Cluster validation is an important issue in fuzzy clustering research and many validity measures, most of which are motivated by intuitive justification considering geometrical features, have been developed. This paper proposes a new validation approach, which evaluates the validity degree of cluster partitions from the view point of the optimality of objective functions in FCM-type clustering....

The well-known fuzzy c-means algorithm is an objective function based fuzzy clustering technique that extends the classical k-means method to fuzzy partitions. By replacing the Euclidean distance in the objective function other cluster shapes than the simple (hyper-)spheres of the fuzzy c-means algorithm can be detected, for instance ellipsoids, lines or shells of circles and ellipses. We propo...

In this paper we investigate various properties of equivalence classes of fuzzy equivalence relations over a complete residuated lattice. We give certain characterizations of fuzzy semipartitions and fuzzy partitions over a complete residuated lattice, as well as over a linearly ordered complete Heyting algebra. In the latter case, for a fuzzy equivalence relation over a linearly ordered comple...

Given the current limitations in fuzzy clustering metric, the aim of this paper is to present new wasserstein metric based adaptive fuzzy clustering methods for partitioning symbolic interval data. Wasserstein metric shows adavantages in digging distribution information in symbolic interval data. Besides, the proposed fuzzy clustering methods also emphasize correlation structure between indices...

Fuzzy implications are one of the most important fuzzy logic connectives. In this work, we conduct a systematic algebraic study on the set I of all fuzzy implications. To this end, we propose a binary operation, denoted by ~, which makes (I,~) a non-idempotent monoid. While this operation does not give a group structure, we determine the largest subgroup S of this monoid and using their represe...

This paper denes and investigates the ergodic proper-ties of the entropy of a countable partition of a fuzzy dynamical sys-tem at different points of the state space. It ultimately introducesthe local fuzzy entropy of a fuzzy dynamical system and proves itto be an isomorphism invariant.

Equivalence relations and orderings are key concepts of mathematics. For both types of relations, formulations within the framework of fuzzy relations have been proposed already in the early days of fuzzy set theory. While similarity (indistinguishability) relations have turned out to be very useful tools, e.g. for the interpretation of fuzzy partitions and fuzzy controllers, the utilization of...

The work presents an extension of the fuzzy approach to 2-D shape recognition [1] through refinement of initial or coarse classification decisions under a two pass approach. In this approach, an unknown pattern is classified by refining possible classification decisions obtained through coarse classification of the same. To build a fuzzy model of a pattern class horizontal and vertical fuzzy pa...

The main purpose of this paper is to prove the existence of the fuzzy core of an exchange economy with a heterogeneous divisible commodity in which preferences of players are concave measures defined on a σ-algebra of admissible pieces of the total endowment of the commodity. The problem is formulated as the partitioning of a measurable space among finitely many players. Applying the Yosida– He...