Showing papers on "Membership function published in 1973"

Cluster Validity with Fuzzy Sets

Abstract: Given a finite, unlabelled set of real vectors X, one often presumes the existence of (c) subsets (clusters) in X, the members of which somehow bear more similarity to each other than to members of adjoining clusters. In this paper, we use membership function matrices associated with fuzzy c-partitions of X, together with their values in the Euclidean (matrix) norm, to formulate an a posteriori method for evaluating algorithmically suggested clusterings of X. Several numerical examples are offered in support of the proposed technique.

On Fuzzy-Mathematical Programming

Abstract: In problems of system analysis, it is customary to treat imprecision by the use of probability theory. It is becoming increasingly clear, however, that in the case of many real world problems involving large scale systems such as economic systems, social systems, mass service systems, etc., the major source of imprecision should more properly be labeled ‘fuzziness’ rather than ‘randomness.’ By fuzziness, we mean the type of imprecision which is associated with the lack of sharp transition from membership to nonmembership, as in tall men, small numbers, likely events, etc. In this paper our main concern is with the application of the theory of fuzzy sets to decision problems involving fuzzy goals and strategies, etc., as defined by R. E. Bellman and L. A. Zadeh [1]. However, in our approach, the emphasis is on mathematical programming and the use of the concept of a level set to extend some of the classical results to problems involving fuzzy constraints and objective functions.

Fuzzy sets and decision theory

Abstract: The problem of making decisions to classify the objects of a certain universe into two or more suitable classes has been considered in the setting of fuzzy sets theory. A measure of the total amount of uncertainty that arises in making decisions has been proposed in the general case. This quantity reduces to the “entropy” of a fuzzy set in the case of two classes. Other quantities which play a relevant role in this theory are the “energy” and the “effective power” of a fuzzy set, defined as ∑ i = 1 N w i f i a n d φ ∑ i = 1 N f i , respectively, where w is a nonnegative weight function and φ a nonnegative constant. If w = constant and φ ≠ 0, the energy is proportional to the effective power and, therefore, to the “power” of the fuzzy set. The maximum of the uncertainty has been calculated in some cases of interest, keeping constant the total energy and effective power. In particular the Maxwell-Boltzmann, Fermi-Dirac, and Bose-Einstein distributions are derived. Some applications to decision theory are considered in the case of both deterministic and probabilistic decisions. Finally, the analogies that exist between the previous concepts and the thermodynamic ones are discussed.

Constructing Fuzzy Measure and Grading Similarity of Patterns by Fuzzy Integral

Abstract: Lately, studies on fuzzy systems as well as applications of fuzzy set theory have attracted the attention of many researchers. The author has suggested the concept of fuzzy measure and fuzzy integral for representing fuzzy systems. The fuzzy measure without additivity may be regarded as a subjective one by which •gfuzziness•h is measured. At first, this paper explains the fuzzy measure extended onto a family of sets including fuzzy sets and the concrete methods for constructing the fuzzy measure. As an example, the fuzzy measure gƒÉ, -1<ƒÉ<•‡, is defined and its characteristics are clarified. Here, gƒÉ, corresponding to the probability measure when ƒÉ=0, is continuous for ƒÉ. Furthermore, the new concept of the ƒÉcomplement of a fuzzy set is proposed. This has flexibility due to the parameter ƒÉ. It is an extension of the complement defined by L.A. Zadeh. Next, this paper considers the problems appearing when a human grades the similarity of several patterns with no sharp boundaries. The human decision mechanism is represented by a macro model where the fuzzy integral is used. Man's subjective characteristics in grading the similarity are obtained through the fuzzy measure identified so that both, human and model's, outputs agree with each other. A simple expriment on the above problem was performed. The experimental results show that the outputs of the model agree approximately with those obtained by human evaluation.