G. Klir and T. Folger. Fuzzy Sets, Uncertainty, and Information. Prentice-Hall, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....may have different degrees of belief in these statements. Therefore, to describe the experts knowledge adequately, we must also describe these degrees of belief. There are several ways to assign a degree of belief d(S) to a statement S [4] One of the natural methods ( 4] IV.1. d; 3] 2] [5]) is to take several (N) experts, and ask each of them whether he believes that S is true. If Y (S; N) of them answer yes , we take d(S) Y (S; N) N as the desired degree of belief. If all the experts believe in S, then this value is 1 ( 100 ) if half of them believe in S, then d(S) 0:5 ....

G. Klir, T. A. Folger, Fuzzy sets, uncertainty, and information, Prentice-Hall, U.K., 1988.

....situations [57] ffl When we have few information to evaluate probabilities [53, 54, 56] ffl When available information is not specific enough. For example, when we extract balls from an urn with 10 balls, where 5 are red and 5 are white or black, but we do not known the exact rate of each one [21, 43, 30]. ffl In robust Bayesian inference, to model uncertainty about a prior distribution [3, 22] This work has 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 ....

G.J. Klir and T.A. Folger. Fuzzy Sets, Uncertainty, and Information. Prentice-Hall, Englewood Cliffs, New Jersey, 1988.

....Of partic ular interest here is the development of information theoretic methods which easily incorporate uncertainties from broad assessments. These informationbased approaches can be shown to provide a foundation to unify approxi mate reasoning techniques with traditional probabilistic methods [5, 4]. Un fortunately, these techniques fail to fully incorporate understanding of the underlying system. The result is an analysis that often provides inconclusive solutions. A novel feature of this work is the ability to incorporate understanding of the system constraints. This paper extends earlier ....

....next section. 3.2 Fuzzy logic Fuzzy sets are useful for representing relationships as linguistic descriptions of classes of objects such as those given by experts. An element of a fuzzy set is an ordered pair consisting of an element and the degree of membership in the fuzzy set for that element [5]: x with fuzzy set A: A: X,A(X) Ix X (7) where X is the universe of discourse and IA(X) represents the degree to which x matches the characteristic feature of the set A. The following definitions of fuzzy set operations are most commonly used. If C = A f B, 8) and if C = A U B, and if C ....

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G.J. Klir and T.A. Folger, Fuzzy Sets, Uncertainty, and Information, Prentice-Hall, New Jersey, 1988.

.... Theta A j . Two exceptions are hyperrectangles and joint Gaussian (or other exponential) set functions with uncorrelated factors. Ellipsoids and most sets do not factor. Yet al..most all real fuzzy systems start with scalar sets A and combine them with product or minimum or some other t norm [2] to form the if part fuzzy sets of rules in a fuzzy system F : R . This yields factorable joint set functions of the form a j (x) a j (x 1 ) Theta Delta Delta Delta Theta j (x n ) or a j (x) min(a j (x 1 ) Delta Delta Delta ; a j (x n ) Factorable joint set functions ....

G. J. Klir and T. A. Folger. Fuzzy Sets, Uncertainty, and Information. Prentice Hall, 1988.

....find the car that best matches the description. In our case, we have chosen to convert symbols to the universe of the measurable parameters. In general, symbolic descriptions contain linguistic terms like red and small that do not denote a unique numerical value. Sticking to a common practice [8, 3], we have chosen to map each linguistic term of this kind to a fuzzy set over the relevant frame. For example, we associate the term red with the fuzzy set shown in Fig 3 (left) for each possible value h, the value of red(h) measures, on a [0; 1] scale, how much h can be regarded as red. 3 ....

....summarized degree of matching. The simplest way to combine our degrees is by using a conjunctive type of combination, where we require that each one of the features matches the corresponding part in the description. Conjunctive combination is typically done in fuzzy set theory by T norm operators [6, 3], whose most used instances are min, product, and the Lukasiewicz T norm max(x y Gamma 1; 0) In our experiments, we have noticed that the latter operator provides the best results. See [1] for an overview of the use of alternative operators with applications to image processing. The overall ....

