en>RomanSpa at 12:16, 24 February 2014

2014-02-24T12:16:40Z

en>Yobot: WP:CHECKWIKI error fixes using AWB (9751)

2013-12-02T10:32:01Z

WP:CHECKWIKI error fixes using AWB (9751)

← Older revision		Revision as of 12:32, 2 December 2013
Line 1:		Line 1:
	58 year-~~old Special Requires Teachers Bert Leif from Port Hope~~, ~~usually spends time~~ with ~~hobbies like bell ringing~~, ~~ganhando dinheiro na internet~~ and ~~eating~~. ~~Gets motivation through travel~~ and ~~just spent 9 days at~~ the ~~Birthplace~~ of the ~~Lord Buddha~~.<br><br>~~my homepage~~ [~~http~~://~~comoconseguirdinheiro~~.~~comoganhardinheiro101~~.~~com ganhar dinheiro~~]		The [[Ordered weighted averaging aggregation operator\|Yager's OWA (ordered weighted averaging) operators]]<ref name="yagerOWA">{{cite journal\|last=Yager\|first=R.R\|title=On ordered weighted averaging aggregation operators in multi-criteria decision making\|journal=IEEE Transactions on Systems, Man and Cybernetics\|year=1988\|volume=18\|pages=183–190}}</ref> have been widely used to aggregate the crisp values in decision making schemes (such as multi-criteria decision making, multi-expert decisin making, multi-criteria multi-expert decision making).<ref>{{cite book\|last=Yager\|first=R. R. and Kacprzyk, J\|title=The Ordered Weighted Averaging Operators: Theory and Applications\|year=1997\|publisher=Kluwer: Norwell, MA}}</ref><ref>{{cite book\|last=Yager\|first=R.R, Kacprzyk, J. and Beliakov, G\|title=Recent Developments in the Ordered Weighted Averaging Operators-Theory and Practice\|year=2011\|publisher=Springer}}</ref> It is widely accepted that fuzzy sets<ref>{{cite journal\|last=Zadeh\|first=L.A\|title=Fuzzy sets\|journal=Information and Control \|year=1965\|volume=8 \|pages=338–353\|doi=10.1016/S0019-9958(65)90241-X}}</ref> are more suitable for representing preferences of criteria in decision making. But fuzzy sets are not crisp values, how can we aggregate fuzzy sets in OWA mechanism?

			The type-1 OWA operators<ref name="fssT1OWA">{{cite journal\|last=Zhou\|first=S. M.\|coauthors=F. Chiclana, R. I. John and J. M. Garibaldi\|title=Type-1 OWA operators for aggregating uncertain information with uncertain weights induced by type-2 linguistic quantifiers\|journal=Fuzzy Sets and Systems\|year=2008\|volume=159\|issue=24\|pages=3281–3296\|doi=10.1016/j.fss.2008.06.018}}</ref><ref name="kdeT1OWA">{{cite journal\|last=Zhou\|first=S. M.\|coauthors=F. Chiclana, R. I. John and J. M. Garibaldi\|title=Alpha-level aggregation: a practical approach to type-1 OWA operation for aggregating uncertain information with applications to breast cancer treatments\|journal=IEEE Transactions on Knowledge and Data Engineering\|year=2011\|volume=23\|issue=10\|pages=1455–1468\|doi=10.1109/TKDE.2010.191}}</ref> have been proposed for this purpose. So the type-1 OWA operators provides us with a new technique for directly aggregating uncertain information with uncertain weights via OWA mechanism in soft decision making and data mining, where these uncertain objects are modelled by fuzzy sets.

			First, there are two definitions for type-1 OWA operators, one is based on Zadeh's Extension Principle, the other is based on <math>\alpha</math>-cuts of fuzzy sets. The two definitions lead to equivalent results.

			==Definitions==

			'''Definition 1.<ref name="fssT1OWA" /> '''
			Let <math>F(X)</math> be the set of fuzzy sets with domain of discourse <math>X</math>, a type-1 OWA operator is defined as follows:

			Given n linguistic weights <math>\left\{ {W^i} \right\}_{i = 1}^n </math> in the form of fuzzy sets defined on the domain of discourse <math>U = [0,1]</math>, a type-1 OWA operator is a mapping, <math>\Phi</math>,

			:<math>\Phi \colon F(X)\times \cdots \times F(X) \longrightarrow F(X)</math>
			:<math>(A^1 , \cdots ,A^n) \mapsto Y</math>

			such that

			:<math>\mu _{Y} (y) =\displaystyle \sup_{\displaystyle \sum_{k =1}^n \bar {w}_i a_{\sigma (i)} = y }\left({\begin{array}{*{1}l}\mu _{W^1 } (w_1 )\wedge \cdots \wedge \mu_{W^n } (w_n )\wedge \mu _{A^1 } (a_1 )\wedge \cdots \wedge \mu _{A^n } (a_n )\end{array}}\right)</math>

			where <math>\bar {w}_i = \frac{w_i }{\sum_{i = 1}^n {w_i } }</math>,and <math>\sigma \colon \{1, \cdots ,n\} \longrightarrow \{1, \cdots ,n\}</math> is a permutation function such that <math>a_{\sigma (i)} \geq a_{\sigma (i + 1)},\ \forall i = 1, \cdots ,n - 1</math>, i.e., <math>a_{\sigma(i)} </math> is the <math>i</math>th highest element in the set <math>\left\{ {a_1 , \cdots ,a_n } \right\}</math>.

