Membership function

About: Membership function is a research topic. Over the lifetime, 15795 publications have been published within this topic receiving 418366 citations.

Putting Rough Sets and Fuzzy Sets Together

Abstract: In this paper we argue that fuzzy sets and rough sets aim to different purposes and that it is more natural to try to combine the two models of uncertainty (vagueness for fuzzy sets and coarseness for rough sets) in order to get a more accurate account of imperfect information. First, the upper and lower approximations of a fuzzy set are defined, when the universe of discourse of a fuzzy sets is coarsened by means of an equivalence relation. We then come close to Caianiello’s C-calculus. Shafer’s concept of coarsened belief functions also belongs to the same line of thought and is reviewed here. Another idea is to turn the equivalence relation relation into a fuzzy similarity relation, for a more expressive modeling of coarseness. New results on the representation of similarity relations by means of a fuzzy partition of fuzzy clusters of more or less indiscernible points are surveyed. The properties of upper and lower approximations of fuzzy sets by similarity relations are thoroughly studied. Lastly the potential usefulness of the fuzzy rough set notions for logical inference in the presence of both fuzzy predicates and graded indiscernibility is indicated. Especially fuzzy rough sets may provide a nice semantic background for modal logic involving fuzzy modalities and/or fuzzy sentences.

On Intuitionistic Fuzzy Sets Theory

Abstract: This book aims to be a comprehensive and accurate survey of state-of-art research on intuitionistic fuzzy sets theory and could be considered a continuation and extension of the authors previous book on Intuitionistic Fuzzy Sets, published by Springer in 1999 (Atanassov, Krassimir T., Intuitionistic Fuzzy Sets, Studies in Fuzziness and soft computing, ISBN 978-3-7908-1228-2, 1999). Since the aforementioned book has appeared, the research activity of the author within the area of intuitionistic fuzzy sets has been expanding into many directions. The results of the authors most recent work covering the past 12 years as well as the newest general ideas and open problems in this field have been therefore collected in this new book.

Fuzzy logic controllers are universal approximators

Abstract: In this paper, we consider a fundamental theoretical question on why does fuzzy control have such a good performance for a wide variety of practical problems. We try to answer this fundamental question by proving that for each fixed fuzzy logic belonging to a wide class of fuzzy logics, and for each fixed type of membership function belonging to a wide class of membership functions, the fuzzy logic control systems using these two and any method of defuzzification are capable of approximating any real continuous function on a compact set to arbitrary accuracy. On the other hand, this result can be viewed as an existence theorem of an optimal fuzzy logic control system for a wide variety of problems. >

Fuzzy Min-Max Neural Networks-Part 1 : Classification

Abstract: A supervised learning neural network classifier that utilizes fuzzy sets as pattern classes is described. Each fuzzy set is an aggregate (union) of fuzzy set hyperboxes. A fuzzy set hyperbox is an n-dimensional box defined by a min point and a max point with a corresponding membership function. The min-max points are determined using the fuzzy min-max learning algorithm, an expansionxontraction process that can learn nonlinear class boundaries in a single pass through the data and provides the ability to incorporate new and refine existing classes without retraining. The use of a fuzzy set approach to pattern classification inherently provides degree of membership information that is extremely useful in higher level decision mak- ing. This paper will describe the relationship between fuzzy sets and pattern classification. It explains the fuzzy min-max classifier neural network implementation, it outlines the learning and recall algorithms, and it provides several examples of operation that demonstrate the strong qualities of this new neural network classifier. AmRN classification is a key element to many engi- P neering solutions. Sonar, radar, seismic, and diagnostic applications all require the ability to accurately classify a situation. Control, tracking, and prediction systems will often use classifiers to determine input-output relationships. Because of this wide range of applicability, pattern classification has been studied a great deal (13), (15), (19). This paper describes a neural network classifier that creates classes by aggregating several smaller fuzzy sets into a single fuzzy set class. This technique, introduced in (42) as an extension of earlier work (41), can learn pattern classes in a single pass through the data, it can add new pattern classes on the fly, it can refine existing pattern classes as new information is received, and it uses simple operations that allow for quick execution. Fuzzy min-max classification neural networks are built using hyperbox fuzzy sets. A hyperbox defines a region of the n-dimensional pattern space that has patterns with full class membership. A hyperbox is completely defined by its min point and its max point, and a membership function is defined with respect to these hyperbox min-max points. The min-max (hyperbox) membership function combination defines a fuzzy set, hyperbox fuzzy sets are aggregated to form a single fuzzy set class, and the resulting structure fits naturally into a neural network framework; hence this classification system is called a fuzzy min-max classification neural network. Learning in the fuzzy min-max classification neural network is performed by properly placing and adjusting hyperboxes in the pattern space.