Wang, Zhenyuan and Klir, George J: (1992) Fuzzy Measure Theory, Plenum Press, New York  Home/Search   Document Not in Database   Summary   Related Articles   Check

....been supported by CICYT under project TIC97 1135 C04 01. ffl To model conflict between several sources of information [52, 36] There is a variety of mathematical models for imprecise probability [53, 57] comparative probability orderings [25, 26] possibility measures [60, 23] fuzzy measures [48, 58, 28], belief functions [43, 47] Choquet capacities [29, 14] interval probabilities [59, 18] convex sets of probabilities [13, 53, 19] Among all these models we think that convex sets of probabilities is the most appropriate to represent and calculate with imprecise probabilities. We think that, ....

Z. Wang and G.J. Klir. Fuzzy Measure Theory. Plenum Press, New York, 1992.

....of the fuzzy set of more or less possible situations in the fuzzy set of antecedents of preferred consequences. Therefore the pessimistic utility can be rewritten as u (f) sup ff2[0;1] min(ff; N(F ff ) with F ff = fs 2 S; u(f(s) ffg, which is a particular case of Sugeno integral [11] [14] Z S Gammah(s) ffi g( Delta) sup ff2[0;1] min(ff; g(H ff ) with H ff = fs 2 S; h(s) ffgand g is a set function, monotonic with respect to set inclusion, such that g( 0 and g(S) 1. Dually, the optimistic qualitative utility is the possibility of the same fuzzy event: u (f) Pi(F ....

Z. Wang, G.J. Klir. Fuzzy Measure Theory, Plenum Press, New York and London, 1992.

....(Smith [33] Kyburg [26] Good [17] Models 3 6 are special types of lower or upper probability: 3. belief functions (Dempster [6] Shafer [30, 31] 4. Choquet capacities (Choquet [4] Huber [20, 21] Denneberg [7] 5. fuzzy measures (Klir and Folger [23] de Campos et al. 3] Wang and Klir [43]) 6. possibility measures (Zadeh [45] Dubois and Prade [9] Klir and Folger [23] 7. sets of probability measures (Levi [28] Berger [1] 8. upper and lower previsions (Williams [44] Walley [35, 36] There are many other models which allow imprecision, including classi catory models ....

Wang, Z. and Klir, G. J. (1992). Fuzzy Measure Theory. Plenum Press, New York.

....a few other notions of uncertainty that they generalize: A belief function B on W is a function B : 2 W # [0, 1] satisfying certain axioms [Sha76] These axioms certainly imply property A1, so a belief function is a plausibility measure. A fuzzy measure (or a Sugeno measure) f on W [WK92] is a function f : 2 W ## [0, 1] that satisfies A1 and some continuity constraints. A possibility measure [DP90] Poss is a fuzzy measure such that Poss(W ) 1, Poss(#) 0, and Poss(A) sup w#A (Poss( w ) An ordinal ranking (or # ranking) on W (as defined by [GP92] based on ideas that ....

Z. Wang and G. J. Klir. Fuzzy Measure Theory. Plenum Press, New York, 1992. 64

.... to model uncertainty about a prior distribution [3, 22] To model the con ict between several sources of information [51, 35] There are various mathematical models for imprecise probability [52, 56] comparative probability orderings [25, 26] possibility measures [59, 23] fuzzy measures [47, 57, 28], belief functions [42, 46] Choquet capacities [30, 16] interval probabilities [58, 6] convex sets of probabilities [15, 52, 7] Out of all these models, we think that convex sets of probabilities are the most suitable for calculating with and representing imprecise probabilities. We think that ....

Z. Wang and G.J. Klir. Fuzzy Measure Theory. Plenum Press, New York, 1992.

....discuss a few other notions of uncertainty that they generalize: A belief function B on W is a function B : 2 W [0; 1] satisfying certain axioms [55] These axioms certainly imply property A1, so a belief function is a plausibility measure. A fuzzy measure (or a Sugeno measure) f on W [59] is a function f : 2 W 7 [0; 1] that satisfies A1 and some continuity constraints. A possibility measure [12] Poss is a fuzzy measure such that Poss(W ) 1, Poss( 0, and Poss(A) sup w2A (Poss(fwg) An ordinal ranking (or ranking) on W (as defined by [31] based on ideas that go ....

