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BL (logic) : Hájek P., 1998, Metamathematics of Fuzzy Logic. Dordrecht: Kluwer. Ono, H., 2003, "Substructural logics and residuated lattices — an introduction". In F.V. Hendricks, J. Malinowski (eds.): Trends in Logic: 50 Years of Studia Logica, Trends in Logic 20: 177–212. Cintula P., 2005, "Short note: On the redunda... |
Combs method : The Combs method is a rule base reduction method of writing fuzzy logic rules described by William E. Combs in 1997. It is designed to prevent combinatorial explosion in fuzzy logic rules. The Combs method takes advantage of the logical equality ( ( p ∧ q ) ⇒ r ) ⟺ ( ( p ⇒ r ) ∨ ( q ⇒ r ) ) . |
Combs method : The simplest proof of given equality involves usage of truth tables: |
Combs method : Suppose we have a fuzzy system that considers N variables at a time, each of which can fit into at least one of S sets. The number of rules necessary to cover all the cases in a traditional fuzzy system is S N , whereas the Combs method would need only S × N rules. For example, if we have five sets and... |
Combs method : Suppose we were designing an artificial personality system that determined how friendly the personality is supposed to be towards a person in a strategic video game. The personality would consider its own fear, trust, and love in the other person. A set of rules in the Combs system might look like this: ... |
Combs method : The Combs Method for Rapid Inference (the original paper by William E. Combs) The Combs Method for Rapid Inference (Archive of the original paper by William E. Combs) |
Construction of t-norms : In mathematics, t-norms are a special kind of binary operations on the real unit interval [0, 1]. Various constructions of t-norms, either by explicit definition or by transformation from previously known functions, provide a plenitude of examples and classes of t-norms. This is important, e.g... |
Construction of t-norms : The method of constructing t-norms by generators consists in using a unary function (generator) to transform some known binary function (most often, addition or multiplication) into a t-norm. In order to allow using non-bijective generators, which do not have the inverse function, the followin... |
Construction of t-norms : Many families of related t-norms can be defined by an explicit formula depending on a parameter p. This section lists the best known parameterized families of t-norms. The following definitions will be used in the list: A family of t-norms Tp parameterized by p is increasing if Tp(x, y) ≤ Tq(x... |
Construction of t-norms : The ordinal sum constructs a t-norm from a family of t-norms, by shrinking them into disjoint subintervals of the interval [0, 1] and completing the t-norm by using the minimum on the rest of the unit square. It is based on the following theorem: Let Ti for i in an index set I be a family of t... |
Construction of t-norms : The construction of t-norms by rotation was introduced by Sándor Jenei (2000). It is based on the following theorem: Let T be a left-continuous t-norm without zero divisors, N: [0, 1] → [0, 1] the function that assigns 1 − x to x and t = 0.5. Let T1 be the linear transformation of T into [t, 1... |
Construction of t-norms : T-norm T-norm fuzzy logics |
Construction of t-norms : Klement, Erich Peter; Mesiar, Radko; and Pap, Endre (2000), Triangular Norms. Dordrecht: Kluwer. ISBN 0-7923-6416-3. Fodor, János (2004), "Left-continuous t-norms in fuzzy logic: An overview". Acta Polytechnica Hungarica 1(2), ISSN 1785-8860 [1] Dombi, József (1982), "A general class of fuzzy ... |
Defuzzification : Defuzzification is the process of producing a quantifiable result in crisp logic, given fuzzy sets and corresponding membership degrees. It is the process that maps a fuzzy set to a crisp set. It is typically needed in fuzzy control systems. These systems will have a number of rules that transform a n... |