G. Klir and T. Folger. Fuzzy sets, uncertainty, and information. Prentice-Hall, 1988.

....data, defining a point in the feature space. This point may be mapped to a symbolic interpretation of the world based on that symbol s neighborhood in the feature space. Such a neighborhood function may be defined by probability theory [101] Dempster Shafer s theory of evidence [88] fuzzy logic [51], neural networks [108] or other means. The prevailing techniques for symbol level fusion include Bayesian (Maximum A Posteriori) estimation, Dempster Shafer evidential reasoning, and fuzzy set theory. Bayesian estimation combines sensory information according to the rules of probability theory. ....

G. J. Klir and T. A. Folger, Fuzzy Sets, Uncertainty, and Information, Prentice Hall, Englewood Cliffs, NJ, 1988.

....integration data flow control flow Figure 1: The concept of active fusion in image understanding. The following mathematical theories are candidates for the active fusion concept, since they can cope with uncertain, imprecise or vague data [3] ffl probability theory [10] ffl fuzzy theory [7], ffl evidence theory [14] This paper presents a concept for active fusion using Dempster Shafer theory of evidence (Section 2) and an application for a multitemporal remote sensing data set (Section 3) Similar concepts using Bayesian networks [13] and fuzzy sets [8] have been investigated by ....

....2: Active fusion module adapted to evidence theory. The selection of the Best Information Source is achieved by a cost benefit analysis of including the evidence of a specific information source. Measures of entropy adapted to evidence theory (measures of dissonance, confusion, nonspecificity [7], pp. 169 188; belief entropy, core entropy [15] are possible candidates for the cost benefit analysis, because they can be used to measure different aspects of information content in a mass distribution (see Eq. 4) Measures of dissonance, E (Eq. 1) and confusion, C (Eq. 2) display the ....

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G.J. Klir and T.A. Folger. Fuzzy Sets, Uncertainty, and Information. Prentice Hall, 1988.

.... widely used because computationally, it is the simplest (for a formal proof, see Kreinovich 1994 [36] Angelov (1994) 2] proposes to use Hamacher s t Gammanorm ab fi (1 Gamma fi) a b Gamma ab) It is also possible to use other Gammaoperations (t Gammanorms; see, e.g. Klir 1988) [29]. These operations are symmetric, but the role of constraint and an objective function can be different, so a non symmetric aggregation operation may be used (Dubois 1994) 23] For 7 example, in (Bellman 1970) 4] a linear combination a; b ffa fib was used to combine C and M : D (x) ff M ....

....reasonable (Kreinovich 1990) 31] Kreinovich 1991) 32] Kreinovich 1992) 35] There are several ways to assign a certainty (truth) value t(S) to a statement S (Dubois 1980) 18] One of the natural methods (Dubois 1980) 18] Section IV.1. d, Blin 1973) 7] Blin 1974) 6] Klir 1988) [29] is to take several (N) experts, and ask each of them whether he believes that S is true If N(S) of them answer yes , we take t(S) N(S) N as the desired certainty value. If all the experts believe in S, then this value is 1 ( 100 ) if half of them believe in S, then t(S) 0:5 (50 ) etc. ....

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G. Klir, T. A. Folger, Fuzzy sets, uncertainty, and information (Prentice-Hall, Englewood Cliffs, NJ, 1988).

....These areas will be developed more fully to highlight the application of these techniques to diagnostic problems. Several examples are given in the following section to clarify the application of these techniques and design issues. More extensive treatment of fuzzy mathematics can be found in [14,15]. 3.1 Fundamentals of fuzzy logic Each element of a fuzzy set is an ordered pair containing a set element and the degree of membership in the fuzzy set. A higher membership value can be said to indicate that an element more closely matches the characteristic feature of the set. For fuzzy set A: ....

....1. Additivity) It is important to emphasize that m A ( is not a measure but rather can be used to generate a measure or conversely, to be generated from a measure. A specific basic assignment over P ( C is often referred to as a body of evidence. Based on the above axioms, it can be shown [14] that: 12 Bel A m B B A ( 11) Pl A m B B A ( f (12) and conversely that: m A A B B A Bel A ( 1 (13) where is set cardinality. These equations show us another view of belief and plausibility. Belief measures the evidence that can completely (from the ....

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G.J. Klir and T.A. Folger, Fuzzy Sets, Uncertainty, and Information, Prentice-Hall, New Jersey, 1988.

