G. J. Klir and B. Yuan. Fuzzy Sets and Fuzzy Logic. Prentice Hall, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....lattice and a sublattice of (F, A, V, 2.3 Fuzzy Relations In the sequel we will often use fuzzy relations on subsets of U (e.g. inclusion, similarity, distance) In this paper, the fuzzy relations of interest are functions of the form R: F x F B. Adapting well known definitions [15] to this context we have the following. Definition 2.4 A fuzzy relation R is called reflexive if[ VA F we have R(A, A) 1. n(A, B) R( A, B) A, B) A fuzzy relation R is called symmetric iff VA, B F we have R(A, B) R(B, A) A fuzzy relation R is called antisymmetric iff VA, ....

....This connection will be reported elsewhere and parallels the work of many authors in obtaining inclusion measures from fuzzy implication operators, in the style of Bandler and Kohout. The L fuzzy implication operator satisfies properties analogous to Klir s axioms for the implication operator [15]. The connection to conditional probability is also worth investigating. A scalar inclusion measure i(A, B) is in many ways analogous to Pr(B]A) compare with inclusion measure no.1 in Section 3.2) it would be interesting to use the ideas presented in this paper to define lattice valued ....

G.J. Klir and B. Yuan. Fuzzy Sets and Fuzzy Logic, Prentice Hall, 1991.

....the user s Web navigation patterns to the final recommendations, including the intermediate stages of logging Web usage, preprocessing and segmenting Web log data into Web user sessions, clustering these sessions, and computing Web user profiles from these clusters. Fuzzy approximate reasoning [12,13] can offer a general framework for the recommendation process. It is this framework that is investigated in this paper. Some previous work in this area has been performed in [7,8,9,14] Pazzani and Billsus [7] presented a collaborative filtering approach to recommendation, based on users rating ....

....overlap, or (iii) of outlying interest, i.e. not identifying with any of the pre discovered strong profiles. The wide spectrum of uncertainties involved in the Web navigation process can be modeled and handled using well studied formal models of uncertainty in fuzzy set theory and soft computing [11,12]. We point, in particular to probabilistic models for case (i) and (ii) and general possibilistic models of uncertainty for cases (ii) and (iii) 3.2 FUZZY APPROXIMATE REASONING BASED RECOMMENDATION ENGINE Fuzzy approximate reasoning is an inference procedure that derives its conclusions from ....

Klir, G. J., and Yuan, B., Fuzzy Sets and Fuzzy Logic, Prentice Hall, 1995, ISBN 0-13-101171-5.

....complete lattice and a sublattice of (F, A, V ) 2.3 Fuzzy Relations In the sequel we will often use fuzzy relations on subsets of U (e.g. inclusion, similarity, distance) In this paper, the fuzzy relations of interest are functions of the form R: F x F B. Adapting well known definitions [15] to this context we have the following. Definition 2.4 A fuzzy relation R is called reflexive iff VA C F we have R(A, A) 1. Definition 2.5 A fuzzy relation R is called symmetric iffVA, B F we have R(A,B) R(B,A) Definition 2.6 A fuzzy relation R is called antisymmetric iff VA, B F we have ....

....This connection will be reported elsewhere and parallels the work of many authors in obtaining inclusion measures from fuzzy implication operators, in the style of Bandlet and Kohout. The L fuzzy implication operator satisfies properties analogous to Klir s axioms for the implication operator [15]. The connection to conditional probability is also worth investigating. A scalar inclusion measure (A, B) is in many ways analogous to Pr(B[A) compare with inclusion measure no.1 in Section 3.2) it would be interesting to use the ideas presented in this paper to define lattice valued ....

G.J. Klir and B. Yuan. Fuzzy Sets and Fuzzy Logic, Prentice Hall, 1991.

....Thus, before we analyze how to process these statements, we must be able to translate them in a language that a computer can understand. This translation of expert statements from natural language into a precise language of numbers is one of the main original objectives of fuzzy logic (see, e.g. [7, 18]) It is therefore important to extend traditional statistical techniques from processing crisp data to processing fuzzy data. In this paper, we provide an overview of our research in this direction, outline our main results and open problems. 2 ESTIMATING PARAMETERS OF A DISTRIBUTION BASED ON ....

G.J. Klir and B. Yuan (1995) Fuzzy Sets and Fuzzy Logic, Prentice Hall, NJ.