Defuzzification : There are many different methods of defuzzification available, including the following: AI (adaptive integration) BADD (basic defuzzification distributions) BOA (bisector of area) CDD (constraint decision defuzzification) COA (center of area) COG (center of gravity) ECOA (extended center of area) EQM ... |
Defuzzification : Fuzzy logic Fuzzy set Fuzzy control == Notes == |
Degree of truth : In classical logic, propositions are typically unambiguously considered as being true or false. For instance, the proposition one is both equal and not equal to itself is regarded as simply false, being contrary to the Law of Noncontradiction; while the proposition one is equal to one is regarded as s... |
Degree of truth : Language Meaning (linguistics) — Semiotics Technology Artificial intelligence Logic Bivalence Fuzzy logic Fuzzy set Half-truth Multi-valued logic Paradox of the heap Truth Truth value Vagueness Books Vagueness and Degrees of Truth |
Degree of truth : == References == |
European Society for Fuzzy Logic and Technology : The European Society for Fuzzy Logic and Technology (EUSFLAT) is a scientific association with the aims to disseminate and promote fuzzy logic and related subjects (sometimes comprised under the collective terms soft computing or computational intelligence) and to provi... |
European Society for Fuzzy Logic and Technology : EUSFLAT was founded in 1998 in Spain as the successor of the National Spanish Fuzzy Logic Society, ESTYLF, with the aim to open the society for members from other European countries. Since then, the society managed to attract a large share of members from outside Spain,... |
European Society for Fuzzy Logic and Technology : Starting with 1999, EUSFLAT has been organizing its biannual conferences in odd years. Previous meetings: Palma de Mallorca, Balearic Islands, Spain, September 22–25, 1999 (jointly with National Spanish conference, ESTYLF) Leicester, United Kingdom, September 5–7, 2001 ... |
European Society for Fuzzy Logic and Technology : EUSFLAT publishes the proceedings of its conferences in an open access manner. Until 2010, Mathware & Soft Computing was the official journal of EUSFLAT. On July 1, 2010, the International Journal of Computational Intelligence Systems (Atlantis Press, ISSN 1875-6891 (pr... |
European Society for Fuzzy Logic and Technology : EUSFLAT is led by the President, who is elected for a two-year period, and cannot serve for more than two consecutive periods. Francesc Esteva (1998–2011) Luis Magdalena (2001–2005) Ulrich Bodenhofer (2005–2009) Javier Montero (2009–2013) Gabriella Pasi (2013–present) |
Evolving intelligent system : In computer science, an evolving intelligent system is a fuzzy logic system which improves the own performance by evolving rules. The technique is known from machine learning, in which external patterns are learned by an algorithm. Fuzzy logic based machine learning works with neuro-fuzzy ... |
Evolving intelligent system : EISs can be implemented, for example, using neural networks or fuzzy rule-based models. The first neural networks which consider an evolving structure were published in. These were later expanded by N. Kasabov and P. Angelov for the neuro-fuzzy models. P. Angelov introduced the evolving fu... |
Functional presence engine : A Functional Presence Engine, or FPE, is a probabilistic parsing mechanism that uses at least four components to respond to input patterns. It comprises a lexing system, a probabilistic fitness function, a knowledge base, and a library of functions that the knowledge base can trigger. The l... |
Functional presence engine : The first Functional Presence Engine was deployed in 2001 by Spectre AI Incorporated. The technology and a number of embodiments were subsequently patented by Spectre AI's cofounder Robert Hust, the FPE's original inventor, and Mark Zartler who was Spectre AI's lead developer. The developme... |
Fuzzy architectural spatial analysis : Fuzzy architectural spatial analysis (FASA) (also fuzzy inference system (FIS) based architectural space analysis or fuzzy spatial analysis) is a spatial analysis method of analysing the spatial formation and architectural space intensity within any architectural organization. Fuz... |