....4 Many of the fundamental techniques of fuzzy sets are widely available so only a fairly brief review is given in the following. The lesser known methodology of fuzzy measures and information theory is developed more fully here. More extensive treatment of the mathematics can be found in [8,9]. A fuzzy set is a set of ordered pairs with each containing an element and the degree of membership for that element. A higher membership value indicates that an element more closely matches the characteristic feature of the set. For fuzzy set A: C = x x x A A m (1) where C is ....

G.J. Klir and T.A. Folger, Fuzzy Sets, Uncertainty, and Information, Prentice-Hall, New Jersey, 1988.

....OF GENERALIZED AGGREGATION OPERATORS Ever since the advent of fuzzy sets [239] and fuzzy control [240, 156] evaluation of fuzzy rules has been widely performed using the MIN and MAX operators for fuzzy intersection and union. Other operators for performing fuzzy intersection and union exist [129]. They fall in the general class of T norms (for intersection) and T conorms (for union) A good overview of the theory of such operators is presented by Gupta [63] Design of fuzzy controllers based on such operators is considered in [64] where it is shown, through simulation studies, that the ....

George Klir and Tina Folger, Fuzzy sets, uncertainty, and information. Englewood Cliffs, N.J.: Prentice Hall, 1988.

....53, 55] 1 This work has been supported by the CICYT under project TIC97 1135 C04 01. 1 When available information is not speci c enough. For example, when we take out balls from an urn with 10 balls, where 5 are red and 5 are white or black (but we do not know the exact number of each one) [21, 42, 31]. In robust Bayesian inference, 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 ....

G.J. Klir and T.A. Folger. Fuzzy Sets, Uncertainty, and Information. Prentice-Hall, Englewood Clis, New Jersey, 1988.

....can be used to choose data which is most beneficial from an information theoretic perspective. There have been a number of discussions on the subject of information theory and neural network training in the literature; Klir and Folger provide an excellent introduction to information theory [15], and specific aspects are contained in [1, 22, 28] Often, it is not possible to collect large data sets because of the cost and time required. If the data set is sparse relative to the number of input and output variables and the complexity of the process to be modeled, special attention must ....

Klir, G. J. and T. A. Folger, Fuzzy Sets, Uncertainty, and Information, Englewood Cliffs, NJ, Prentice-Hall (1988).

....described in this thesis are well described by the third, set based approach to planning a meeting quoted above. Fuzzy sets are, after all, a generalization of ordinary crisp sets. Many researchers have used fuzzy sets to represent imprecision in decision making outside of engineering design [6, 16, 24, 27, 40, 41, 72]. Most of these formulations are based on fuzzy and and or operators and are directed at modeling linguistic uncertainty and fuzzy logic. Although the design appropriate P min and P# aggregation functions are used to combine fuzzy sets, two classes of functions that do not in general ....

G. J. Klir and T. A. Folger. Fuzzy Sets, Uncertainty, and Information. Prentice Hall, Englewood Cliffs, New Jersey, 1988.

.... experiments and or through simple deductions based on the agents structure (similar to how their PAGE descriptions are determined see [6] When facing the uncertainty in the available information about MAS clustering dynamics, two aspects should be considered: vagueness and ambiguity [4]. It is already well known that among the other uncertainty facets, vagueness deals with information that is inconsistent. In the context of MAS, this means that the clear distinction between a possible plan reaching the imposed goal and a plan which, on the contrary, leads the MAS in an ....

....optimize it in order to select the least vague source plan from the family K k k , 1 = P P : opt arg , 1 0 k K k k V P P = where K k o , 1 . 1) 3. THE LEAST VAGUE SOURCE PLAN The solution proposed here is based on the Theory of fuzzy sets and uses measures of fuzziness [4] to model the vagueness aspect of the uncertainty in the initial information about MAS behavior. Recall that the only available information consists of a family of occurrence degrees for some configurations. The optimization problem stated in Eq. 1) is solved within the next four steps. Step 1: ....

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Klir G.J., Folger T.A. - Fuzzy sets, Uncertainty, and Information, Prentice Hall, 1984.

....2. FRAMEWORK AND CONSIDERATIONS description. In our case, we have chosen to convert symbols to the universe of the measurable parameters. In general, symbolic descriptions contain linguistic terms like red and small that do not denote a unique numerical value. Sticking to a common practice [26, 40], we have chosen to map each linguistic term of this kind to a fuzzy set over the relevant frame. For example, we associate the term red with the fuzzy set shown in Fig 2.6 (left) for each possible value h, the value of red(h) measures, on a [0; 1] scale, how much h can be regarded as red. ....