Z. Wang and G. J. Klir. Fuzzy Measure Theory. Plenum Press, New York, 1992. 63

....the probability theoretic additivity requirement can probably be replaced by the less restrictive sub additive or super additive requirement. This would require the ideas of mutually singular, absolutely continuous, and RadonNikodym derivative to be defined in terms of the resulting fuzzy measure [68]. Such a development would provide a theoretical tie not only between fuzzy logic controllers and the satisficing principle, but also between neural networks and the satisficing principle. ffl Instead of relying solely upon quadratic cost functions to formulate controllers, a hybrid controller ....

Chen yuan Wang. Fuzzy Measure Theory. Plenum Press, 1992.

....of fuzzy sets, we refer to [11, 19] as typical examples and recommend the recent expository volume [1] and the references therein for an overview. 2 Finite element method with vague and uncertain parameters 2.1 Preliminaries We begin by introducing some preliminary de nitions, c.f. [7, 30]. Let X R k be a nonempty set, F = fF 1 ; Fn g a nite set of distinct subsets (focal sets) of X and the set function m : F [0; 1] a basic probability assignment on F . Then P (A) Bel(A) X F i A F i 2F m(F i ) 1) is the belief measure on (X; X) or the lower probability ....

Z. Wang and G. J. Klir. Fuzzy Measure Theory. Plenum Press, New York, 1992.

....and discuss some of their more important properties, amongst which a monotone convergence theorem. 1. Introduction Since possibility measures were introduced by Zadeh [18] in 1978, the measuretheoretic aspects of possibility measures and possibility integrals have been studied by various authors [2, 4, 5, 6, 8, 9, 11, 12, 13, 14, 15, 16, 17, 19, 20]. The most general de nition of a possibility measure that can be distilled from this work is the following: a possibility measure is a mapping de ned on a complete Boolean algebra [3] B and taking values in a complete lattice L, that is furthermore supremumpreserving , in that (sup j2J a j ) ....

.... (sup j2J a j ) sup j2J (a j ) for any family fa j j j 2 Jg of elements of B . Most of the time, the complete lattice L is taken to be the real unit interval [0; 1] linearly ordered by the natural ordering of real numbers, and B is a complete Boolean algebra of sets (also called ample eld [7, 16]) which is always atomic 1 . In such a context, possibility measures are completely characterised by their distribution, that is, by the values they assume on the atoms of their domains. But sometimes, and most notably in the work of Dubois and Prade on possibilistic logic [10] possibility ....

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Z. Wang and G. J. Klir. Fuzzy Measure Theory. Plenum Press, New York, 1992.

....x, we may associate a number of special events: fX xg = f 2 : X( xg is the strict cut set , fX xg = f 2 : X( xg the dual cut set and fX xg = f 2 : X( xg the strict dual cut set of X at level x. We shall make frequent use of special classes of events, called ample elds [11, 31, 32]. An ample eld R on is a class of subsets of that is closed under arbitrary unions and complementation. It is therefore also closed under arbitrary intersections. For any in the atom [ R of R containing is de ned as [ R = T fA 2 R : 2 Ag. The atoms of R make up a partition ....

....the smallest ample eld for which the sets (s) s 2 S, are all measurable. Solving (4) amounts to extending P to a possibility measure, and (4) is therefore called a possibilistic extension problem. We summarise what is known about this problem in the following de nition and theorem, due to Wang [32]. Generalisations of these results in a more general context have been given by Boyen et al. 2] We want to mention in passing that the distribution g in Theorem 3 is also the basic tool in the construction of a semantics for the possibilistic logic of Dubois et al. 17] In possibilistic ....

