Granular, Soft and Fuzzy Approaches for Intelligent Systems

Probability has, as a mathematical theory that is an important part of pure mathematics, a long and distinguished history of more than 300 years, with fertile applications in almost all domains of science and technology; but the history of fuzzy sets only lasts 50 years, during which it was theoretically developed and successfully applied in many fields. From the very beginning there was, and there still is, a controversy on the nature of fuzzy sets viewed by its researchers far from randomness, and instead close by probabilists. This paper only goal is nothing else than trying to contribute to the clarification on the differences its authors see between fuzzy sets and probabilities and through the representation, or scientific domestication, of meaning by quantities.

The cube of opposition is a structure that extends the traditional square of opposition originally introduced by Ancient Greek logicians in relation with the study of syllogisms. This structure, which relates formal expressions, has been recently generalized to non Boolean, graded statements. In this paper, it is shown that the cube of opposition applies to well-known families of idempotent, monotonically increasing aggregation operations, used in multiple criteria decision making, which qualitatively or quantitatively provide evaluations between the minimum and the maximum of the aggregated quantities. This covers weighted minimum and maximum, and more generally Sugeno integrals on the qualitative side, and Choquet integrals, with the important particular case of Ordered Weighted Averages, on the quantitative side. The main appeal of the cube of opposition is its capability to display the various possible aggregation attitudes in a given setting and to show their complementarity.

Among the countless papers written by Ronald R. Yager, those on Entropies and measures of information are considered, keeping in mind the notion of view of a set, in order to point out a similarity between the quantities introduced in various frameworks to evaluate a kind of entropy. We define the concept of entropy measure and we show that its main characteristic is a form of monotonicity, satisfied by quantities scrutinised by R.R. Yager.

In this chapter, we make use of the notion of admissible order between intervals to extend the definition of OWA operators and Choquet integrals to the interval-valued setting. We also present an algorithm for decision making based on these developments.

An overview of information theory metrics and the ranges of their values for extreme probability cases is provided. Imprecise database models including similarity based fuzzy models and rough set models are described. Various entropy measures for these database models’ content and responses to querying is provided. Aggregation of uncertainty representations are also considered. In particular the possibilistic conditioning of probability aggregation is examined. Information measures are used to compare the resultant conditioned probability to the original probability for three cases of possibility distributions.

Many different types of aggregation operators have been suggested as decision functions for multicriteria fuzzy decision making. This paper investigates the practical applicability of generalized averaging operator as decision functions in modeling human decision behavior. Previously published numerical data is used in the analysis and the results are compared with those obtained from compensatory operators. The numerical data suggests that the generalized averaging operator may be used for modeling human decision behavior.

This chapter suggests an evolving possibilistic fuzzy modeling approach for value-at-risk modeling and estimation. The modeling approach is based on an extension of the possibilistic fuzzy c-means clustering and functional fuzzy rule-based systems. It employs memberships and typicalities to update clusters centers and creates new clusters using a statistical control distance-based criteria. Evolving possibilistic fuzzy modeling (ePFM) also uses an utility measure to evaluate the quality of the current cluster structure. The fuzzy rule-based model emerges from the cluster structure. Market risk exposure plays a key role for financial institutions in risk assessment and management. A way to measure risk exposure is to evaluate the losses likely to incur when the prices of the portfolio assets decline. Value-at-risk (VaR) estimate is amongst the most prominent measure of financial downside market risk. Computational experiments are conducted to evaluate ePFM for value-at-risk estimation using data of the main equity market indexes of United States (S&P 500) and Brazil (Ibovespa) from January 2000 to December 2012. Econometric models benchmarks such as GARCH and EWMA, and state of the art evolving approaches are compared against ePFM. The results suggest that ePFM is a potential candidate for VaR modeling and estimation because it achieves higher performance than econometric and alternative evolving approaches.