....and antisymmentric iff R(x,y) 0 (y,x) 0 = x= y, Vx, y Finally, R is called sup t transitive (or simply, transi t tive) iff R o R G R A transitive closure of a relation is a transitive rela tion that contains the original relation and has the fewest possible members. It can be proved [3] that if R is reflexive, then its transitive closure is given by the formula R r = R (n4) where n denotes the number of entities and g =R . R. n Similar operations for reflexive, anti reflexive and symmetric properties can be defmed, and they are trivial to compute: R(x, x) 1, Vx ....

George J. Klir and Bo Yuan, Fuzzy Sets and Fuzzy Logic, Theory andApplications, Prentice Hall, New Jersey, 1995

....and union. With the min max system proposed by Zadeh [23] fuzzy set operations are defined component wise as: #A (x) 1 A (x) A#B (x) min[A (x) B (x) A#B (x) max[A (x) B (x) 12) In general, fuzzy set intersection and union may be defined in terms of t norms and t conorms [6]. By choosing di#erent pairs of t norms and t conorms, one can derive distinct fuzzy set systems. A crisp set can be regarded as a degenerated fuzzy set in which the membership function is restricted to the extreme points 0, 1 of [0, 1] In this case, the membership function is also referred ....

....be reconstructed from its # level sets as follows: A (x) sup # # A# . 16) This observation is commonly summarized by a representation theorem of fuzzy sets, which states that there is an one to one relationship between a fuzzy set and a family of crisp sets satisfying certain conditions [6, 12, 13]. An implication of the min max system is that fuzzy set operations can be defined by set operations on # level sets. They can be expressed by: # A)# (1 #) 17) The # level sets of fuzzy sets for intersection and union are obtained from the same # level sets of the fuzzy ....

Klir, G.J., and Yuan, B., 1995, Fuzzy Sets and Fuzzy Logic, Theory and Applications, Prentice Hall, New Jersey.

....predictions in the interval of a single day. One promising alternative to work with continuous variables and to overcome this inconvenience is the use of fuzzy logic. Besides expressing knowledge in a more natural way, fuzzy logic is also a flexible and powerful method for uncertainty management [13], 6] In the literature several techniques have been used for discovery of fuzzy IFTHEN rules. Several recent projects have proposed the use of evolutionary algorithms for fuzzy rule discovery [2] 10] 11] 19] 21] 17] because it allows a global search in the state space, increasing the ....

....temperature = low) In our system the function set contains the logical operators AND, OR, NOT . Since each individual represents fuzzy rules, a fuzzy version of these logical operators must be used. We have used the standard fuzzy AND (intersection) OR (union) and NOT (complement) operators [13]. More precisely, let g (x) denote the membership degree of an element x in the fuzzy set A, i.e. the degree to which x belongs to the fuzzy set A. The standard AND of two fuzzy sets A and B, denoted A AND B, is defined as g NO B(x) min[g (x) x) where min denotes the minimum operator. The ....

G.J. Klir and B. Yuan. Fuzzy Sets and Fuzzy Logic. Prentice-Hall, 1995.

....lattice and a sublattice of (F, #, #, #, # ) 2.3 Fuzzy Relations In the sequel we will often use fuzzy relations on subsets of U (e.g. inclusion, similarity, distance) In this paper, the fuzzy relations of interest are functions of the form R : F F # B. Adapting well known definitions [15] to this context we have the following. Definition 2.4 A fuzzy relation R is called reflexive i# #A # F we have R(A, A) 1. Definition 2.5 A fuzzy relation R is called symmetric i# #A, B # F we have R(A, B) R(B,A) Definition 2.6 A fuzzy relation R is called antisymmetric i# #A, B # F ....

....This connection will be reported elsewhere and parallels the work of many authors in obtaining inclusion measures from fuzzy implication operators, in the style of Bandler and Kohout. The L fuzzy implication operator satisfies properties analogous to Klir s axioms for the implication operator [15]. The connection to conditional probability is also worth investigating. A scalar inclusion measure i(A, B) is in many ways analogous to Pr(B A) compare with inclusion measure no.1 in Section 3.2) it would be interesting to use the ideas presented in this paper to define lattice valued ....

G.J. Klir and B. Yuan. Fuzzy Sets and Fuzzy Logic, Prentice Hall, 1991.

