By Simona Salicone
ISBN-10: 0387306552
ISBN-13: 9780387306551
ISBN-10: 0387463283
ISBN-13: 9780387463285
The expression of uncertainty in size is a hard element for researchers and engineers operating in instrumentation and size since it includes actual, mathematical and philosophical matters. This challenge is intensified via the constraints of the probabilistic procedure utilized by the present average (GUM). this article is the 1st to make complete use of the mathematical concept of facts to specific the uncertainty in measurements. It provides an summary of the present usual, then pinpoints and constructively resolves its boundaries via its new angle. The textual content offers numerous instruments for comparing uncertainty, starting with the probabilistic strategy and concluding with the expression of uncertainty utilizing random-fuzzy variables. The exposition is pushed via various examples. The publication is designed for instant use and alertness in learn and laboratory paintings. necessities for college kids contain classes in facts and dimension technology. except a lecture room surroundings, this ebook can be utilized by way of practitioners in numerous fields (including utilized arithmetic, utilized likelihood, electric and machine engineering, and experimental physics), and by way of such associations because the IEEE, ISA, and nationwide Institute of criteria and know-how.
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Additional info for Measurement Uncertainty: An Approach Via the Mathematical Theory of Evidence
Example text
Hence, within the possibility theory, the focal elements of X are some or all of the subsets in the complete sequence in Eq. 20). 20), then two indexes i and j exist so that A = Ai and B = Aj . Hence, Ai ∩ Aj = Ai if i < j Aj if i > j Ai if i > j Aj if i < j and Ai ∪ Aj = Then, by applying Eq. 10): Bel(A ∩ B) = Bel(Ai ∩ Aj ) min(i,j) j i m(Ak ) = min = k=1 m(Ak ), k=1 m(Ak ) k=1 = min [Bel(Ai ), Bel(Aj )] = min [Bel(A), Bel(B)] and by applying Eq. 1 Necessity and possibility functions When a belief function satisfies Eq.
This situation may happen only if the considered set A is a focal element; that is, ∃i so that A = Ai . Under this situation, Eqs. 33) apply; and Eq. 38) is thus proved. If Pos(A) < 1, it means that, in Eq. 32), not all focal elements Ai , for i = 1, . . , n, are considered. This situation may happen only if the considered set A is not a focal element; that is, A = Ai for each i = 1, . . , n. 20) is a subset of A; hence, Nec(A) = 0 and Eq. 39) is proved. 2 The possibility distribution function An important property of the possibility theory is that a frame of discernment X may be completely determined by the plausibilities assigned to singletons, that is, to subsets that include only one single element x of X.
25), Pos(A) = max[Pos({x1 , x2 , . . , xn−1 }), Pos(xn )] = max[max[Pos(x1 ), Pos(x2 ), . . , Pos(xn−1 )], Pos(xn )] = max[Pos(x1 ), Pos(x2 ), . . , Pos(xn−1 ), Pos(xn )] = max r(x) x∈A 2 When X is not finite, Eq. 3 Possibility theory 55 If Eq. 42) x∈X Let us also consider that the normalization condition, which applies to the universal set X, is also valid for all nested focal elements of X, as shown in Fig. 9. It is possible to prove that each basic probability assignment function m represents exactly one possibility distribution function r, and vice versa.
Measurement Uncertainty: An Approach Via the Mathematical Theory of Evidence by Simona Salicone
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