An interesting and important problem of how similar and/or dissimilar voting procedures (social choice functions) are dealt with. We extend our previous qualitative type analysis based on rough sets theory which make it possible to partition the set of voting procedures considered into some subsets within which the voting procedures are indistinguishable, i.e. (very) similar. Then, we propose an extension of those analyses towards a quantitative evaluation via the use of degrees of similarity and dissimilarity, not necessarily metrics and dual (in the sense of summing up to 1). We consider the amendment, Copeland, Dodgson, max-min, plurality, Borda, approval, runoff, and Nanson, voting procedures, and the Condorcet winner, Condorcet loser, majority winner, monotonicity, weak Pareto winner, consistency, and heritage criteria. The satisfaction or dissatisfaction of the particular criteria by the particular voting procedures are represented as binary vectors. We use the Jaccard–Needham, Dice, Correlation, Yule, Russell–Rao, Sockal–Michener, Rodgers–Tanimoto, and Kulczyński measures of similarity and dissimilarity. This makes it possible to gain much insight into the similarity/dissimilarity of voting procedures.

In order to effectively plan both preparedness and response to emergency situations it is necessary to access and analyse timely information on plausible scenarios of occurrence of ongoing events. Scenario maps representing the estimated susceptibility, hazard or risk of occurrence of an event on a territory are hardly generated real time. In fact the application of physical or statistical models using environmental parameters representing current dynamic conditions is time consuming on low cost hardware equipment. To cope with this practical issue we propose an off line generation of scenario maps under diversified environmental dynamic parameters, and a geo-Spatial Data Infrastructure (SDI) to allow people in charge of emergency preparedness and response activities to flexibly discover, retrieve, fuse and visualize the most plausible scenarios that may happen given some ongoing or forecasted dynamic conditions influencing the event. The novelty described in this chapter is related with both the ability to interpret flexible queries in order to retrieve risk scenario maps that are related to the current situation and to show the most plausible worst and best scenarios that may occur in each elementary area of the territory. Although, the SDI proposal has been conceived and designed to support the management of distinct natural and man-made risks, in the proof of concept prototypal implementation the scenarios maps target wild fire events.

Look-up table (LUT) controllers are among the most widely utilized control tools in engineering practice. The reasons for their popularity include simplicity, easy to use, inexpensive hardware implementation, and strong nonlinearity and multimodal behaviors that can be formalized, in many cases, only by experimentally measured data. In a previous paper, we showed that the two-dimensional (2D) LUT controllers and one special type of two-input Mamdani fuzzy controllers are connected in that they have the identical input-output mathematical relation. We also demonstrated how to represent the LUT controllers by the fuzzy controllers. Finally, we showed how to determine the local stability of the LUT control systems. In the present work, we extend these results to the n-dimensional LUT controllers and the special type of the n-input Mamdani fuzzy controllers.

An inherent limitation of Linear Programming is the need to know precisely all the conditions concerning the problem being modeled. This is not always possible as there exist uncertainty situations which require a more suitable approach. Fuzzy Linear Programming allows working with imprecise data and constraints, leading to more realistic models. Despite being a consolidated field with more than 30 years of existence, almost no software has been developed for public use that solves fuzzy linear programming problems. Here we present an open-source R package to deal with fuzzy constraints, fuzzy costs and fuzzy coefficients in linear programming. The theoretical foundations for solving each type of problem are introduced first, followed by code examples. The package is accompanied by a user manual and can be freely downloaded, employed and extended by any R user.

This study presents a bibliometric analysis of the publications of Ronald R. Yager available in Web of Science. Currently Professor Yager has more than 500 publications in this database. He is recognized as one of the most influential authors in the World in Computer Science. The bibliometric review considers a wide range of issues including a specific analysis of his publications, collaborators and citations. The VOS viewer software is used to visualize his publication and citation network though bibliographic coupling and co-citation analysis. The results clearly show his strong influence in Computer Science although it also shows a strong influence in Engineering and Applied Mathematics.