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 “fuzziness” 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 gλ, -1<λ<∞, is defined and its characteristics are clarified. Here, gλ, corresponding to the probability measure when λ=0, is continuous for λ. Furthermore, the new concept of the λ-complement of a fuzzy set is proposed. This has flexibility due to the parameter λ. 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.