....NEURObjects. I. Introduction Neural networks play an important role in machine learning, in particular they permitt to efficently face problems such as regression and classification [13] Moreover, neural networks are often relevant components of complex systems used in inductive learning tasks [12]. Nowadays, the relatively limited diffusion of neural network technology in industrial applications mainly depends on the high costs related to the long development time necessary when neural networks algorithms are implemented from scratch in order to embed those tools in new software products. ....

Klir G.J., Yuan B. Fuzzy sets and fuzzy logic, Prentice Hall, chapter 12, 1998.

.... for a particular cluster then they are considered similar to each other [Wind 82] If the fuzzy membership functions j takes values from set f0; 1g, ie either 0 or 1, then each vector will exclusively belong to one cluster only and the membership functions are known as characteristic functions [Klir 95] 8 Definations of Proxity Measure 8.1 Dissimilarity measure Dissimilarity measure (DM) d on X is a function defined below d : X Theta X Here X is defined as a feature vector and a set of real numbers. In other words d is a function that takes in two feature vectors and returns a ....

Klir G., Yuan B., Fuzzy sets and fuzzy logic, Prentice Hall, 1995

....measure p(S) defined on the class of sets S. A natural way to do this is to assign, to each such probability measure, the values (s) P (s 2 S) P (fS j s 2 Sg) X S3s p(S) These values belong to the interval [0; 1] and do not necessarily add up to 1. In fuzzy formalism (see, e.g. [10,24]) numbers (s) 2 [0; 1] s 2 A (that do not necessarily add up to 1) form a membership function of a fuzzy set. Interpretation in terms of random sets. Hung T. Nguyen has shown [22] that every membership function can be thus interpreted, and that, moreover, standard (initially ad hoc) operations ....

....seems to be a reasonable approach to choosing and and or operations. It has indeed lead to successful expert systems (see, e.g. Pearl [26] Unfortunately, the resulting operations are not always the ones we need. For example, if we are interested in fuzzy control applications (see, e.g. [23,10,24]) then it is natural to choose and and or operations for which the resulting control is the most stable or the most smooth. Let us describe this example in some detail. In some control problems (e.g. in tracking a spaceship) we are interested in the most stable control, i.e. in the ....

G.J. Klir and Bo Yuan, Fuzzy Sets and Fuzzy Logic, Prentice Hall, NJ, 1995.

.... conclusion (Klir and Folger, 1988; Fresco and Kroonenberg, 1992; Pelt et al. 1995) Fuzzy uncertainty, in contrast, relates to events that have no well defined, unambiguous meaning (Kosko, 1992) Fuzzy set theory is based on multi valued logic (McNeill and Freiberger, 1993; Pedrycz, 1993; Klir and Yuan, 1995; Zimmermann, 1996) Multi valued logic enables intermediate assessment between strictly sustainable and strictly unsustainable; i.e. fuzziness describes the degree to which an event occurs, not whether it occurs (Kosko, 1990; Kosko, 1992) We propose, therefore, that fuzzy set theory offers a ....

....models, which use such functions to operate linguistic variables. In fuzzy set theory, a linguistic variable is characterized by: 1) base variable x of , 2) name of , 3) linguistic value i of (i = 1, n) and (4) membership function i of i (adopted from: Zadeh, 1975a; Zadeh, 1975b; Klir and Yuan, 1995). Characteristics of a linguistic variable are in Figure 2. Consider the example of housing systems for laying hens. The amount of ammonia emission x, which is a measurement of the SI Ammonia Emission, defines U SI ; hence, x is the base variable of . If the contribution of Ammonia Emission ....

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Klir, G.J., Yuan, B., 1995. Fuzzy Sets and Fuzzy Logic. Theory and Application.

....vagueness in di erent ways [2, 3, 4, 5] In this section, the basic aspects of FRBSs will be introduced: the di erent existing types, their composition and functioning will be described. Nevertheless, we shall not focus on the basic principles of Fuzzy Logic, that are to be found in texts like [6, 7]. II.A Framework: Fuzzy Logic and Fuzzy Systems As it is known, Rule Based Systems (production rule systems) have been successfully used to model human problem solving activity and adaptive behavior, where a classical way to represent the human knowledge is the use of IF THEN rules. The ....

....trapezoid or bell shaped ones. The function of the nodes in this layer are determined and formulated by applications. Layer 3 (AND layer) Each node in layer 3 represents a possible IF part for the fuzzy rules. In fuzzy set theory, there are many di erent operators for fuzzy intersection [6, 7, 27, 29, 30]. The authors choose the most commonly used one, i.e. the minoperator, which is simple and e ective and has strong characteristics of competition. Layer 4 (OR layer) Each node in layer 4 represents a possible THEN part for the fuzzy rules. The operation performed by an OR node is to combine ....