Fuzzy architectural spatial analysis : Fuzzy architectural spatial analysis was developed by Burcin Cem Arabacioglu (2010) from the architectural theories of space syntax and visibility graph analysis, and is applied with the help of a fuzzy system with a Mamdami inference system based on fuzzy logic within any archite... |
Fuzzy architectural spatial analysis : Arabacioglu, Burcin Cem (2010). "Using fuzzy inference system for architectural space analysis". Applied Soft Computing. 10 (3): 926–937. doi:10.1016/j.asoc.2009.10.011. Cekmis, Asli; Hacihasanoglu, Isis; Ostwald, Michael J (2013). "A computational model for accommodating spatial ... |
Fuzzy architectural spatial analysis : Fuzzy logic Spatial analysis Space syntax Spatial network analysis software Visibility graph Visibility graph analysis Boundary problem (in spatial analysis) |
Fuzzy associative matrix : A fuzzy associative matrix expresses fuzzy logic rules in tabular form. These rules usually take two variables as input, mapping cleanly to a two-dimensional matrix, although theoretically a matrix of any number of dimensions is possible. From the perspective of neuro-fuzzy systems, the mathe... |
Fuzzy associative matrix : In the context of game AI programming, a fuzzy associative matrix helps to develop the rules for non-player characters. Suppose a professional is tasked with writing fuzzy logic rules for a video game monster. In the game being built, entities have two variables: hit points (HP) and firepower... |
Fuzzy associative matrix : There is no inherent pattern in the matrix. It appears as if the rules were just made up, and indeed they were. This is both a strength and a weakness of fuzzy logic in general. It is often impractical or impossible to find an exact set of rules or formulae for dealing with a specific situati... |
Fuzzy classification : Fuzzy classification is the process of grouping elements into fuzzy sets whose membership functions are defined by the truth value of a fuzzy propositional function. A fuzzy propositional function is analogous to an expression containing one or more variables, such that when values are assigned t... |
Fuzzy classification : Intuitively, a class is a set that is defined by a certain property, and all objects having that property are elements of that class. The process of classification evaluates for a given set of objects whether they fulfill the classification property, and consequentially are a member of the corres... |
Fuzzy classification : Fuzzy logic == References == |
Fuzzy concept : A fuzzy concept is an idea of which the boundaries of application can vary considerably according to context or conditions, instead of being fixed once and for all. This means the idea is somewhat vague or imprecise. Yet it is not unclear or meaningless. It has a definite meaning, which can be made more... |
Fuzzy concept : Problems of vagueness and fuzziness have probably always existed in human experience. In the West, ancient texts show that philosophers and scientists were already thinking about those kinds of problems in classical antiquity. Kit Fine states that "when a philosopher talks of vagueness he has in mind a ... |
Fuzzy concept : The ordinary scholarly definition of a concept as "fuzzy" has been in use from the 1970s onward. |
Fuzzy concept : The technique of fuzzy concept lattices is increasingly used in programming for the formatting, relating and analysis of fuzzy data sets. |
Fuzzy concept : There have been many academic debates about the meaning, relevance and utility of fuzzy concepts, as well as their appropriate use. Rudolf E. Kálmán stated in 1972 that "there is no such thing as a fuzzy concept... We do talk about fuzzy things but they are not scientific concepts". The suggestion is th... |
Fuzzy concept : The idea of fuzzy concepts has also been applied in the philosophical, sociological and linguistic analysis of human behaviour. |
