By Zhenyuan Wang, Rong Yang, Kwong-Sak Leung

Concerning the set of all function attributes in a given database because the common set, this monograph discusses a variety of nonadditive set features that describe the interplay one of the contributions from characteristic attributes in the direction of a thought of objective characteristic. Then, the appropriate nonlinear integrals are investigated. those integrals might be utilized as aggregation instruments in details fusion and knowledge mining, akin to man made evaluate, nonlinear multiregressions, and nonlinear classifications. a few equipment of fuzzification also are brought for nonlinear integrals such that fuzzy information might be handled and fuzzy details is retrievable. The e-book is acceptable as a textual content for graduate classes in arithmetic, machine technological know-how, and knowledge technological know-how. it's also precious to researchers within the suitable region.

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Extra info 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)

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14 Let X = (–∞, ∞) and ≅ be the relation on X × X defined by (x1, y1) ≅ (x2, y2) iff x1 + y1 = x2 + y2 . Prove that ≅ is an equivalence relation. 15 Let X = {a, b, c, d } and C = {∅, A, {a, c}, B, X } . Find all possible set pairs A and B such that C is a chain with respect to set inclusion ⊆. 16 Let E be the set of all irrational numbers in [0, 1]. (1) Find sup E and inf E. (2) Find sequences {ai | ai ∈ E , i = 1, 2, L} and {bi | bi ∈ E , i = 1, 2, L} lim i → ∞ ai = sup E and lim i → ∞ bi = inf E .

So, Aα is convex. 3, Fuzzy sets Y, O, M, and Y are convex fuzzy sets. The membership function of a convex fuzzy set is not necessarily convex (also called concave down, in some books) in the meaning discussed in calculus. 10 Let X be the real line R. 7). However, 2 function e − x is not convex (concave down) on R in calculus. 5 Decomposition Theorems In this section, we discuss how to express a fuzzy set by its α-cuts. Let A be a fuzzy subset of universal set X. For any crisp set E and any real number α ∈ [0, 1] , we use αE to denote the fuzzy set having membership function 38 Nonlinear Integrals and Their Applications in Data Mining m 1 α -2 -1 0 1 x 2 Fig.

13 The class of all countable subsets of X and their complements is a σ-algebra. The power set of X is a σ-algebra; any σ-algebra is a σ-ring as well as an algebra; any σ-ring or algebra is a ring; and any ring is a semi-ring. 1. Power set σ-Algebra σ-Ring Algebra Ring Semiring Fig. 1 Relations among classes of sets. 16 Nonlinear Integrals and Their Applications in Data Mining For any given semiring S, there exists at least one set E such that E ∈ S. Since E ⊆ E , for any finite class {C0 , C1 , L, Cn } of sets in S satisfying E = C0 ⊆C1⊆ L ⊆ Cn = E , we must have C0 =C1= L= Cn = E , such that D i = Ci − Ci −1 = ∅ .

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