Klir, G. J., and Yuan, B. (1995). \Fuzzy sets and fuzzy logic." Prentice-Hall.

....requirement (expression (5) evaluates to 0 then no trade off occurs and the entire alternative is evaluated to 0. A geometric mean is used to aggregate the individual ratings. This method obeys the aggregation axioms of monotonicity, continuity, symmetry, idempotent, boundary, and annihilation [Klir and Yuan, 1995]. It is, h n i i n n ( 1 2 1 1 = 8) Expression (8) is termed a compensatory operator since higher satisfaction of one objective will partially offset a lower satisfaction of another objective. This aggregate treats all the objectives as if they are of equal importance. ....

Klir, G.J., and Yuan, B., (1995). Fuzzy Sets and Fuzzy Logic, Prentice Hall, NJ, 1995.

....of quadratic concepts as energies is not the only possibility. Their quantitative relation to probabilities may be di erent. For example, combination of concepts may also be modeled by fuzzy logical operations. Those exist in many variations but coincide on their boundaries with Boolean operations [37, 38]. We choose for concept distances d(h; t) the logical interpretation of d 2 (h; t) 0, i.e. h = t, as true and of d 2 (h; t) 1 as false . For such variables a typical implementation of a logical OR is a product d 2 (AORB) d 2 A d 2 B : 96) A product implementation for ....

Klir, G.J. & Yuan, B.: Fuzzy Sets and Fuzzy Logic. Prentice Hall, 1995.

.... conclusion (Klir and Folger, 1988; Fresco and Kroonenberg, 1992; Pelt et al. 1995) Fuzzy uncertainty, in contrast, relates to events that have no well defined, unambiguous meaning (Kosko, 1992) Fuzzy set theory is based on multi valued logic (McNeill and Freiberger, 1993; Pedrycz, 1993; Klir and Yuan, 1995; Zimmermann, 1996) Multi valued logic enables intermediate assessment between strictly sustainable and strictly unsustainable; i.e. fuzziness describes the degree to which an event occurs, not whether it occurs (Kosko, 1990; Kosko, 1992) We propose, therefore, that fuzzy set theory offers a ....

....models, which use such functions to operate linguistic variables. In fuzzy set theory, a linguistic variable is characterized by: 1) base variable x of , 2) name of , 3) linguistic value i of (i = 1, n) and (4) membership function i of i (adopted from: Zadeh, 1975a; Zadeh, 1975b; Klir and Yuan, 1995). Characteristics of a linguistic variable are in Figure 2. Consider the example of housing systems for laying hens. The amount of ammonia emission x, which is a measurement of the SI Ammonia Emission, defines U SI ; hence, x is the base variable of . If the contribution of Ammonia Emission ....

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Klir, G.J., Yuan, B., 1995. Fuzzy Sets and Fuzzy Logic. Theory and Application.

....the fuzzy membership functions low and medium truncated, respectively, by the 0.90 and 0.70 membership degrees. That is: Max(min(0:90; f low ) min(0:70; fmedium ) The result is shown in figure 5. Finally defuzzifies this result by computing the COA (Center of Area) of the combined function (Klir Yuan 1995). The defuzzification step gives the precise value for the tempo to be applied to the initially inexpressive note, in this example the obtained result is 123. An analogous process is applied to the other expressive parameters. The advantage of such fuzzy combination is that the resulting ....

Klir, G., and Yuan, B. 1995. Fuzzy Sets and Fuzzy Logic. Prentice Hall.

....course other intepretations have been and could be formalized [3, 10] 4 Fuzzy logic to model weighted Boolean models 4. 1 Some concepts of fuzzy logic Fuzzy logic is a logic of vagueness, which has been formalized to the aim of dealing with vague knowledge and supporting approximate reasoning [6, 7, 9, 22]. The concept of vague predicate is a central one in fuzzy logic. In classical logic a unary predicate identifies a subset of the universe of discourse; analogously, in fuzzy logic a vague predicate identifies a fuzzy subset of the universe of discourse. For example, the unary predicate young(x) ....

