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| {{Expert-subject|Mathematics|date=November 2008}}
| | The title of the author is Numbers. The preferred pastime for my kids and me is to play baseball but I haven't made a dime with it. Puerto Rico is where he's usually been living but she requirements to move because of her family members. For many years I've been working as a payroll clerk.<br><br>My weblog [http://ece.modares.ac.ir/mnl/?q=node/1234839 home std test kit] |
| In [[mathematics]], '''fuzzy measure theory''' considers generalized [[measure (mathematics)|measure]]s 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 <ref>{{cite journal
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| |author = [[Gustave Choquet]]
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| |title = Theory of Capacities
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| |journal = Annales de l'Institut Fourier
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| |volume = 5
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| |pages = 131–295
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| |year = 1953
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| }}</ref>) which was introduced by [[Gustave Choquet|Choquet]] in 1953 and independently defined by Sugeno in 1974 in the context of [[Sugeno integral|fuzzy integrals]]. There exists a number of different classes of fuzzy measures including [[Dempster-Shafer theory|plausibility/belief]] measures; [[Possibility theory|possibility/necessity]] measures; and [[probability measure|probability]] measures which are a subset of [[measure (mathematics)|classical]] measures.
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| ==Definitions==
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| Let <math>\mathbf{X}</math> be a [[universe of discourse]], <math>\mathcal{C}</math> be a [[Class (mathematics)|class]] of [[subset]]s of <math>\mathbf{X}</math>, and <math>E,F\in\mathcal{C}</math>. A [[function (mathematics)|function]] <math>g:\mathcal{C}\to\mathbb{R}</math> where
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| # <math>\emptyset \in \mathcal{C} \Rightarrow g(\emptyset)=0</math>
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| # <math>E \subseteq F \Rightarrow g(E)\leq g(F)</math>
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| is called a ''fuzzy measure''. | |
| A fuzzy measure is called ''normalized'' or ''regular'' if <math>g(\mathbf{X})=1</math>.
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| Adding some examples of fuzzy measures-->
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| ==Properties of fuzzy measures==
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| For any <math> E,F \in \mathcal{C} </math>, a fuzzy measure is:
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| * '''additive''' if <math> g(E \cup F) = g(E) + g(F). </math> for all <math> E \cap F = \emptyset </math>;
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| * '''supermodular''' if <math>g(E \cup F) + g(E \cap F) \geq g(E) + g(F)</math>;
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| * '''[[submodular]]''' if <math>g(E \cup F) + g(E \cap F) \leq g(E) + g(F)</math>;
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| * '''superadditive''' if <math>g(E \cup F) + g(E \cap F) \geq g(E) + g(F)</math> for all <math>E \cap F = \emptyset</math>;
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| * '''subadditive''' if <math>g(E \cup F) + g(E \cap F) \leq g(E) + g(F)</math> for all <math>E \cap F = \emptyset</math>;
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| * '''symmetric''' if <math>|E| = |F|</math> implies <math> g(E) = g(F)</math>;
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| * '''Boolean''' if <math> g(E) = 0 </math> or <math> g(E) = 1 </math>.
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| Understanding the properties of fuzzy measures is useful in application. When a fuzzy measure is used to define a function such as the [[Sugeno integral]] or [[Choquet integral]], these properties will be crucial in understanding the function's behavior. For instance, the Choquet integral with respect to an additive fuzzy measure reduces to the [[Lebesgue integral]]. In discrete cases, a symmetric fuzzy measure will result in the [[Ordered weighted averaging aggregation operator|ordered weighted averaging]] (OWA) operator. Submodular fuzzy measures result in convex functions, while supermodular fuzzy measures result in concave functions when used to define a Choquet integral.
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| ==Möbius representation==
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| Let ''g'' be a fuzzy measure, the Möbius representation of ''g'' is given by the set function ''M'', where for every <math> E,F \subseteq X </math>,
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| :<math>M(E) = \sum_{F \subseteq E} (-1)^{|E \backslash F|} g(F).</math>
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| The equivalent axioms in Möbius representation are:
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| # <math> M(\emptyset)=0</math>.
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| # <math> \sum_{F \subseteq E|i \in F} M(F) \geq 0</math>, for all <math> E \subseteq \mathbf{X} </math> and all <math> i \in E </math>
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| A fuzzy measure in Möbius representation ''M'' is called ''normalized''
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| if <math>\sum_{E \subseteq \mathbf{X}}M(E)=1.</math>
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| Möbius representation can be used to give an indication of which subsets of '''X''' interact with one another. For instance, an additive fuzzy measure has Möbius values all equal to zero except for singletons. The fuzzy measure ''g'' in standard representation can be recovered from the Möbius form using the Zeta transform:
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| :<math> g(E) = \sum_{F \subseteq E} M(F), \forall E \subseteq \mathbf{X} .</math>
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| ==Simplification assumptions for fuzzy measures==
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| Fuzzy measures are defined on a [[semiring|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 as large as 2<sup>|'''X'''|</sup>. For this reason, in the context of [[multi-criteria decision analysis]] and other disciplines, simplification assumptions on the fuzzy measure have been introduced so that it is less computationally expensive to determine and use. For instance, when it is assumed the fuzzy measure is ''additive'', it will hold that <math> g(E) = \sum_{i \in E} g(\{i\}) </math> and the values of the fuzzy measure can be evaluated from the values on '''X'''. Similarly, a ''symmetric'' fuzzy measure is defined uniquely by |'''X'''| values. Two important fuzzy measures that can be used are the Sugeno- or <math>\lambda</math>-fuzzy measure and ''k''-additive measures, introduced by Sugeno<ref>{{cite journal
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| |author = M. Sugeno
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| |title = Theory of fuzzy integrals and its applications. Ph.D. thesis
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| |journal = Tokyo Institute of Technology, Tokyo, Japan
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| |year = 1974
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| }}</ref> and Grabisch<ref>{{cite journal
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| |author = M. Grabisch
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| |title = ''k''-order additive discrete fuzzy measures and their representation
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| |journal = Fuzzy Sets and Systems
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| |volume = 92
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| |pages = 167–189
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| |year = 1997
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| |issue = 2
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| |doi = 10.1016/S0165-0114(97)00168-1
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| }}</ref> respectively.
