New PDF release: Aggregation and Fusion of Imperfect Information

By Sergei Ovchinnikov (auth.), Dr. Bernadette Bouchon-Meunier (eds.)

ISBN-10: 3662110733

ISBN-13: 9783662110737

ISBN-10: 3790818895

ISBN-13: 9783790818895

This booklet offers the most instruments for aggregation of data given via numerous participants of a bunch or expressed in a number of standards, and for fusion of information supplied via a number of resources. It specializes in the case the place the provision wisdom is imperfect, this means that uncertainty and/or imprecision needs to be taken under consideration. The e-book includes either theoretical and utilized reviews of aggregation and fusion equipment mainly frameworks: chance concept, facts thought, fuzzy set and risk idea. The latter is extra built since it permits to control either obscure and unsure wisdom. functions to decision-making, photograph processing, keep an eye on and category are defined. The reader can discover a cutting-edge of the most methodologies, in addition to extra complicated effects and outlines of utilizations.

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44 5 Idempotency In this section we study the idempotency of T -S aggregation functions defined from a t-norm and a t-conorm. That is, we consider the following two main classes: F1(x,y) = S(T(k,S1(x,y)),T1(x,y)), F2(x, y) = T(S(k, T1(x, y)), S 1(x, y)), x,yE [0,1], x, y E [0, 1], (8) {9) where T1 is a t-norm and 81 is a t-conorm, and determine conditions so that either F1 or F2 is idempotent: F1(x, x) = x or F2(x, x) = x for all x E [0, 1]. We distinguish the usual cases as follows, starting with the extremes 'min' and 'max'.

N indicates a permutation of ai, i = 1, ... , n such that a(l) a(2) ~ ... ~a(n)•andA(i) ~ := {X(i), ... ,X(n)}· Another definition was proposed later by Murofushi and Sugeno [28], using a concept introduced by Choquet in capacity theory [2]. Definition 3 Let p, be afuzzy measure on X. The discrete Choquet integral ofa~. , an with respect to p, is defined by n Cl'(a~, ... ,an):= ~)a(i)- a(i-l))p,(A(i)), i=l with the same notations as above, and a(o) = 0. An equivalent expression of (3) is n Cl'(a,, ...

Hence, we have F1 = F2 = max. Suppose 0 < k < 1. If there is an x E [0, 1] such that S 1(x,x) < 1- k then we have X= Tl(x, x) = max(O, k + Tl(x, x) + sl (x, x)- 1), which is possible in the present case if and only if T1(x,x) = 0 = x. Similarly, T1(x, x) > 1- k implies x = 1. Therefore, we have the inequality Tl (x, X) ~ 1- k ~ sl (x, X) for all X E (0, 1)' whence we have T1(x,y) ~ 1- k ~ S1(x,y) for all x,y E (0, 1). ) of T1 cannot be continuous at the point 1 when x > 1-k. ) cannot be continuous at 0 when x < 1 - k, whence discontinuity of F 1 and F2 at the corresponding points follow.

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Aggregation and Fusion of Imperfect Information by Sergei Ovchinnikov (auth.), Dr. Bernadette Bouchon-Meunier (eds.)

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