.... which a; b 2 [0; 1] The fuzzy (multi valued) implication operator constitues an extension of the implication operator of classical logic, in the sense that its behaviour (under a given interpretation) for the values 0 and 1 reduces to the behaviour of the implication operator in classical logic [8, 9]. It is a binary operator (fuzzy relation) defined on [0; 1] Theta [0; 1] and taking values on [0,1] Several definitions of the fuzzy implication operator have been given in the literature, among which the following ones: a RG b = 1 if a b, 0 otherwise a Gd b = 1 if a b, b otherwise a Gg ....

G. J. Klir, B. Yuan. Fuzzy Sets and Fuzzy Logic. Prentice Hall, 1995.

....Section 5 is concerned with SUMPY s implementation. The last section describes SUMPY in action, and contained plans for the future. 2. Some possible architectures Briefly descriptions of subsumption architecture and of fuzzy controllers are given here. More details can be found in [1,2,3] and [8]. 2.1 Subsumption Architecture Brooks introduced the subsumption architecture for building autonomous mobile robots [1,2,3] The architecture enables a tight connection of perception to action [6] A subsumption architect begins by decomposing the problem into a series of task achieving behaviors ....

....channels. Such structures also have the advantages of robustness and extensibility. 2. 2 Fuzzy controllers Fuzzy controllers are capable of utilizing knowledge elicited from human operators to solve control problems for which precise mathematical models are difficult or even impossible to construct[8,9]. Such a fuzzy controller employs a knowledge base, expressed in terms of relevant fuzzy inference rules, and an appropriate inference engine to solve a given control problem. A general fuzzy controller (Fig. 2.2) consists of four modules: a fuzzy rule base, a fuzzy inference engine, a ....

Klir, George and Bo Yuan (1995) Fuzzy Sets and Fuzzy Logic, Upper Saddle River, NJ: Prentice Hall Press.

....anything about the meaning of the signal. There is thus a growing semiotic theory of information, where issues of the semantics, interpretation of signals, and the groundings of signals in measurements are finally being seriously considered [26] Second, since the introduction of fuzzy 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] ....

.... since the introduction of fuzzy 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 ....

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Klir, George and Yuan, Bo: (1995) Fuzzy Sets and Fuzzy Logic, Prentice-Hall, New York

....depending on its inclination or orientation( 8] The theory of fuzzy sets deals with a subset F of the universe of discourse X , where the transition between full membership and no membership is gradual rather than abrupt. Fuzzy set theory is a generalization of the conventional set theory([6]) Definition 1 Let X be a collection of objects x, which could be discrete or continuous, then a fuzzy set F in X is represented as a set of ordered pairs: F = f(x; mF (x) jx 2 Xg; 1) where mF (x) is the membership function or degree of membership of the element x in the fuzzy set F and mF ....

G.Klir and B.Yuan:"Fuzzy Sets and Fuzzy Logic",

....denoted by expression (5) evaluates to zero then no trade off occurs and the entire alternative is not compatible. A geometric mean is used to aggregate the individual ratings. This method obeys the aggregation axioms of monotonicity, continuity, symmetry, idempotent, boundary, and annihilation (Klir and Yuan, 1995). For n criteria the geometric mean aggregate is, h n i i n n ( 1 2 1 1 = 10) This aggregate was also separately developed by Yu, et al. 1993) based on empirical studies with engineers in industry who wanted a metric that evaluated to zero when one of ....

Klir, G.J. and Yuan, B. (1995) Fuzzy Sets and Fuzzy Logic, Prentice Hall, NJ.

....as size capability. Discrete domains are where only a single option of a set of options is possible, such as rotational versus prismatic shape. The individual compatibility ratings are aggregated to obtain a single measure for the overall part. A compensatory operator, the geometric mean, is used [Klir and Yuan, 1995]. The match between the product profile and the process capabilities determine the feasibility of using the production process to fabricate the artifact. A low compatibility C for a certain process capability indicates that the related features could be changed to improve the design from a ....

Klir, G.J., and Yuan, B., (1995). Fuzzy Sets and Fuzzy Logic, Prentice Hall, NJ, 1995.

....are P L and P R and are defined by the standard approximation for multiplication. The approximation is defined by expression (4) and the a cuts are P L = b a)a a P R = b c)a c [P L , P R ] define a set of closed intervals for a [0, 1] which define a fuzzy number according to definition 1 [14, 23]. There are three possible cases to examine. When a = 1, a = 0, and 0 a 1. i) When a = 1, G(a, n) 0. Then P N(L) P L = P N(R) P R . ii) When a = 0, G(a, n) 0. Then P N(L) P L and P N(R) P R . iii) When 0 a 1 and n 2 (the conditions when the new approximation is used) then ....