....summarized degree of matching. The simplest way to combine our degrees is by using a conjunctive type of combination, where we require that each one of the features matches the corresponding part in the description. Conjunctive combination is typically done in fuzzy set theory by T norm operators [26, 37], whose most used instances are min, product, and the ukasiewicz T norm max(x y 1; 0) In our experiments, we have noticed that the latter operator provides the best results. See [7] for an overview of the use of alternative operators with applications to image processing. The overall degree ....

G. Klir and T. Folger. Fuzzy sets, uncertainty, and information. Prentice-Hall, 1988.

.... it represents one of the most successful applications of fuzzy logic based systems (Bonissone et al. 1995) 12.5 Complementarity The distinction between probability and fuzziness has been presented and analyzed in many different publications, such as (Bezdek, 1994, Dubois and Prade, 1993, Klir and Folger, 1988) to mention a few. Most researches in probabilistic reasoning and fuzzy logic have reached the same conclusion about the complementarity of the two theories (Bonissone, 1991a) This complementarity was first noted by Zadeh (Zadeh, 1968) who, in 1968, introduced the concept of the probability ....

G. Klir and T.A. Folger. Fuzzy Sets, Uncertainty, and Information. Prentice Hall, Englewood Cliffs, New Jersey, 1988.

....= 1) that is, c has not been explored. This information can be used to plan further exploration [85] Feature based representations of maps can also profit from the ability of fuzzy logic to represent and reason with weak knowledge, and to distinguish between different facets of uncertainty [59, 60]. For example, we may wish to distinguish between the vagueness or inaccuracy in the position of the feature, and the uncertainty in its very existence e.g. the map may be wrong, the feature may have been removed from the environment, or its existence may have been inferred from a spurious ....

G. J. Klir and T. A. Folger. Fuzzy sets, uncertainty, and information. Prentice-Hall, 1988.

....minimal conjunctive and disjunctive forms of a given formula. 6 Although the choice of the operations max, min and complementation to 1 as the basic operations for fuzzy sets is not mandatory, they are the only functions that present some interesting properties like idempotence and continuity [18]. 820 Concurrent Inference through Dual Transformation 3.6 The Algorithm Complexity A complete analysis of the complexity of the algorithm has not yet been undertaken, but we can compare its performance with the nave algorithm in a general case. Consider the dual transformation of a set W c ....

G.J. Klir and T.A. Folger. Fuzzy Sets, Uncertainty, and Information. Prentice Hall, Englewood Cli#s, N.J, 1988.

....match these data to the high level description of an object. In order to improve the reliability of the anchoring process, this uncertainty has to be taken into account in the proper way. Research in fuzzy logic has produced a number of techniques for dealing with different facets of uncertainty [4, 1] . In this work, we propose to use these techniques to define a degree of matching between a perceptual signature and an object description. The possibility to distinguish between objects that match a given description at different degrees is pivotal to the ability to discriminate perceptually ....

....and the data coming from the vision system are affected by several types of inexactness. Symbolic descriptions use linguistic terms like red and small that do not denote a unique numerical value. Fuzzy sets are commonly considered to be an adequate representation of linguistic terms [10, 4], so in our system we have chosen to map each symbol of this kind to a fuzzy set over the relevant space. For example, we associate the term red to three fuzzy sets: one for the hue characterizing the tint of color, one for the saturation characterizing the purity of the color, and one for value ....

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G. Klir and T. Folger. Fuzzy sets, uncertainty, and information. Prentice-Hall, 1988.

....location of an object by a fuzzy subset of a given space, read under a possibilistic interpretation [17] if P o is a fuzzy set representing the approximate location of object o, then we read the value of P o (x) 2 [0; 1] as the degree of possibility that o be actually located at x. See [16, 7] on fuzzy sets, and [11] for some foundation issues. This representation allows us to model different aspects of locational uncertainty. Figure 2 shows six approximate locations in one dimension: a) is a crisp (certain) location; in (b) we know that the object is located at approximately 5 ....

G. J. Klir and T. A. Folger. Fuzzy sets, uncertainty, and information. PrenticeHall, 1988.

....reach the work place after leaving the transportation mode. e) the (mean) travel time (without considering stops for shopping, gas fill up, buy transport ticket, etc) f) the travel cost (in franks) per trip or per month (e.g. in case of season ticket) g) an appreciation (very good [1] good [2], middle [3] bad [4] very bad [5] without opinion [6] of the transportation mode with regard to comfort, ease of use, safety, availability (of the mode) and the parking availability (for car) 2. Then an alternative transportation mode is specified for cases in which the main mode becomes ....