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| ===Sugeno ''λ''-measure===
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| The Sugeno <math>\lambda</math>-measure is a special case of fuzzy measures defined iteratively. It has the following definition:
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| ====Definition====
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| Let <math>\mathbf{X} = \left\lbrace x_1,\dots,x_n \right\rbrace </math> be a finite set and let <math>\lambda \in (-1,+\infty)</math>. A '''Sugeno <math>\lambda</math>-measure''' is a function <math>g:2^X\to[0,1]</math> such that
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| # <math>g(X) = 1</math>.
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| # if <math>A, B\subseteq \mathbf{X}</math> (alternatively <math>A, B\in 2^{\mathbf{X}}</math>) with <math>A \cap B = \emptyset</math> then <math>g(A \cup B) =g(A)+g(B)+\lambda g(A)g(B)</math>.
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| As a convention, the value of g at a singleton set <math>\left\lbrace x_i \right\rbrace </math>
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| is called a density and is denoted by <math>g_i = g(\left\lbrace x_i \right\rbrace)</math>. In addition, we have that <math>\lambda</math> satisfies the property
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| :<math> \lambda +1 = \prod_{i=1}^n (1+\lambda g_i) </math>. | |
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| Tahani and Keller <ref>{{cite journal
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| |author = H. Tahani and J. Keller
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| |title = Information Fusion in Computer Vision Using the Fuzzy Integral
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| |journal = IEEE Transactions on Systems, Man and Cybernetic
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| |volume = 20
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| |pages = 733–741
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| |year = 1990
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| |doi = 10.1109/21.57289
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| |issue = 3
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| }}</ref> as well as Wang and Klir have showed that once the densities are known, it is possible to use the previous [[polynomial]] to obtain the values of <math>\lambda</math> uniquely.
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| ===''k''-additive fuzzy measure===
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| The ''k''-additive fuzzy measure limits the interaction between the subsets <math> E \subseteq X </math> to size <math>|E|=k</math>. This drastically reduces the number of variables needed to define the fuzzy measure, and as ''k'' can be anything from 1 (in which case the fuzzy measure is additive) to '''X''', it allows for a compromise between modelling ability and simplicity.
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| ====Definition====
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| A discrete fuzzy measure ''g'' on a set '''X''' is called ''k-additive'' (<math> 1 \leq k \leq |\mathbf{X}|</math>) if its Möbius representation verifies <math>M(E) = 0 </math>, whenever <math> |E| > k </math> for any <math> E \subseteq \mathbf{X} </math>, and there exists a subset ''F'' with ''k'' elements such that <math> M(F) \neq 0 </math>.
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| ==Shapley and interaction indices==
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| In [[game]] theory, the [[Shapley value]] or Shapley index is used to indicate the weight of a game. Shapley values can 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 singleton.
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| For a given fuzzy measure ''g'', and <math>|\mathbf{X}|=n</math>, the Shapley index for every <math> i,\dots,n \in X </math> is:
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| :<math> \phi (i) = \sum_{E \subseteq \mathbf{X} \backslash \{i\}} \frac{(n-|E|-1)!|E|!}{n!} [g(E \cup \{i\}) - g(E)]. </math>
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| The Shapley value is the vector <math> \mathbf{\phi}(g) = (\psi(1),\dots,\psi(n)).</math>
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| ==See also==
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| *[[Probability theory]]
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| *[[Possibility theory]]
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| ==References==
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| {{reflist}}
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| * Beliakov, Pradera and Calvo, ''Aggregation Functions: A Guide for Practitioners'', Springer, New York 2007.
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| * Wang, Zhenyuan, and, [[George J. Klir]], ''Fuzzy Measure Theory'', Plenum Press, New York, 1991.
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| ==External links==
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| *[http://pami.uwaterloo.ca/tizhoosh/measure.htm Fuzzy Measure Theory at Fuzzy Image Processing]
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| [[Category:Exotic probabilities]]
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| [[Category:Measure theory]]
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| [[Category:Fuzzy logic]]
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The title of the author is Numbers. The preferred pastime for my kids and me is to play baseball but I haven't made a dime with it. Puerto Rico is where he's usually been living but she requirements to move because of her family members. For many years I've been working as a payroll clerk.
My weblog home std test kit