Fuzzy concept : Fuzzy concepts can generate uncertainty because they are imprecise (especially if they refer to a process in motion, or a process of transformation where something is "in the process of turning into something else"). In that case, they do not provide a clear orientation for action or decision-making ("w... |
Fuzzy concept : Ordinary language, which uses symbolic conventions and associations which are often not logical, inherently contains many fuzzy concepts – "knowing what you mean" in this case depends partly on knowing the context (or being familiar with the way in which a term is normally used, or what it is associated... |
Fuzzy concept : Various different aspects of human experience commonly generate concepts with fuzzy characteristics. |
Fuzzy concept : Fuzzy concepts often play a role in the creative process of forming new concepts to understand something. In the most primitive sense, this can be observed in infants who, through practical experience, learn to identify, distinguish and generalise the correct application of a concept, and relate it to o... |
Fuzzy concept : Fuzzy concepts can be used as a practical method to describe something of which a complete description would be an unmanageably large undertaking, or very time-consuming; thus, a simplified indication of what is at issue is regarded as sufficient, although it is not exact. |
Fuzzy concept : Common use of this sort of approach (combining words and numbers in programming), has led some logicians to regard fuzzy logic merely as an extension of Boolean logic (a two-valued logic or binary logic is simply replaced with a many-valued logic). However, Boolean concepts have a logical structure whic... |
Fuzzy concept : In mathematical logic, computer programming, philosophy and linguistics fuzzy concepts can be analyzed and defined more accurately or comprehensively, by describing or modelling the concepts using the terms of fuzzy logic or other substructural logics. With the accelerating development of computer progr... |
Fuzzy concept : A process of defuzzification is said to occur, when fuzzy concepts can be logically described in terms of fuzzy sets, or the relationships between fuzzy sets, which makes it possible to define variations in the meaning or applicability of concepts as quantities. Effectively, qualitative differences are ... |
Fuzzy concept : James F. Brule, Fuzzy systems tutorial "Fuzzy Logic", Stanford Encyclopedia of Philosophy "Vagueness", Stanford Encyclopedia of Philosophy Calvin College Engineering Department, Getting Started with Fuzzy Logic Archived 2018-02-21 at the Wayback Machine 2009 Benjamin Franklin Medal Winner: Lotfi A. Zade... |
Fuzzy Control Language : Fuzzy Control Language, or FCL, is a language for implementing fuzzy logic, especially fuzzy control. It was standardized by IEC 61131-7. It is a domain-specific programming language: it has no features unrelated to fuzzy logic, so it is impossible to even print "Hello, world!". Therefore, one ... |
Fuzzy Control Language : RULE 0: IF (temperature IS cold) THEN (output IS low) RULE 1: IF (temperature IS very cold) THEN (output IS high) |
Fuzzy Control Language : FCL is not an entirely complete fuzzy language, for instance, it does not support "hedges", which are adverbs that modify the set. For instance, the programmer cannot write: RULE 0: If (Temperature is VERY COLD) then (Output is VERY HIGH) However, the programmer can simply define new sets for "... |
Fuzzy Control Language : fuzzyTECH, a commercial fuzzy logic development system containing the specification document for IEC1131-7 (select Fuzzy Application Library) IEC 1131-7 CD1 Archived 2021-03-04 at the Wayback Machine IEC 1131-7 CD1 PDF fuzzylite, A fuzzy logic controller library written in C++. Free Fuzzy Logic... |
Fuzzy control system : A fuzzy control system is a control system based on fuzzy logic – a mathematical system that analyzes analog input values in terms of logical variables that take on continuous values between 0 and 1, in contrast to classical or digital logic, which operates on discrete values of either 1 or 0 (tr... |