Z. Wang and G. J. Klir. Fuzzy Measure Theory. Plenum Press, New York, NY, 1992.

....for understanding the formal developments in the rest of the paper. 2.1. Plump classes, ample elds and measurability. Throughout we shall denote by X a nonempty set. A subset D of the power set (X) of X is called a plump class on X i it is closed under arbitrary unions and intersections [20]. The atom [x] D of D containing the element x of X is de ned as [x] D = T fA j A 2 D and x 2 Ag, and a subset A of X is a called an atom of D i (9x 2 X) A = x] D ) The set of the atoms of D is denoted by XD . Note that XD D and that (8x 2 X) x 2 [x] D ) Also, for any subset A of X, A 2 D ....

.... ) possibility measure, that is, if there exists an ample eld R on X and a (L; possibility measure on (X; R) such that coincides with on S: 8A 2 S) A) A) It is clear that we must at least have that X (S) R. This so called possibilistic extension problem was solved by Wang [19, 20] for [0; 1] valued set functions. Quite recently, Boyen et al. 2] considered and partially solved the more general problem for (L; valued set functions. In this section, we brie y recall the most relevant results in their study. Boyen et al. have generalised Wang s de nition of P consistency ....

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Z. Wang and G. J. Klir. Fuzzy Measure Theory. Plenum Press, New York, 1992.

....between vagueness and uncertainty. Following Hajek [33] we can say that the key concepts of vagueness and uncertainty identify two classes, both characterized by functions from the set of events to the real interval [0, 1] the interpretation function for the first class and the fuzzy measures [58] (possibility, probability and belief function) for the second class. Deep and important properties di#erentiate the two classes: 1. the interpretation functions for the class of vague reasoning are compositional,ortruth functional, i.e. the interpretation function of a sentence is the ....

Z. Wang and G. J. Klir. Fuzzy Measure Theory. Plenum Press, New York, 1992.

.... fuzzy measure g on a continuous set Y is a function g : 2 Y [0; 1] which assigns to each subset of Y a number in the unit interval [0; 1] In order to qualify g as a fuzzy measure, the function g must satisfy certain properties, as the axioms of fuzzy measures de ned in (Klir Folger, 1988; Wang Klir, 1992). Two special types of fuzzy measures are the probability measures and possibility measures. Let P : 2 Y [0; 1] be a probability measure P . By de nition P (A) Z A p(y)dy (7) is the probability measure of A for all A 2 2 Y . The function p : Y is the probability density ....

Wang Z. & Klir G. J. 1992. Fuzzy Measure Theory. New York: Plenum Press.

.... set functions on the power class of a non empty set X [6, 7, 8, 11, 19] In an e ort to de ne such set functions on more general domains, a number of authors have investigated which set functions, de ned on arbitrary classes of subsets, can be extended to possibility measures on the power class [2, 17]. From this study, ample elds emerged as natural domains for possibility measures. Formally, a subclass D of the power class (X) of X is called a plump class on X i it is closed under arbitrary unions and intersections [17] The atom of D containing the element x of X is de ned as [x] D = T ....

....subsets, can be extended to possibility measures on the power class [2, 17] From this study, ample elds emerged as natural domains for possibility measures. Formally, a subclass D of the power class (X) of X is called a plump class on X i it is closed under arbitrary unions and intersections [17]. The atom of D containing the element x of X is de ned as [x] D = T fA j A 2 D and x 2 Ag. The set of the atoms of D is denoted by XD . Clearly, XD D and (8x 2 X) x 2 [x] D ) For any subset A of X, A 2 D , A = S x2A [x] D . An ample eld R on X is a plump class on X that is closed under ....

Z. Wang and G. J. Klir, Fuzzy Measure Theory (Plenum Press, New York, 1992).

.... = fg: g (A 1 [ A 2 ) g (A 1 ) g (A 2 ) Delta g (A 1 ) Delta g (A 2 ) 5) Gamma1 1 To each element of the power set P (F ) an unambiguous degree of importance is assigned (for further details, e.g. the determination of or cross references to other types of fuzzy measures, see [SW90, WK92]) Using fuzzy measures thus enables the incorporation of knowledge about the sensitivity of the individual colour channels into the search process by first determining the relative performance of the channels and subsequently assigning these importance values to the different measures. One of ....

