Javier Montero

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.

A Historical Account of Types of Fuzzy Sets and Their Relationships

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.

Fuzzy classification systems

In this paper it is pointed out that a classification is always made taking into account all the available classes, i.e., by means of a classification system. The approach presented in this paper generalizes the classical definition of fuzzy partition as defined by Ruspini, which is now conceived as a quite often desirable objective that can be usually obtained only after a long learning process. In addition, our model allows the evaluation of the resulting classification, according to several indexes related to covering, relevance and overlapping.

Ignorance functions. An application to the calculation of the threshold in prostate ultrasound images

In 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 prostate ultrasound images. The experimental results on ultrasound and natural images also allow us to determine the best choice of the function to represent the ignorance.

Recursive connective rules

An associative binary connective allows the evaluation of arbitrary finite sequences of items by means of a one‐by‐one sequential process. In this paper we develop an alternative approach for those nonassociative connectives, allowing a sequential definition by means of binary fuzzy connectives. It will be then stressed that a connective rule should be understood as a consistent sequence of binary connective operators. ©1999 John Wiley & Sons, Inc.

Representation of consistent recursive rules

This paper develops the recursive model for connective rules (as proposed in V. Cutello, E. Molina, J. Montero, Associativeness versus recursiveness, in: Proceedings of the 26th IEEE International Symposium on Multiple-valued Logic, Santiago de Compostela, Spain, 29-31 May, 1996, pp. 154-159; V. Cutello, E. Molina, J. Montero, Binary operators and connective rules, in: M.H. Smith, M.A. Lee, J. Keller, J. Yen (Eds.), Proceedings of NAFIPS 96, North American Fuzzy Information Processing Society, IEEE Press, Piscataway, NJ, 1996, pp. 46-49), where a particular solution in the Ordered Weighted Averaging (OWA) context (see V. Cutello, J. Montero, Recursive families of OWA operators, in: P.P. Bonissone (Ed.), Proceedings of the Third IEEE Conference on Fuzzy Systems, IEEE Press, Piscataway, NJ, 1994, pp. 1137-1141; V. Cutello, J. Montero, Recursive connective rules, International Journal of Intelligent Systems, to appear) was translated into a more general framework. In this paper, some families of solutions for the key recursive equation are obtained, based upon the general associativity equation as solved by K. Mak (Coherent continuous systems and the generalized functional equation of associativity, Mathematics of Operations Research 12 (1987) 597-625). A context for the representation of families of binary connectives is given, allowing the characterization of key families of connective rules

Fuzzy rationality measures

Fuzzy preference relations formalize intensity of individual preferences over fixed sets of alternatives. It is therefore natural to extend to fuzzy preferences the notion of rationality or consistency. With this goal in mind, we address in this paper the problem of giving an axiomatic basis for defining the concept of fuzzy rationality. Specifically, we establish a collection of conditions that any fuzzy rationality measure must satisfy. Some examples of fuzzy rationality measures are then given and analyzed.

Migrativity of aggregation functions

In this paper 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.

A coloring fuzzy graph approach for image classification

One of the main problems in practice is the difficulty in dealing with membership functions. Many decision makers ask for a graphical representation to help them to visualize results. In this paper, we point out that some useful tools for fuzzy classification can be derived from fuzzy coloring procedures. In particular, we bring here a crisp grey coloring algorithm based upon a sequential application of a basic black and white binary coloring procedure, already introduced in a previous paper [D. Gomez, J. Montero, J. Yanez, C. Poidomani, A graph coloring algorithm approach for image segmentation, Omega, in press]. In this article, the image is conceived as a fuzzy graph defined on the set of pixels where fuzzy edges represent the distance between pixels. In this way, we can obtain a more flexible hierarchical structure of colors, which in turn should give useful hints about those classes with unclear boundaries.

A general model for deriving preference structures from data

In this paper we comment upon the integrated model for valued preferences introduced by Fodor, Ovchinnikov and Roubens. In particular, while on one hand we revise basic assumptions and point out their intuitive meaning, on the other hand we propose an alternative mathematical justification of such a model which allows not only a better understanding of the obtained results, but also a functional characterization of the whole family of solutions.