G.J. Klir and T.A. Folger. Fuzzy Sets, Uncertainty, and Information, Prentice Hall, Englewood Cliffs, USA, 1988.

....the needed basic data for probability is missed. Most of these theories are special cases of Zadeh s fuzzy set theory [21] so we give a short description of his basic ideas. Additionally we mention the most important application of this theory, fuzzy control. The interested reader is referred to [23, 15] for more details. 2.1 Fuzzy Logic In classical set theory an element x of the universe Omega is member of a set F Omega or it is not. The characteristic function F maps from Omega to the two element set f0; 1g, indicating the membership x 2 F . Zadeh [21] extends this mapping to the ....

....He uses maximum for union, minimum for intersection and complement for negation as indicated in fig. 2 A B C A B u A B u C Figure 2: Operations on fuzzy sets These operators are not the only valid operations on fuzzy sets. Other authors define different operations during the past decades (cf. [15, 23]) The use of a specific operator is often context sensitive or author sensitive but all defined operators include the binary (conventional) case. Zadeh extends his idea of fuzzification to an extension principle which generalizes crisp mathematical concepts to fuzzy sets. One application ....

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G.J Klir and T.A. Folger. Fuzzy Sets, Uncertainty, and Information. Prentice Hall, Englewood Cliffs, New Jersey, 1988.

.... can easily be implemented with the use of a cuts of fuzzy sets (this a is not to be confused with the degree of applicability) An a cut of a fuzzy set, A, defined over a universe U, is a crisp set, A a , containing all the elements of U with membership grade in A greater than or equal to a [9]. Formally, A a = x X A (x) a (3) where A ( X [0,1] is a membership function of fuzzy set A. To utilize a cuts for implementing thresholding behavior activation, we consider the rulestrength of an applicability rule. If the a cut of the fuzzy set resulting from evaluation of the ....

G.J. Klir and T.A. Folger, Fuzzy Sets, Uncertainty, and Information, Prentice Hall, Englewood Cliffs, NJ, 1988.

....a x Gamma 1 year old person is said to be not old . A possible solution to this problem is to generalize the definition of the characteristic function in a way that it yields values from the interval [0; 1] and not just the two values of the set f0; 1g. This leads to the notion of a fuzzy set [32, 17, 20]. Definition 2.1 A fuzzy set of X is a function that maps from the universe X into the unit interval, i.e. X [0; 1] F (X) denotes the set of all fuzzy sets of X. The value (x) denotes the membership degree of x to the fuzzy set . Figure 1b shows a (subjectively defined) membership ....

Klir, G.J., Folger, T.A.: Fuzzy Sets, Uncertainty, and Information. Prentice Hall, 1988.

....confidence interval, developed in statistics to report uncertainty in the value of a parameter, is used to build fuzzy sets from scatter information. For a given confidence level, the interval provides an estimation of the region within which the segment is likely to lie. In the same way, ff cuts [6] in the fuzzy set define intervals (regions) within which all segments can be considered similar at least to the degree ff (Figs. 2 c and 2 d) Relating both concepts, we obtain the degrees of membership to the fuzzy sets in terms of the confidence intervals (Fig. 2 e) A similar analysis is ....

G. J. Klir and T. A. Folger. Fuzzy sets, uncertainty, and information. Prentice-Hall, 1988.

....memberships functions: C = A B ) C (x) A B (x) min[ A (x) B (x) 2 8x 2 X (2) Similarly, the union of two fuzzy sets can be represented by the maximum operation. These operations are not unique. Other operators for performing fuzzy intersection, union, and complementation exist [5]. However the min and max operations are special in the sense that they are the only continuous and idempotent fuzzy set intersection and union operators respectively [5] 1.2 Fuzzy Inference Fuzzy inference is based on the concept of the fuzzy conditional statement: If A Then B, or, for short ....