Fuzzy control system : Fuzzy logic was proposed by Lotfi A. Zadeh of the University of California at Berkeley in a 1965 paper. He elaborated on his ideas in a 1973 paper that introduced the concept of "linguistic variables", which in this article equates to a variable defined as a fuzzy set. Other research followed, wi... |
Fuzzy control system : The input variables in a fuzzy control system are in general mapped by sets of membership functions similar to this, known as "fuzzy sets". The process of converting a crisp input value to a fuzzy value is called "fuzzification". The fuzzy logic based approach had been considered by designing two... |
Fuzzy control system : As an example, consider an anti-lock braking system, directed by a microcontroller chip. The microcontroller has to make decisions based on brake temperature, speed, and other variables in the system. The variable "temperature" in this system can be subdivided into a range of "states": "cold", "c... |
Fuzzy control system : In spite of the appearance there are several difficulties to give a rigorous logical interpretation of the IF-THEN rules. As an example, interpret a rule as IF (temperature is "cold") THEN (heater is "high") by the first order formula Cold(x)→High(y) and assume that r is an input such that Cold(r... |
Fuzzy control system : Before an Artificial Intelligence system is able to plan the action sequence, some kind of model is needed. For video games, the model is equal to the game rules. From the programming perspective, the game rules are implemented as a Physics engine which accepts an action from a player and calcula... |
Fuzzy control system : Fuzzy control systems are suitable when the process complexity is high including uncertainty and nonlinear behavior, and there are no precise mathematical models available. Successful applications of fuzzy control systems have been reported worldwide mainly in Japan with pioneering solutions sinc... |
Fuzzy control system : Dynamic logic Bayesian inference Function approximation Fuzzy concept Fuzzy markup language Hysteresis Neuro-fuzzy Fuzzy control language Type-2 fuzzy sets and systems |
Fuzzy control system : Kevin M. Passino and Stephen Yurkovich, Fuzzy Control, Addison Wesley Longman, Menlo Park, CA, 1998 (522 pages) Archived 2008-12-15 at the Wayback Machine Kazuo Tanaka; Hua O. Wang (2001). Fuzzy control systems design and analysis: a linear matrix inequality approach. John Wiley and Sons. ISBN 97... |
Fuzzy control system : Robert Babuska and Ebrahim Mamdani, ed. (2008). "Fuzzy control". Scholarpedia. Retrieved 31 December 2022. Introduction to Fuzzy Control Archived 2010-08-05 at the Wayback Machine Fuzzy Logic in Embedded Microcomputers and Control Systems IEC 1131-7 CD1 Archived 2021-03-04 at the Wayback Machine ... |
Fuzzy differential equation : Fuzzy differential equation are general concept of ordinary differential equation in mathematics defined as differential inclusion for non-uniform upper hemicontinuity convex set with compactness in fuzzy set. d x ( t ) / d t = F ( t , x ( t ) , α ) , for all α ∈ [ 0 , 1 ] . |
Fuzzy differential equation : A first order fuzzy differential equation with real constant or variable coefficients x ′ ( t ) + p ( t ) x ( t ) = f ( t ) where p ( t ) is a real continuous function and f ( t ) : [ t 0 , ∞ ) → R F ,\infty )\rightarrow R_ is a fuzzy continuous function y ( t 0 ) = y 0 )=y_ such that y ... |
Fuzzy differential equation : A system of equations of the form x ( t ) n ′ = a n 1 ( t ) x 1 ( t ) + . . . . . . + a n n ( t ) x n ( t ) + f n ( t ) =a_1(t)x_(t)+......+a_n(t)x_(t)+f_(t) where a i j j are real functions and f i are fuzzy functions x n ′ ( t ) = ∑ i = 0 1 a i j x i . (t)=\sum _^a_x_. |
Fuzzy differential equation : A fuzzy differential equation with partial differential operator is ∇ x ( t ) = F ( t , x ( t ) , α ) , for all α ∈ [ 0 , 1 ] . |
