By Dmitri A. Viattchenin

ISBN-10: 3642355358

ISBN-13: 9783642355356

ISBN-10: 3642355366

ISBN-13: 9783642355363

The current publication outlines a brand new method of possibilistic clustering within which the sought clustering constitution of the set of items is predicated at once at the formal definition of fuzzy cluster and the possibilistic memberships are made up our minds at once from the values of the pairwise similarity of items. The proposed process can be utilized for fixing varied type difficulties. right here, a few strategies that may be precious at this goal are defined, together with a technique for developing a collection of categorized items for a semi-supervised clustering set of rules, a strategy for decreasing analyzed characteristic area dimensionality and a equipment for uneven information processing. in addition, a method for developing a subset of the main acceptable possible choices for a suite of susceptible fuzzy choice kinfolk, that are outlined on a universe of 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 attracted to and paintings with fuzzy clustering and its applications

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Let X = {x1 , , x n } be the finite universe and R be a fuzzy relation on X with μ R ( x i , x j ) being its membership function. 54) for all α ∈ (0, Proj( R)] . Proof. 23) can be rewritten as Proj( R) = max μ R ( xi , x j ) , xi , x j ∀xi , x j ∈ X . If the fuzzy relation R is the subnormal fuzzy relation, then Proj( R ) < 1 . So, the values of the membership function μ R ( x i , x j ) are absent on the interval (Proj( R ),1] and α ∉ (Proj( R),1] . On the other hand, if the fuzzy R is the α ∈ (0, Proj( R) = 1] .

Proofs of these corollaries are obvious and are omitted here. 1. 102), then the supports { Aα1 ,, Aαc } of the α -cores { A(1α ) ,, A(cα ) } of the fuzzy clusters form a partition of a set of objects into disjoint sets. 2. 104), then the crisp coverage which is formed by the supports { Aα1 ,, Aαc } of the α -cores { A(1α ) ,, A(cα ) } of fuzzy clusters of the fuzzy c partition P ( X ) = { A1 ,, Ac } is minimal. 38 1 Introductioon So, the notion of the α -cores of fuzzy clusters allows to include each object oof X = {x1 ,, xn } in the smallest s number ~ c , 1 ≤ c~ ≤ c of fuzzy clusters of thhe fuzzy c -partition P (X ) = { A1 ,, A c } , which is the result of classification.

Hence, the condition c xi ∈ Aαl , ∀i ∈ {1,, n} is met. 17). Thus, the condition μ A(α~ ) ( xi ) = 0 will be met for some value α~ > αˆ . Hence, the condition c xi ∉ Supp( A(α~ ) ) will be met and thecondition xi ∉ Aαl~ will also be satisfied. l =1 That is why the theorem is correct. □ Some propositions are corollaries of this theorem and these corollaries were formulated also in [128]. Let us consider two most important corollaries. Proofs of these corollaries are obvious and are omitted here.

### A Heuristic Approach to Possibilistic Clustering: Algorithms and Applications by Dmitri A. Viattchenin

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