			'''Definition 2.<ref name="kdeT1OWA" /> '''

			The definition below is based on the alpha-cuts of fuzzy sets:

			Given the n linguistic weights <math>\left\{ {W^i} \right\}_{i =1}^n </math> in the form of fuzzy sets defined on the domain of discourse <math>U = [0,\;\;1]</math>, then for each <math>\alpha \in [0,\;1]</math>, an <math>\alpha </math>-level type-1 OWA operator with <math>\alpha </math>-level sets <math>\left\{ {W_\alpha ^i } \right\}_{i = 1}^n </math> to aggregate the <math>\alpha </math>-cuts of fuzzy sets <math>\left\{ {A^i} \right\}_{i =1}^n </math> is given as

			: <math>
			\Phi_\alpha \left( {A_\alpha ^1 , \ldots ,A_\alpha ^n } \right) =\left\{ {\frac{\sum\limits_{i = 1}^n {w_i a_{\sigma (i)} } }{\sum\limits_{i = 1}^n {w_i } }\left\| {w_i \in W_\alpha ^i ,\;a_i } \right. \in A_\alpha ^i ,\;i = 1, \ldots ,n} \right\}</math>

			where <math>W_\alpha ^i= \{w\| \mu_{W_i }(w) \geq \alpha \}, A_\alpha ^i=\{ x\| \mu _{A_i }(x)\geq \alpha \}</math>, and <math>\sigma :\{\;1, \cdots ,n\;\} \to \{\;1, \cdots ,n\;\}</math> is a permutation function such that <math>a_{\sigma (i)} \ge a_{\sigma (i + 1)} ,\;\forall \;i = 1, \cdots ,n - 1</math>, i.e., <math>a_{\sigma (i)} </math> is the <math>i</math>th largest
			element in the set <math>\left\{ {a_1 , \cdots ,a_n } \right\}</math>.

			== Representation theorem of Type-1 OWA operators<ref name="kdeT1OWA" />==

			Given the ''n'' linguistic weights <math>\left\{ {W^i} \right\}_{i =1}^n </math> in the form of fuzzy sets defined on the domain of discourse <math>U = [0,\;\;1]</math>, and the fuzzy sets <math>A^1, \cdots ,A^n</math>, then we have that<ref name="kdeT1OWA" />
			:<math>Y=G</math>

			where <math>Y</math> is the aggregation result obtained by Definition 1, and <math>G</math> is the result obtained by in Definition 2.

			==Programming problems for Type-1 OWA operators==

			According to the '''''Representation Theorem of Type-1 OWA Operators''''',a general type-1 OWA operator can be decomposed into a series of <math>\alpha</math>-level type-1 OWA operators. In practice, these series of <math>\alpha</math>-level type-1 OWA operators are used to construct the resulting aggregation fuzzy set. So we only need to compute the left end-points and right end-points of the intervals <math>\Phi _\alpha \left( {A_\alpha ^1 , \cdots ,A_\alpha ^n } \right)</math>. Then, the resulting aggregation fuzzy set is constructed with the membership function as follows:

			:<math>\mu _{G} (x) = \mathop \vee \limits_{\alpha :x \in \Phi _\alpha \left( {A_\alpha ^1 , \cdots
			,A_\alpha ^n } \right)_\alpha } \alpha </math>

			For the left end-points, we need to solve the following programming problem:
			:<math> \Phi _\alpha \left( {A_\alpha ^1 , \cdots ,A_\alpha ^n } \right)_{-} = \mathop {\min }\limits_{\begin{array}{l} W_{\alpha - }^i \le w_i \le W_{\alpha + }^i A_{\alpha - }^i \le a_i \le A_{\alpha + }^i \end{array}} \sum\limits_{i = 1}^n {w_i a_{\sigma (i)} / \sum\limits_{i = 1}^n {w_i } } </math>

			while for the right end-points, we need to solve the following programming problem:
			:<math>\Phi _\alpha \left( {A_\alpha ^1 , \cdots , A_\alpha ^n } \right)_{+} = \mathop {\max }\limits_{\begin{array}{l} W_{\alpha - }^i \le w_i \le W_{\alpha + }^i A_{\alpha - }^i \le a_i \le A_{\alpha + }^i \end{array}} \sum\limits_{i = 1}^n {w_i a_{\sigma (i)} / \sum\limits_{i =
			1}^n {w_i } } </math>

			A fast method has been presented to solve two programming problem so that the type-1 OWA aggregation operation can be performed efficiently, for details, please see the paper.<ref name="kdeT1OWA" />