Fuzzy differential equation : A fuzzy differential equation with fractional differential operator is d n x ( t ) d t n = F ( t , x ( t ) , α ) , x(t)=F(t,x(t),\alpha ), for all α ∈ [ 0 , 1 ] where n is a rational number. == References == |
Fuzzy differential inclusion : Fuzzy differential inclusion is the extension of differential inclusion to fuzzy sets introduced by Lotfi A. Zadeh. x ′ ( t ) ∈ [ f ( t , x ( t ) ) ] α with x ( 0 ) ∈ [ x 0 ] α ]^ Suppose f ( t , x ( t ) ) is a fuzzy valued continuous function on Euclidean space. Then it is the collecti... |
Fuzzy differential inclusion : The second order differential is x ″ ( t ) ∈ [ k x ] α where k ∈ [ K ] α , K is trapezoidal fuzzy number ( − 1 , − 1 / 2 , 0 , 1 / 2 ) , and x 0 is a trianglular fuzzy number (-1,0,1). |
Fuzzy differential inclusion : Fuzzy differential inclusion (FDI) has applications in Cybernetics Artificial intelligence, Neural network, Medical imaging Robotics Atmospheric dispersion modeling Weather forecasting Cyclone Pattern recognition Population biology == References == |
Fuzzy electronics : Fuzzy electronics is an electronic technology that uses fuzzy logic, instead of the two-state Boolean logic more commonly used in digital electronics. Fuzzy electronics is fuzzy logic implemented on dedicated hardware. This is to be compared with fuzzy logic implemented in software running on a conv... |
Fuzzy electronics : The first fuzzy electronic circuit was built by Takeshi Yamakawa et al. in 1980 using discrete bipolar transistors. The first industrial fuzzy application was in a cement kiln in Denmark in 1982. The first VLSI fuzzy electronics was by Masaki Togai and Hiroyuki Watanabe in 1984. In 1987, Yamakawa bu... |
Fuzzy electronics : Defuzzification Fuzzy set Fuzzy set operations |
Fuzzy electronics : Ibrahim, Ahmad M. (1997). Introduction to Applied Fuzzy Electronics. Prentice Hall. ISBN 0-13-206400-6. Abraham Kandel, Gideon Langholz (eds), Fuzzy Hardware: Architectures and Applications, Springer Science & Business Media, 2012 ISBN 1461540909. Russo, Marco (1998). "Fuzzy Hardware Research from H... |
Fuzzy electronics : Yamakawa, T.; Inoue, T.; Ueno, F.; Shirai, Y., "Implementation of Fuzzy Logic hardware systems-Three fundamental arithmetic circuits", Transactions of the Institute of Electronics and Communications Engineers, vol. 63, 1980, pp. 720–721. Togai, M.; Watanabe, H., "A VLSI implementation of a fuzzy inf... |
Fuzzy electronics : Applications of Fuzzy logic in electronics |
Fuzzy finite element : The fuzzy finite element method combines the well-established finite element method with the concept of fuzzy numbers, the latter being a special case of a fuzzy set. The advantage of using fuzzy numbers instead of real numbers lies in the incorporation of uncertainty (on material properties, par... |
Fuzzy markup language : Fuzzy Markup Language (FML) is a specific purpose markup language based on XML, used for describing the structure and behavior of a fuzzy system independently of the hardware architecture devoted to host and run it. |
Fuzzy markup language : FML was designed and developed by Giovanni Acampora during his Ph.D. course in Computer Science, at University of Salerno, Italy, in 2004. The original idea inspired Giovanni Acampora to create FML was the necessity of creating a cooperative fuzzy-based framework aimed at automatically controlli... |
Fuzzy markup language : FML allows fuzzy systems to be coded through a collection of correlated semantic tags capable of modeling the different components of a classical fuzzy controller such as knowledge base, rule base, fuzzy variables and fuzzy rules. Therefore, the FML tags used to build a fuzzy controller represen... |
Fuzzy markup language : Lee, Chang-Shing; et al. (December 2010). "Diet assessment based on type-2 fuzzy ontology and fuzzy markup language". International Journal of Intelligent Systems. 25 (12): 1187–1216. doi:10.1002/int.20449. S2CID 13570946. (subscription required) Acampora, G.; Loia, V. (2005). "Fuzzy control int... |