Z. Wang and G.J. Klir. Fuzzy measure theory. Plenum Press, 1992.

.... sets [25] and evidence theory [4, 28] in the mid 1960 s there has been a proliferation of mathematical methods for the representation of uncertainty which generalize beyond classical probability theory [24] In addition to a fully developed fuzzy systems theory [25] there are also fuzzy measures [32], rough sets [27] random sets [8, 21] Dempster Shafer bodies of evidence [9, 28] and possibilistic systems [2] There is a pressing need to synthesize these fields within a collective as General Information Theory (git) 23] searching out larger formal frameworks within which to place these ....

....a triangular conorm t: 0; 1] 2 7 [0; 1] resp. triangular norm u: 0; 1] 2 7 [0; 1] as an associative, commutative, monotonic operator with identity 0 (resp. 1) R : ht; ui is a conorm semiring if u distributes over t. The function : 2 Omega 7 [0; 1] is a (finite) fuzzy measure [32] if ( 0 and 8A; B Omega ; A B (A) B) is called distributional if there exists a conorm t such that 8A Omega ; F i 2A q ( i ) A) where q : Omega 7 [0; 1] with q ( i ) f i g) the distribution of . Furthermore, is normal when ( Omega Gamma = 1, so that F ....

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Wang, Zhenyuan and Klir, George J: (1992) Fuzzy Measure Theory, Plenum Press, New York

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Wang, Z. and Klir, G. J., Fuzzy Measure Theory. Plenum Press, New York, 1992.

....measures. They require a more general measure theory, which can deal with nonadditive measures. It is usually required that the measures be monotonic with respect to set inclusion and continuous or semicontinuous; a theory dealing with measures of this kind is usually called a fuzzy measure theory [22]. 2 Precise and Imprecise Probabilities In classical probability theory, elementary events are required to be pairwise disjoint and the probability of each is required to be expressed precisely by a real number in the unit interval [0, 1] This precision requirement of classical probability ....

....proposed candidate (12) for the Shannon like measure in evidence theory is that it is fully formulated in terms of singletons or their complements, but a belief function of evidence theory is not determined, in general, by its values on singletons and their complements. However, as is well known [22], a significant subset of belief functions and plausibility functions can be represented by the l measures proposed by Sugeno [20] For this subset of belief functions, any belief function is uniquely determined by its values on singletons. Uncertainty represented in terms of l measures is one ....

Wang, Z. and G. J. Klir, Fuzzy Measure Theory. Plenum Press, New York, 1992.

....is applied. 4.2.1 Non additive set function Let be a nonempty set and ) be the power set of . We use the symbol to denote a non negative set function defined on ) with the properties 0 ) If , 1 ) is said to be regular. It is a generalization of classic measure [9]. When is finite, is usually called a fuzzy measure if it satisfies monotonicity, i.e. # , for For a non negative set function , there are some associated concepts. is said to be additive if ) for ) to be subadditive ....

....fuzzy measure is determined. 4.2. 3 Using Choquet integral to compute overall group competence Due to the non additivity of the set function , some new types of integrals (known as non linear integrals) have to be used, which can be considered to be the generalization of the weighted mean [9]. The advantage of using fuzzy integrals is that the interactions of all factors in a factor space can be taken into account. Fuzzy integrals have found a few applications in CBR systems [11 12] X. Z. Wang and D. S. Yeung used fuzzy integrals to compute the overall similarity between problems and ....

Z. Wang and G.J.Klir, Fuzzy Measure Theory, Plenum, New York, 1992

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Wang, Zhenyuan and Klir, George J: (1992) Fuzzy Measure Theory, Plenum Press, New York

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Wang, Zhenyuan and Klir, George J: (1992) Fuzzy Measure Theory, Plenum Press, New York

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Wang, Zhenyuan and Klir, George J: (1992) Fuzzy Measure Theory, Plenum Press, New York

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Z. Wang, G.J. Klir. Fuzzy Measure Theory, Plenum Press, New York and London, 1992.

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Z. Wang and G.J. Klir, Fuzzy Measure Theory (Plenum Press, New York, 1992).

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Wang, Z. and Klir, G. J. Fuzzy Measure Theory. Plenum Press, 1992

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