			== Alpha-level approach to Type-1 OWA operation<ref name="kdeT1OWA" />==
			* '''Step 1'''.To set up the <math>\alpha </math>- level resolution in [0, 1].
			* '''Step 2'''. For each <math>\alpha \in [0,1]</math>,
			''Step 2.1.'' To calculate <math>\rho _{\alpha +} ^{i_0^\ast } </math>
			# Let <math>i_0 = 1</math>;
			# If <math>\rho _{\alpha +} ^{i_0 } \ge A_{\alpha + }^{\sigma (i_0 )} </math>, stop, <math>\rho _{\alpha +} ^{i_0 } </math> is the solution; otherwise go to ''Step 2.1-3''.
			# <math>i_0 \leftarrow i_0 + 1</math>, go to ''Step 2.1-2''.

			''Step 2.2.'' To calculate<math>\rho _{\alpha -} ^{i_0^\ast } </math>
			# Let <math>i_0 = 1</math>;
			# If <math>\rho _{\alpha -} ^{i_0 } \ge A_{\alpha - }^{\sigma (i_0 )} </math>, stop, <math>\rho _{\alpha -} ^{i_0 } </math> is the solution; otherwise go to ''Step 2.2-3.''
			#<math>i_0 \leftarrow i_0 + 1</math>, go to step ''Step 2.2-2.''

			'''Step 3.'''To construct the aggregation resulting fuzzy set <math>G</math> based on all the available intervals <math>\left[ {\rho _{\alpha -} ^{i_0^\ast } ,\;\rho _{\alpha +} ^{i_0^\ast } } \right]</math>:

			:<math>\mu _{G} (x) = \mathop \vee \limits_{\alpha :x \in \left[ {\rho _{\alpha -} ^{i_0^\ast } ,\;\rho _{\alpha +} ^{i_0^\ast } } \right]} \alpha </math>

			==Special cases of Type-1 OWA operators==
			* Any OWA operators, like maximum, minimum, mean operators;<ref name="yagerOWA" />
			* Join operators of (type-1) fuzzy sets,<ref name="MT">{{cite journal\|last=Mizumoto\|first=M.\|coauthors=K. Tanaka\|title=Some Properties of fuzzy sets of type 2\|journal=Information and Control\|year=1976\|volume=31\|pages=312–40}}</ref><ref name="zadehJ">{{cite journal\|last=Zadeh\|first=L. A.\|title=The concept of a linguistic variable and its application to approximate reasoning-1\|journal=Information Sciences\|year=1975\|volume=8\|pages=199–249}}</ref> i.e., fuzzy maximum operators;
			* Meet operators of (type-1) fuzzy sets,<ref name="MT"/><ref name="zadehJ"/> i.e., fuzzy minimum operators;
			* Join-like operators of (type-1) fuzzy sets;<ref name="kdeT1OWA"/><ref name="bookT1OWA">{{cite journal\|last=Zhou\|first=S. M.\|coauthors=F. Chiclana, R. I. John, and J. M. Garibaldi\|title=Fuzzificcation of the OWA Operators in Aggregating Uncertain Information\|journal=R. R. Yager, J. Kacprzyk and G. Beliakov (ed): Recent Developments in the Ordered Weighted Averaging Operators-Theory and Practice\|year=2011\|volume=Springer\|pages=91–109\|doi=10.1007/978-3-642-17910-5_5}}</ref>
			* Meet-like operators of (type-1) fuzzy sets.<ref name="kdeT1OWA"/><ref name="bookT1OWA"/>

			==Generalizations==
			Type-2 OWA operators<ref>{{cite journal\|last=Zhou\|first=S.M.\|coauthors=R. I. John, F. Chiclana and J. M. Garibaldi\|title=On aggregating uncertain information by type-2 OWA operators for soft decision making\|journal=International Journal of Intelligent Systems\|year=2010\|volume=25\|issue=6\|pages=540–558\|doi=10.1002/int.20420}}</ref> have been suggested to aggregate the [[Type-2 fuzzy sets and systems\|type-2 fuzzy sets]] for soft decision making.

			== References ==
			{{reflist}}

			[[Category:Artificial intelligence]]
			[[Category:Logic in computer science]]
			[[Category:Fuzzy logic]]
			[[Category:Information retrieval]]

en>Crowsnest: corect wikilink

2011-11-11T13:09:16Z

corect wikilink

New page

58 year-old Special Requires Teachers Bert Leif from Port Hope, usually spends time with hobbies like bell ringing, ganhando dinheiro na internet and eating. Gets motivation through travel and just spent 9 days at the Birthplace of the Lord Buddha.<br><br>my homepage [http://comoconseguirdinheiro.comoganhardinheiro101.com ganhar dinheiro]

Conjugate depth - Revision history

en>RomanSpa at 12:16, 24 February 2014

en>Yobot: WP:CHECKWIKI error fixes using AWB (9751)

en>Crowsnest: corect wikilink