....operation. These operations are not unique. Other operators for performing fuzzy intersection, union, and complementation exist [5] However the min and max operations are special in the sense that they are the only continuous and idempotent fuzzy set intersection and union operators respectively [5]. 1.2 Fuzzy Inference Fuzzy inference is based on the concept of the fuzzy conditional statement: If A Then B, or, for short A ) B, where the antecedent A and the consequent B are fuzzy sets. A general fuzzy inference system consists of three parts (see Fig. 1) A crisp input is fuzzified by ....

[Article contains additional citation context not shown here]

G. Klir and T. Folger, Fuzzy Sets, Uncertainty, and Information. Englewood Cliffs, N.J.: Prentice Hall, 1988.

....A n j . Two exceptions are hyperrectangles and joint Gaussian (or other exponential) set functions with uncorrelated factors. Ellipsoids and most sets do not factor. Yet al..most all real fuzzy systems start with scalar sets A i j and combine them with product or minimum or some other t norm [2] to form the if part fuzzy sets of rules in a fuzzy system F : R n R p . This yields factorable joint set functions of the form a j (x) a 1 j (x 1 ) Theta Delta Delta Delta Theta a n j (x n ) or a j (x) min(a 1 j (x 1 ) Delta Delta Delta ; a n j (x n ) Factorable joint ....

G. J. Klir and T. A. Folger. Fuzzy Sets, Uncertainty, and Information. Prentice Hall, 1988.

....set theory, a relation is considered to be a set of tuples or ordered pairs that represents the absence or presence of association between the elements of two or more crisp sets. This concept can be generalized to allow for various degrees or strengths of relation between elements of fuzzy sets [16]. That is, a fuzzy relation is a fuzzy set of tuples where each tuple has a membership degree in [0, 1] Its formal definition is Definition 1. Let X 1 , X 2 , X n be continuous universes of discourse (universal sets) and R (x) X 1 X 2 . X n [0,1] then R = R (x 1 ....

Klir, G.J. and Folger, T.A. Fuzzy Sets, Uncertainty, and Information. Prentice-Hall, Englewood Cliffs, NJ, (1988).

....are. Our first extension aims to solve the dominance problem. We propose to generalize the CH framework by allowing a constraint to belong to possibly more than one level in the hierarchy to a varying degree. In the following, we briefly introduce some basic machineries of the fuzzy set theory [4], followed by the definition of the Fuzzy Constraint Hierarchies (FCH) framework and its formal properties. 3.1 Basic Fuzzy Set Theory Every crisp set A is associated with a characteristic function A : U f0; 1g, which determines whether an individual from the universal set U is a member or ....

....equal, that is the question. Our second extension aims to solve the stiff equality problem. We formalize the notion of approximately equal to and define new valuation comparators using the modified equality relation. The ff approximate equality relation = a(ff) is obtained by computing the ff cut [4] of the fuzzified equality relation, where ff is a user defined constant that controls the degree of approximation or tolerance. A relation is a set of tuples. To fuzzify a crisp relation R, we need to devise a meaningful membership function that conforms to the intended interpretation of the ....

G.J. Klir and T.A. Folger. Fuzzy Sets, Uncertainty, and Information. Prentice Hall, 1992.

....new. Uncertainty may be: uncontrolled stochastic variations in variable values, design imprecision as described above, variable values to be chosen by optimization, etc. Probability and Bayesian inferencing [37, 95, 104] Dempster Shafer theory [83, 87] fuzzy sets and triangular norms in general [20, 42, 44, 54, 56, 113], and finally utility theory [24, 26, 43] are among the existing formal 3 methods for representing uncertainty. These methods all represent uncertainty with a range for each variable and a function defined on that range. An illustration is shown in Figure 1, where d is an uncertain variable and ....

....property and annihilation cannot be simultaneously satisfied. This implies that utility theory will not permit a worst case analysis, which is required in many instances in engineering design [33, 49] Fuzzy Sets. Fuzzy sets have been used to represent imprecision in (non design) decisionmaking [20, 42, 44, 54, 56, 113]. Fuzzy sets are intended to model subjective uncertainty for use in logic, constructing subjectively uncertain versions of and and or of classical logic. In the first paper describing the use of fuzzy sets for decision making [9] a decision was defined as a convolution of the constraints and ....

KLIR, G. J., AND FOLGER, T. A. Fuzzy Sets, Uncertainty, and Information. Prentice Hall, Englewood Cliffs, New Jersey, 1988.