Fuzzy mathematics : Fuzzy mathematics is the branch of mathematics including fuzzy set theory and fuzzy logic that deals with partial inclusion of elements in a set on a spectrum, as opposed to simple binary "yes" or "no" (0 or 1) inclusion. It started in 1965 after the publication of Lotfi Asker Zadeh's seminal work F... |
Fuzzy mathematics : A fuzzy subset A of a set X is a function A: X → L, where L is the interval [0, 1]. This function is also called a membership function. A membership function is a generalization of an indicator function (also called a characteristic function) of a subset defined for L = . More generally, one can use... |
Fuzzy mathematics : The evolution of the fuzzification of mathematical concepts can be broken down into three stages: straightforward fuzzification during the sixties and seventies, the explosion of the possible choices in the generalization process during the eighties, the standardization, axiomatization, and L-fuzzif... |
Fuzzy mathematics : Fuzzy subgroupoids and fuzzy subgroups were introduced in 1971 by A. Rosenfeld. Analogues of other mathematical subjects have been translated to fuzzy mathematics, such as fuzzy field theory and fuzzy Galois theory, fuzzy topology, fuzzy geometry, fuzzy orderings, and fuzzy graphs. |
Fuzzy mathematics : Fuzzy measure theory Fuzzy subalgebra Monoidal t-norm logic Possibility theory T-norm |
Fuzzy mathematics : Zadeh, L.A. Fuzzy Logic - article at Scholarpedia Hajek, P. Fuzzy Logic - article at Stanford Encyclopedia of Philosophy Navara, M. Triangular Norms and Conorms - article at Scholarpedia Dubois, D., Prade H. Possibility Theory - article at Scholarpedia Center for Mathematics of Uncertainty Fuzzy Mat... |
Fuzzy measure theory : In mathematics, fuzzy measure theory considers generalized measures in which the additive property is replaced by the weaker property of monotonicity. The central concept of fuzzy measure theory is the fuzzy measure (also capacity, see ), which was introduced by Choquet in 1953 and independently ... |
Fuzzy measure theory : Let X be a universe of discourse, C be a class of subsets of X , and E , F ∈ C . A function g : C → R \to \mathbb where ∅ ∈ C ⇒ g ( ∅ ) = 0 \Rightarrow g(\emptyset )=0 E ⊆ F ⇒ g ( E ) ≤ g ( F ) is called a fuzzy measure. A fuzzy measure is called normalized or regular if g ( X ) = 1 )=1... |
Fuzzy measure theory : A fuzzy measure is: additive if for any E , F ∈ C such that E ∩ F = ∅ , we have g ( E ∪ F ) = g ( E ) + g ( F ) . ; supermodular if for any E , F ∈ C , we have g ( E ∪ F ) + g ( E ∩ F ) ≥ g ( E ) + g ( F ) ; submodular if for any E , F ∈ C , we have g ( E ∪ F ) + g ( E ∩ F ) ≤ g ( E ) + g (... |
Fuzzy measure theory : Let g be a fuzzy measure. The Möbius representation of g is given by the set function M, where for every E , F ⊆ X , M ( E ) = ∑ F ⊆ E ( − 1 ) | E ∖ F | g ( F ) . (-1)^g(F). The equivalent axioms in Möbius representation are: M ( ∅ ) = 0 . ∑ F ⊆ E | i ∈ F M ( F ) ≥ 0 M(F)\geq 0 , for all E ⊆ X ... |
Fuzzy measure theory : Fuzzy measures are defined on a semiring of sets or monotone class, which may be as granular as the power set of X, and even in discrete cases the number of variables can be as large as 2|X|. For this reason, in the context of multi-criteria decision analysis and other disciplines, simplification... |
Fuzzy measure theory : In game theory, the Shapley value or Shapley index is used to indicate the weight of a game. Shapley values can be calculated for fuzzy measures in order to give some indication of the importance of each singleton. In the case of additive fuzzy measures, the Shapley value will be the same as each... |
Fuzzy measure theory : Probability theory Possibility theory |
Fuzzy measure theory : Beliakov, Pradera and Calvo, Aggregation Functions: A Guide for Practitioners, Springer, New York 2007. Wang, Zhenyuan, and, George J. Klir, Fuzzy Measure Theory, Plenum Press, New York, 1991. |
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