....9. RELATION TO TYPE II FUZZY SETS Generally, exact values for membership in a set are not realistic and only approximate values provided as a lower and upper bound can be provided. This generalization of ordinary fuzzy sets is called a type 2 fuzzy set where the membership function is fuzzy [14]. The approximation used here has an interval or percent error associated with each a cut and consequently it is related to type 2 fuzzy sets. 10. CONCLUSIONS We have found that engineers will easily accept using TFNs in models. They are more tractable than performing fuzzy arithmetic on ....

G.J. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic (Prentice Hall, New Jersey, 1995).

....(thick) and t i = t(dashes with dots) After one iteration step the sum n 2 t 1 (h) n 2 t 2 (h) 1 (thin) meaning that the solution moves along the one dimensional line connecting the two low temperature limits t1 and t2 . Right: The negative error GammaE M (h) during iteration. tions [35,36]. We choose for concept distances d(h; t) the logical interpretation of d 2 (h; t) 0, i.e. h = t, as true and of d 2 (h; t) 1 as false . For such variables a typical implementation of a logical OR is a product d 2 (AORB) d 2 A d 2 B : 95) A product implementation for ....

Klir, G.J. & Yuan, B.: Fuzzy Sets and Fuzzy Logic. Prentice Hall, 1995.

.... ; Pr(A [ B) Pr(A) Pr(B) Gamma Pr(A B) 2) Then p = hp i i : Pl is a probability distribution with additive normalization and operator X i p i = 1; Pr(A) X i 2A p i : It is well known that statistical entropy is the canonical measure of information in probability theory [13]. In general, the probability distribution with maximal entropy is the maximally uninformative probability distribution denoted p , and results when 8i; p i = 1=n [9] Given a random set S, then the Maximum Entropy Principle (mep) 12] has been applied [2] to derive a canonical probability ....

....set consistency, we will use compatibility . 3 f is a natural probability distribution with normalization P i f i = 1, and P is a natural probability measure as in (2) Many methods are available to convert a given probability distribution to a possibility distribution, and vice versa [13]. One of the most prominent is the maximum normalization or ratio scale method [11] Given a frequency distribution f , then let m : Omega 7 [0; 1] be a possibility distribution where m ( i ) m i : c i W i c i : It follows [9] that m i = f i W f i ; f i = m i ....

Klir, George and Yuan, Bo: (1995) Fuzzy Sets and Fuzzy Logic, Prentice-Hall, New York

....forms. Additive, maximal, and interval constraints then complete the characterization of the most important hybrid forms. 1 Introduction Recent years have seen a proliferation of methods in addition to probability theory to represent information and uncertainty, including fuzzy sets and systems [14], fuzzy measures [20] rough sets [15] random sets [11] Dempster Shafer bodies of evidence [6] possibility distributions [2] imprecise probabilities [19] etc. We can identify these fields collectively as General Information Theory (git) 12] So it is clear that there is a pressing need for ....

Klir, George and Yuan, Bo: (1995) Fuzzy Sets and Fuzzy Logic, Prentice-Hall, New York

....clearly reflects our observables. We also show by discussing two examples that this semantics lends itself to be extended so as to model infinite computation. Since our intent is to randomise programs rather than data, we do not need to base our language on any kind of fuzzy or belief system [14]. Therefore, it is sufficient for our purpose to maintain the same standard (cylindric) constraint system as in CCP. Up to now probabilistic algorithms have been usually written using imperative languages extended with some random facility (e.g. pseudo random number generators) and have been ....

George J. Klir and Bo Yuan. Fuzzy Sets and Fuzzy Logic. Prentice Hall, New Jersey, 1995.

....on the combination of GAs and FL collected in [1] It contains the keywords and the number of papers on each of them. With these keywords the application of FL based tools to GA (with the name of fuzzy genetic algorithms) and the different areas of the fuzzy logic and fuzzy sets theory, [6], where GAs have been applied are covered. 1 Fuzzy genetic algorithms 16 2 Fuzzy clustering 11 3 Fuzzy optimization 23 4 Fuzzy neural networks 23 5 Fuzzy relational equations 4 6 Fuzzy expert systems 5 7 Fuzzy classifier systems 19 8 Fuzzy information retrieval 5 and Database Quering 9 Fuzzy ....

