Nonlinear Integrals And Their Applications In Data Mining by Zhenyuan Wang, Rong Yang, Kwong-Sak Leung PDF

By Zhenyuan Wang, Rong Yang, Kwong-Sak Leung

ISBN-10: 9812814671

ISBN-13: 9789812814678

Concerning the set of all function attributes in a given database because the common set, this monograph discusses quite a few nonadditive set features that describe the interplay one of the contributions from function attributes in the direction of a thought of aim characteristic. Then, the proper nonlinear integrals are investigated. those integrals might be utilized as aggregation instruments in details fusion and knowledge mining, corresponding to artificial overview, nonlinear multiregressions, and nonlinear classifications. a few equipment of fuzzification also are brought for nonlinear integrals such that fuzzy info will be taken care of and fuzzy details is retrievable. The booklet is appropriate as a textual content for graduate classes in arithmetic, desktop technology, and knowledge technological know-how. it's also helpful to researchers within the suitable zone.

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Additional resources for Nonlinear Integrals And Their Applications In Data Mining (Advances in Fuzzy Systems-Applications and Theory - Vol. 24) (Advances in Fuzzy Systemss - Applications and Theory)

Example text

9 (Decomposition Theorem I) A= U α ∈[ 0, 1] αAα = U αAα . α ∈( 0, 1] Proof. On one hand, m A ( x) ≥ αχ Aα ( x) = mαAα ( x) for every x ∈ X and every α ∈ [0, 1] . , A ⊇ Uα ∈[ 0, 1]αAα . On the other hand, for any given x ∈ X , denoting mA (x) by α (x) , we have m A ( x) = α ( x) = α ( x) χ Aα ( x ) ( x) ≤ sup αχ Aα ( x) = m α ∈[ 0 , 1] U αAα ( x) α∈[ 0 , 1] since x ∈ Aα ( x ) . This means A ⊆ Uα ∈[ 0, 1]αAα . Consequently, A = Uα ∈[ 0, 1]αAα . The second equality Uα ∈[ 0, 1]αAα = Uα ∈( 0, 1]αAα is □ evident.

7 Categorize class C given in the following descriptions as of a semiring, a ring, an algebra, a σ-ring, a σ-algebra, or none of them: (1) X = (–∞, ∞), C is the class of all bounded, left open, and right closed intervals; (2) X = {1, 2, L} , C = { A | A ⊆ X , | A |≤ 3} where | A | denotes the number of points in set A and called the cardinality of A ; (3) X is a nonempty set with | X |≥ 2 , E is a nonempty proper subset of X, C = {F | E ⊂ F ⊂ X}; (4) X is a nonempty set, E is a proper subset of X, C = {F | F ⊆ E}; (5) X is a nonempty set, E is a nonempty subset of X, C = {E}.

Let { At } = { At | t ∈ T } be a family of sets where T is a nonempty index set. We may define the union and the intersection of { At } as well. 15 The union of { At | t ∈ T } , denoted by U t∈T At , is the set consisting of all points that belong to At for at least one t ∈ T . That is, U t∈T At = {x | x ∈ At for at least one t ∈ T } . 16 The intersection of { At | t ∈ T } , denoted by I t∈T At , is the set consisting of all points that belong to all At for t ∈ T . That is, I t∈T At = {x | x ∈ At for all t ∈ T } .

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Nonlinear Integrals And Their Applications In Data Mining (Advances in Fuzzy Systems-Applications and Theory - Vol. 24) (Advances in Fuzzy Systemss - Applications and Theory) by Zhenyuan Wang, Rong Yang, Kwong-Sak Leung


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