft Computing was coined by L.A. Zadeh in the early 90's. Since that time many researchers have tried to define it considering different approaches: main constituents, properties, abilities, etc. In addition, the term Computational Intelligence has also gained popularity having a somehow quite close meaning to that of Soft Computing. The central idea of this paper is to present, analyze, compare and discuss a few of the definitions that can be found on literature; not trying to find the best but to offer the reader arguments to make his/her own decision.
Borgelt C.,European Center for Soft Computing
Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery | Year: 2012
Frequent item set mining is one of the best known and most popular data mining methods. Originally developed for market basket analysis, it is used nowadays for almost any task that requires discovering regularities between (nominal) variables. This paper provides an overview of the foundations of frequent item set mining, starting from a definition of the basic notions and the core task. It continues by discussing how the search space is structured to avoid redundant search, how it is pruned with the a priori property, and how the output is reduced by confining it to closed or maximal item sets or generators. In addition, it reviews some of the most important algorithmic techniques and data structures that were developed to make the search for frequent item sets as efficient as possible.© 2012 Wiley Periodicals, Inc.
Trillas E.,European Center for Soft Computing
International Journal of Intelligent Systems | Year: 2012
In the setting of a general type of fuzzy algebras, this paper deals with a new theoretic view on the commonsense reasoning, consisting of a kind of Popper's search for conjectures and refutations. It is supposed that the reasoning is done in natural language, but only with nonambiguous precise and imprecise terms, respectively, represented by crisp and fuzzy sets. © 2012 Wiley Periodicals, Inc.
Trutschnig W.,European Center for Soft Computing
Fuzzy Sets and Systems | Year: 2012
It is shown that pointwise convergence of a sequence ( An) n∈N of copulas to a copula A is equivalent (1) to the convergence of the corresponding endographs and (2) to the convergence of the corresponding upper (or lower) α-levels for all but at most countably many α in [0,1] (all with respect to the Hausdorff metric). Examples are given that show that the countably many exceptions in (2) cannot be omitted. It is furthermore shown that the main results also hold on the bigger class of quasi-copulas. © 2011 Elsevier B.V. All rights reserved.
Seising R.,European Center for Soft Computing
Information Sciences | Year: 2010
About 60 years ago Norbert Wiener and Claude Elwood Shannon established the new scientific discipline of information theory. However, it is very probable that Shannon's article A Mathematical Theory of Communication would not have become famous without the help of Warren Weaver, whose popular text on "The Mathematics of Communication" re-interpreted Shannon's work for broader scientific audiences. Weaver's "preface" and Shannon's article were published together in the book The Mathematical Theory of Communication. Norbert Wiener's Cybernetics was an even more popular event, when it appeared in print. However publications were influential on two scientific areas with concepts unmentioned or unelaborated within the texts themselves: Systems Theory and information theory. A "General System Theory" had already been created by Ludwig von Bertalanffy in the late 1920s for biological and philosophical research. This approach melded in North America in the 1950s with cybernetics, as well as a new system theoretical approach in engineering sciences in the 1950s. Bertalanffy's "General System Theory" - or simply "systems theory" was used, became even more famous in humanities. In the 1960s attempts to yield both systems theory took root in the humanities, with mixed success. This paper will review the links across these fields showing the influences across cybernetics, system(s) theory and information theory throughout the 1950s and the theory of Fuzzy Sets and Systems. Then we focus to the non-technical but philosophical aspects of information theory. When Weaver emphasized not the technical but the semantic and influential problems of communication, his arguments were very similar to Charles W. Morris' foundations of the Theory of Signs (1938) - Semiotics. We will show some interesting ideas of Weaver related to semiotic thinking and we will advocate a "fuzzy information theory" that has to be appropriate to cover this "semiotic concept of information". Finally, the paper presents epistemological reflections in historical perspective on the concept of "information" as a "fluctuating object" that we take as a fuzzy concept. © 2010 Elsevier Inc. All rights reserved.
Sugeno M.,European Center for Soft Computing
Fuzzy Sets and Systems | Year: 2013
This paper deals with the Choquet integral on the non-negative real line. First it gives a representation of the Choquet integral of a non-negative, continuous and increasing function with respect to a fuzzy measure. Next, restricting fuzzy measures to a class of distorted Lebesgue measures, it considers Choquet integral equations. In order to solve Choquet integral equations, a concept of the derivatives of functions with respect to fuzzy measures is introduced. For distorted Lebesgue measures, it is shown that Choquet integral equations are formulated as Volterra integral equations of the first kind. The differentiability of functions with respect to fuzzy measures is also discussed. It further shows a relation of a Choquet integral equation with the Abel integral equation. Finally this paper introduces simple differential equations with respect to fuzzy measures and gives their solutions. © 2012 Elsevier B.V. All rights reserved.