....trivial and wrong choices can result in a seemingly successful termination even though highly unwanted output is produced. Only if Q(F ) really reflects improvements in quality can we rely on the algorithm. There is an abundance of suggestions for possible information measures in the literature [6] and some of them can provide quality measures in certain tasks. One promising idea would be to learn the best choice for Q(F ) from failure success analyses in supervised situations. Here again the modular structure of the scheme might become useful because the single expert modules can be ....

G.J. Klir and T.A. Folger. Fuzzy Sets, Uncertainty, and Information. Prentice Hall, 1988.

....then so is R n x. In order to connect with the logic, we associate with R the set of formulas Rj def = fx j y j (x; y) 2 Rg. The notions of closures and removal extend to Rj in the natural way: Rj n x is (R n x)j etc. The algorithm is conveniently described using Kleene s three valued logic [11], the three truth values being t for true , f for false and u for undefined unknown . The Kleene truth tables for conjunction, K , disjunction, K , and implication, K , are: K t f u t t f u f f f f u u f u K t f u t t t t f t f u u t u u K t f u t t f u f t t t u t u u The ....

G.J. Klir and T.A. Folger. Fuzzy Sets, Uncertainty, and Information. Prentice-Hall, 1988.

....among these approaches are reported in literature [16] In our paper we take the position that it is more general and appropriate to approach diagnostic reasoning to represent real world with fuzzy sets and beliefs. As well known fuzzy sets are a generalization of the classical set theory [19]. They make it possible to describe vagueness of the information. Fuzzy measures in general and the belief theory in particular generalize the probability theory. In probability theory the observers know that there are different hypotheses but each one is separate from the others and all of them ....

G.Klir, T.Folger. Fuzzy Sets, Uncertainty, and Information. Prentice Hall , 1988

....confidence in the results obtained from the models. Proper representation of uncertainty and its incorporation into the models could be of genuine use in constructing more realistic models of real world problems. The mathematical representation of uncertainty has received considerable attention [2, 6, 7] since the introduction of Fuzzy Set Theory by Zadeh [13] The concept of Fuzzy Sets is based on the grouping of the elements into classes that do not have sharply defined boundaries. These classes are used to describe ambiguity and vagueness in mathematical models of empirical phenomena. A binary ....

G. J. Klir and T. A. Folgers. Fuzzy Sets, Uncertainty, and Information. Prentice-Hall, Englewood Cliffs, NJ, 1988.

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G. Klir and T. Folger. Fuzzy Sets, Uncertainty, and Information. Prentice-Hall, 1988.

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G. Klir and T. Folger. Fuzy Sets, Uncertainty, and Information. Prentice Hall, New Jersey, 1988.

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G. J. Klir and T. A. Folger. Fuzzy sets, uncertainty, and information. Prentice-Hall, 1988.

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G. J. Klir and T. A. Folger. Fuzzy sets, uncertainty, and information. Prentice-Hall, 1988.

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G. J. Klir and T. A. Folger, Fuzzy sets, uncertainty, and information, Prentice Hall, 1988.

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Klir, G., and Folger, T. Fuzzy Sets, Uncertainty, and Information. Prentice Hall, Englewood Cli#s, NJ, USA, 1987.

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Klir, G.J. and Folger, T.A. Fuzzy Sets, Uncertainty, and Information. Prentice-Hall, Englewood Cliffs, NJ, (1988).

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Klir, George G. and Folger, Tina A., Fuzzy Sets, Uncertainty, and Information, Prentice Hall, Englewood Cliffs,

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KLIR G.J AND FOLGER T.A. Fuzzy sets, uncertainty, and information, PrenticeHall, New York (1988). 27

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G.J.Klir y T.A.Folger ,Fuzzy Sets, Uncertainty, and Information. Englewood Cliffs, NJ:Prentice Hall, 1988

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KLIR G.J AND FOLGER T.A. Fuzzy sets, uncertainty, and information, PrenticeHall, New York (1988).

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G. J. Klir and T. A. Folger. Fuzzy sets, uncertainty, and information. Prentice-Hall, 1988.

No context found.

G. J. Klir and T. A. Folger. Fuzzy sets, uncertainty, and information. Prentice Hall, Englewood Cliffs, N.J., 1988.

No context found.

G.J. Klir and T.A. Folger. Fuzzy Sets, Uncertainty, and Information. Prentice Hall, 1988.

No context found.

Klir, G. and Folger, Fuzzy Sets, Uncertainty, and Information, Prentice Hall, 1988.

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