Humberto Bustince

Vague sets are intuitionistic fuzzy sets

Abstract We recapitulate the definition given by Atanassov (1983) of intuitionistic fuzzy sets as well as the definition of vague sets given by Gau and Byehrer (1993) and see that both definitions coincide.

A Review on Ensembles for the Class Imbalance Problem: Bagging-, Boosting-, and Hybrid-Based Approaches

Classifier learning with data-sets that suffer from imbalanced class distributions is a challenging problem in data mining community. This issue occurs when the number of examples that represent one class is much lower than the ones of the other classes. Its presence in many real-world applications has brought along a growth of attention from researchers. In machine learning, the ensemble of classifiers are known to increase the accuracy of single classifiers by combining several of them, but neither of these learning techniques alone solve the class imbalance problem, to deal with this issue the ensemble learning algorithms have to be designed specifically. In this paper, our aim is to review the state of the art on ensemble techniques in the framework of imbalanced data-sets, with focus on two-class problems. We propose a taxonomy for ensemble-based methods to address the class imbalance where each proposal can be categorized depending on the inner ensemble methodology in which it is based. In addition, we develop a thorough empirical comparison by the consideration of the most significant published approaches, within the families of the taxonomy proposed, to show whether any of them makes a difference. This comparison has shown the good behavior of the simplest approaches which combine random undersampling techniques with bagging or boosting ensembles. In addition, the positive synergy between sampling techniques and bagging has stood out. Furthermore, our results show empirically that ensemble-based algorithms are worthwhile since they outperform the mere use of preprocessing techniques before learning the classifier, therefore justifying the increase of complexity by means of a significant enhancement of the results.

Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets

Abstract We recall the definitions of intuitionistic fuzzy sets and interval-valued fuzzy sets with the relation between these sets established by K. Atanassov. We define the distance measure between intuitionistic fuzzy sets and we give an axiom definition of intuitionistic fuzzy entropy and a theorem which characterizes it. Finally, we study a very special entropy and recall that all we have done for intuitionistic fuzzy sets is also good for interval-valued fuzzy sets.

Correlation of interval-valued intuitionistic fuzzy sets

Abstract In this paper we introduce the concepts of correlation and correlation coefficient of interval-valued intuitionistic fuzzy sets and we study their first properties. We also introduce two decomposition theorems of the correlation of interval-valued intuitionistic fuzzy sets, one in terms of the correlation of interval-valued fuzzy sets and the entropy of the intuitionistic fuzzy sets, and the other theorem in terms of the correlation of intuitionistic fuzzy sets.

Automorphisms, negations and implication operators

In this paper we study the conditions in which the implication operators satisfy the property I(x,c(x)) = c(x) for all x ∈ [0, 1], c being any strong negation. This study has led us to present different implication operator characterization theorems from automorphisms, obtaining a theorem similar to the one presented by Smets and Magrez (Internat. J. Approx. Reason. 1 (1987) 327), in which the strong negation c used is not generated by the same automorphism that generates the implication.

On averaging operators for Atanassov's intuitionistic fuzzy sets

Atanassovs intuitionistic fuzzy set (AIFS) is a generalization of a fuzzy set. There are various averaging operators defined for AIFSs. These operators are not consistent with the limiting case of ordinary fuzzy sets, which is undesirable. We show how such averaging operators can be represented by using additive generators of the product triangular norm, which simplifies and extends the existing constructions. We provide two generalizations of the existing methods for other averaging operators. We relate operations on AIFS with operations on interval-valued fuzzy sets. Finally, we propose a new construction method based on the Lukasiewicz triangular norm, which is consistent with operations on ordinary fuzzy sets, and therefore is a true generalization of such operations.

Interval-valued fuzzy sets constructed from matrices: Application to edge detection

In this paper we present a method to construct interval-valued fuzzy sets (or interval type 2 fuzzy sets) from a matrix (or image), in such a way that we obtain the length of the interval representing the membership of any element to the new set from the differences between the values assigned to that element and its neighbors in the starting matrix. Using the concepts of interval-valued fuzzy t-norm, interval-valued fuzzy t-conorm and interval-valued fuzzy entropy, we are able to detect big enough jumps (edges) between the values of an element and its neighbors in the starting matrix. We also prove that the unique t-representable interval-valued fuzzy t-norms and the unique s-representable interval-valued fuzzy t-conorms that preserve the length zero of the intervals are the ones generated by means of the t-norm minimum and the t-conorm maximum.

Indicator of inclusion grade for interval-valued fuzzy sets. Application to approximate reasoning based on interval-valued fuzzy sets

We begin the paper studying the axioms that the indicators of the grade of inclusion of a fuzzy set in another fuzzy set must satisfy. Next, we present an expression of such indicator, first for fuzzy sets and then for interval-valued fuzzy sets, analyzing in both cases their main properties. Then, we suggest an expression for the similarity measure between interval-valued fuzzy sets. Besides, we study two methods for inference in approximate reasoning based on interval-valued fuzzy sets, the inclusion grade indicator and the similarity measure. Afterwards, we expose some of the most important properties of the methods of inference presented and we compare these methods to Gorzalczany’s. Lastly, we use the indicator of the grade of inclusion for interval-valued fuzzy sets as an element that selects from the diAerent methods of inference studied, the one that will be executed in each case. ” 2000 Elsevier Science Inc. All rights reserved.

Structures on intuitionistic fuzzy relations

Abstract In this paper we study the structures of the intuitionistic fuzzy relations. We analyse the existent relations between the structures of a relation and the structures of its complementary one. We finish characterizing certain structures of intuitionistic relations according to the structures of two concrete fuzzy relations.

On the relevance of some families of fuzzy sets

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 Atanassovs intuitionistic fuzzy sets. We postulate that introducing this graph allows a natural explanation of the different visions underlying Atanassovs model and interval valued fuzzy sets, despite both models have been proven equivalent when such a structure in the valuation space is not assumed.