By Dmitri A. Viattchenin

ISBN-10: 3642355358

ISBN-13: 9783642355356

ISBN-10: 3642355366

ISBN-13: 9783642355363

The current ebook outlines a brand new method of possibilistic clustering within which the sought clustering constitution of the set of gadgets is predicated without delay at the formal definition of fuzzy cluster and the possibilistic memberships are decided without delay from the values of the pairwise similarity of items. The proposed strategy can be utilized for fixing diversified category difficulties. right here, a few concepts that would be worthwhile at this objective are defined, together with a technique for developing a collection of categorized items for a semi-supervised clustering set of rules, a technique for decreasing analyzed characteristic house dimensionality and a equipment for uneven info processing. additionally, a method for developing a subset of the main applicable possible choices for a suite of vulnerable fuzzy choice kinfolk, that are outlined on a universe of possible choices, is defined intimately, and a mode for speedily prototyping the Mamdani’s fuzzy inference structures is brought. This ebook addresses engineers, scientists, professors, scholars and post-graduate scholars, who're drawn to and paintings with fuzzy clustering and its applications

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Extra resources for A Heuristic Approach to Possibilistic Clustering: Algorithms and Applications

Example text

Xn } is the set of objects. So, the two-way data matrix can be represented as follows: Xˆ n×m1  xˆ11   xˆ 2 = 2   xˆ 1  n xˆ12  xˆ1m1   xˆ 22  xˆ 2m1  . 69) Therefore, the two-way data matrix can be represented as Xˆ = ( xˆ 1 ,  , xˆ m1 ) using n -dimensional column vectors xˆ 1 , t1 = 1,  , m1 , composed of the t elements of the t1 -th column of Xˆ . In the relational approach to fuzzy clustering, the problem of the data classification is solved by expressing a relation which quantifies either similarity or dissimilarity between pairs of objects.

3. Compute different fuzzy c -partitions for c = 2,  , c max ; Compute the value of a validity criterion; Seek for the extreme value of validity criterion and set the optimal number of clusters to its correspondent c value. On the other hand, some fuzzy clustering algorithms were proposed that do not require the pre-definition of the number of clusters. The CA-algorithm which was proposed by Frigui and Krishnapuram [37] and the E-FCM-algorithm which was developed by Kaymak and Setnes [62] are good examples of such clustering procedures.

56), then a condition R(α 0 ) = R is met. Proof. 60). 1 Fundamentals of Fuzzy Sets Theory 21 Thus, the α -level fuzzy relations R(α  ) can be constructed from the fuzzy relation R for all α  ∈ (0,1] . The theorem of decomposition for the fuzzy relations was formulated in [117]. 3. Let X = {x1 ,  , x n } be a finite universe and R be a fuzzy relation on X with μ R ( x i , x j ) being its membership function. 25) for all  α  ∈ (0,1] . Proof. Let R(α  ) be fuzzy relations on X with their membership functions μ R(α ) ( xi , x j ) for  μ R ( xi , x j ) = μ R(α 0) all ∪∪ R(α Z ) threshold values α  ∈ (0,1] .

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