George Klir, Bo Yuan "Fuzzy Sets and Fuzzy Logic", Prentice Hall, 1995.

....recent years have seen a proliferation of new, non probabilistic mathematical methods for the representation of uncertainty and information in systems models. Following Klir [1991] we call these methods collectively General Information Theory (git) which includes fuzzy sets, systems, and logic [Klir and Yuan, 1995]; fuzzy measures [Wang and Klir, 1992] random set [Kendall, 1974] and Dempster Shafer evidence theory [Dempster, 1967; Shafer, 1976] possibility theory [de Cooman et al. 1995] imprecise probabilities [Walley, 1990] probability bounds [Ferson et al. 1996] rough set theory [Pawlak, 1991] and ....

Klir, George and Bo Yuan [1995], Fuzzy Sets and Fuzzy Logic, Prentice-Hall, New York.

....takes place. Based on this we define a notion of observables which captures the (finite) results of computation together with their probability of being computed. Since our intent is to randomise programs rather than data, we do not need to base our language on any kind of fuzzy or belief system [12]. Therefore, it is sufficient for our purpose to maintain the same standard (cylindric) constraint system as in CCP. The embedding of randomness within the semantics of a well structured programming paradigm, like CCP, also aims at providing a sound framework for reasoning about randomised ....

George J. Klir and Bo Yuan. Fuzzy Sets and Fuzzy Logic. Prentice Hall, New Jersey, 1995.

....recent years have seen a proliferation of new, non probabilistic mathematical methods for the representation of uncertainty and information in systems models. Following Klir [18] we call these methods collectively General Information Theory (git) which includes fuzzy sets, systems, and logic [19]; fuzzy measures [28] random set [17] and Dempster Shafer evidence theory [25] possibility theory [6] and others. Each of these involves some form of generalization or extension away from stochastic representations, and probability is retained in git as a limiting case. Within git, possibility ....

....defined on random sets; possibility and their dual necessity measures represent extreme ranges of probability intervals; and the distributions of all these generally non additive fuzzy measures are fuzzy sets. A detailed description of fuzzy and possibility theory is not possible here (see [5, 14, 19]) but I will introduce some basic concepts. Assume a finite universe Omega = f i g; 1 i n. A function : 2 Omega 7 [0; 1] is a (finite) fuzzy measure [28] if ( 0 and 8A; B Omega ; A B (A) B) A fuzzy subset denoted e A e Omega is defined by its membership function e ....

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Klir, George and Yuan, Bo: (1995) Fuzzy Sets and Fuzzy Logic, Prentice-Hall, New York

....we are investigating the use of features such as DC component and motion vectors that can be obtained directly from the MPEG data. Fuzzy set theory (and clustering) has been extensively used in pattern recognition and computer vision. A classic text in this area is due to Bezdek[3] while Klir[17] presents more recent and comprehensive information. Also germane to our effort are the work of Frigui and Krishnapuram [6] who combined fuzzy clustering and robust statistical estimators in order to find the number of cluster and obtain the prototype parameter from noisy data. Krishnapuram and ....

G. Klir and Bo Yuan, Fuzzy Sets and Fuzzy Logic,

....qualitative mbd, we plan to use the Generalized Information Theory (git) computational paradigm, and especially its possibilistic modeling techniques. git is the synthesis of modern mathematical theories of uncertainty, including fuzzy systems, random sets, evidence theory, and possibility theory [9, 16, 17]. git promises to provide a key generalizing technology for qm methods in systems theory [9, 11] This paper describes the Data Analysis and Systems Modeling Environment (dasme) which is being used as the development environment for this approach, and our proposed cast based extensions to dasme ....

....theory is logically independent of probability theory, they are related: both arise in Dempster Shafer evidence theory as fuzzy measures defined on random sets; and their distributions are both fuzzy sets. So possibility theory is a component of a git, which includes all of these fields [9, 16, 17]. But at a semantic level, probability and possibility are radically different. Probability represents division of knowledge among a distinct set of point outcomes. Possibility, on the other hand, is inherently non additive. Possibility represents coherence of knowledge around a central core of ....

Klir, George and Yuan, Bo: (1995) Fuzzy Sets and Fuzzy Logic, Prentice-Hall, New York

....of probability theory, they are related: both arise in Dempster Shafer evidence theory as fuzzy measures defined on random sets; and their distributions are both fuzzy sets. So possibility theory is a component of a broader Generalized Information Theory (git) which includes all of these fields [18]. Possibility theory was originally developed in the context of fuzzy systems theory [28] More recently, possibility theory is being developed independently of both fuzzy sets and probability. In particular, the author is developing the mathematics and semantics of possibility theory [9, 12, 13] ....

....an object oriented environment. The approach is based on the mathematics of consistent random sets as the basis for the representation of measured possibility distributions and possibilistic processes. 2 Possibility Theory Mathematical possibility theory can only be briefly introduced here. See [4, 12, 18] for details and proofs. 2.1 Mathematical Possibility in git Given a finite universe Omega : f i g; 1 i n, the function m: 2 Omega 7 [0; 1] is an evidence function (otherwise known as a basic probability assignment) when m( 0 and P A Omega m(A) 1. Denote a random set generated ....

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Klir, George and Yuan, Bo: (1995) Fuzzy Sets and Fuzzy Logic, Prentice-Hall, New York

....takes place. Based on this we define a notion of observables which captures the (finite) results of computation together with their probability of being computed. Since our intent is to randomise programs rather than data, we do not need to base our language on any kind of fuzzy or belief system [14]. Therefore, it is sufficient for our purpose to maintain the same standard (cylindric) constraint system as in CCP. The embedding of randomness within the semantics of a well structured programming paradigm, like CCP, also aims at providing a sound framework for reasoning about randomised ....

George J. Klir and Bo Yuan. Fuzzy Sets and Fuzzy Logic. Prentice Hall, New Jersey, 1995.

.... Pr(B) Gamma Pr(A B) 3) and p = hp i i : ae is a probability distribution with additive normalization P i p i = 1 and operator Pr(A) P i 2A p i . Statistical entropy H( p ) Gamma P n i=1 p i log 2 (p i ) is then the canonical measure of information in probability theory (Klir and Yuan 1995), and the probability distribution with maximal entropy is the maximally uninformative probability distribution denoted p which results when 8i; p i = 1=n (Joslyn 1994) Given a random set S, then the Maximum Entropy Principle (mep) Klir 1993) has been applied (Dubois and Prade 1982) to ....

....B Omega ; Pi(A [ B) Pi(A) Pi(B) where is the maximum operator. Now = h i i : Pl is a possibility distribution with maximal normalization and operator, respectively i i = 1; 5) Pi(A) i 2A i : 6) In possibility theory entropy is replaced by nonspecificity (Klir and Yuan 1995) N( n X i=1 ( i Gamma i 1 ) log 2 (i) n X i=2 i log 2 i i Gamma 1 ; 7) where n 1 : 0 by convention. The possibility distribution with maximal nonspecificity is the maximally uninformative possibility distribution (Klir 1993) denoted , resulting when 8i; ....

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Klir, G., and Yuan, B., Fuzzy Sets and Fuzzy Logic, Prentice-Hall, New York, 1995.

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G. J. Klir and B. Yuan. Fuzzy Sets and Fuzzy Logic. Prentice Hall, 1995.

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G. J. Klir and B. Yuan. Fuzzy Sets and Fuzzy Logic. Prentice Hall, 1995.

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G. J. Klir and B. Yuan. Fuzzy Sets and Fuzzy Logic. Prentice Hall, 1995.

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B. Yuan G. Klir, Fuzzy Sets and Fuzzy Logic, PrenticeHall International, New Jersey, 1995.

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Klir, George and Yuan, Bo: (1995) Fuzzy Sets and Fuzzy Logic, Prentice-Hall, New York

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G. J. Klir and B. Yuan. Fuzzy Sets and Fuzzy Logic. Prentice Hall, 1995.

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G. J. Klir and B. Yuan. Fuzzy Sets and Fuzzy Logic. Prentice Hall, 1995.

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Klir, G.J. & Yuan, B. 1995. Fuzzy Sets and Fuzzy Logic., Prentice Hall, New Jersey.

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Klir, J.G. & Yuan, B. (1995) Fuzzy Sets and Fuzzy Logic. Upper Saddle River, NJ: Prentice Hall.

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Klir, J.G. & Yuan, B. (1995) Fuzzy Sets and Fuzzy Logic. Upper Saddle River, NJ: Prentice Hall.

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G. J. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic, Prentice-Hall, 1995.

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