Two Fuzzy Probability Measures

Transcription

1 Two Fuzzy robablty Measures Zdeěk Karíšek Isttute of Mathematcs Faculty of Mechacal Egeerg Bro Uversty of Techology Techcká Bro Czech Reublc e-mal: karsek@umfmevutbrcz Karel Slavíček System dmstrato Deartmet Isttute of Comuter Scece Masaryk Uversty Bro Botacká 68a 62 Bro Czech Reublc e-mal: karel@csmucz bstract 2 Zadeh-tye fuzzy robablty The aer deals wth two methods of a fuzzfcato of the Borel feld of evets ad too the robablty measure The frst aroach geeralzes the Zadeh defto of a crs robablty of fuzzy evet The secod method s based o the Yager defto of a fuzzy robablty of fuzzy evet The theoretcal results obtaed ca be aled to modelg stochastc heomea wth ucerta character Keywords: fuzzy evet fuzzy σ-algebra fuzzy robablty measure deedet fuzzy evets MS classfcato: E5 6B99 Itroducto The basc oto the theory of robablty s a radom evet (a subset of the basc sace) whch may or may ot occur deedg o the mlemetato of a radom exermet We assume that for a artcular mlemetato we ca decde whether ths evet has or has ot occurred However ractce ths requremet may ot be comled wth a smle way Such evets may be sutably terreted by fuzzy sets [] O the other had the exresso of a robablty value tself may be of a vague ature These accurate values may also be descrbed by a fuzzy sets ad fuzzy umbers [8] The followg geeral symbols are emloyed as eeded throughout the text: : the set of all real umbers; : the set of all atural umbers; : the exteded arthmetc oeratos wth fuzzy real umbers The frst fuzzfcato s based o Zadeh s defto [] of the robablty of fuzzy evet : ( ) = d m R where s a robablty measure ad = ( m ) s a fuzzy set I ths secto ths defto s geeralzed by meas of Zadeh s exteso rcle [2] Defto 2 Let Ω be a uversal set (basc sace) fuzzy radom evet s the fuzzy set = = Ω wth the membersh fucto : Ω ( ) [ ;] Ω s the certa evet ( ) the mossble evet ( ) evet wth a Borel measurable membersh fucto ad s fuzzy radom s called a fuzzy evet Defto 22 oemty set Σ of fuzzy evets = ( Ω ) s called a fuzzy Borel feld of fuzzy evets over the uversal set Ω f Σ has the followg roertes [4]: ( ω) = α ω Ω α [ ; ] Σ 2 Σ Σ 3 2 Σ Σ = 4 2 Σ 2 Σ m Defto 23 Let Ω = where m be a uversal set Σ a crs Borel feld of evets over Ω Π a oemty set of robablty measures o ( Ω Σ ) Σ a fuzzy Borel feld of fuzzy evets o

2 = Ω Σ a fuzzy evet Let further Ω ad ( ) = ( Π ) be such a fuzzy buch o Π that Π ad ( ) = The the fuzzy buch s called a fuzzy robablty measure o Ω ad the fuzzy robablty of a fuzzy evet s the fuzzy set ( ) = ([ ; ] ( ) ) where ) = su ( ) = = {} = = Defto 32 The system of fuzzy sets M F ( U ) {} are de- Defto 24 Fuzzy evets B Σ edet f for Π B d = d d B Ω Fuzzy evets Σ = are mutually deedet f for { } { } Π d = d j j a) = ( ) = = b) = {} {} ( ) = = 3 Yager-tye fuzzy robablty The secod fuzzfcato s based o Yager s defto [] of the fuzzy robablty of fuzzy evet : ( )( Π d= ( ) = α { ( Ω α )} α [ ;] for [ ;] If o measure Π exsts such that where s a robablty measure ad d α s the = we ut ( )( ) = The trlet α cut of fuzzy set I ths secto ths defto Ω ( m Σ s geeralzed ad the ecessary fuzzy structures are m ) s called a fuzzy robablty sace o descrbed [9] Let U be a uversal set F ( U ) the Theorem 2 For ay fuzzy evet Σ we have: system of all fuzzy sets o U a) ( ) = { } ( Ω ) = { } Defto 3 The system of fuzzy sets M b) ( ) = { } ( ) for Σ F ( U ) s called a fuzzy set rg f for M c) ( ) ( ) = ( )( ) for [ ;] M has the followg roertes: j M ( ) fuzzy umber ) d) ( fu zzy umber 2 j M Theorem 22 For ay set of fuzzy evets Σ 3 M = we have: a) b) = = k Ω Ω Ω j= j= Ω k k s called a fuzzy set σ algebra f for M M has the followg roertes: M j 2 U M 3 M Theorem 23 If fuzzy evets B Σ are deedet the the fuzzy evets B; B; B are x y x2 we have y α covex or quas- x x2 x x2 the for y su ad also deedet covex fuzzy set s called a seudo-covex fuzzy Theorem 24 For ay set of mutually deedet set fuzzy evets Σ = we have: The reaso for troducg the term quas-covex s the demad for roer defto of covexty o a Defto 33 Let a fuzzy set F ( U) have the fte suort su Let U be a comlete ordered set wth as the orderg relato The fuzzy set s called a quas-covex f α cut α has for α (;] the followg roerty: f x x 2 α

3 dscrete uverse We eed ths for radom varables wth dscrete dstrbuto laws Defto 34 The ormal ad seudo-covex fuzzy set a = ( a ) s called a geeralzed fuzzy umber The set of all geeralzed fuzzy umbers we deote by Defto 35 Let M ( U ) F be a fuzzy set σ rg o U fuzzy set fucto λ : M s called a fuzzy measure o M f λ has the followg roertes: suλ ( ) + for M 2 For M where λ = λ( ) 3 λ ( ) = { } Let M be a fuzzy set σ algebra o U fte fuzzy measure such that U ( ) = { } s called a fuzzy robablty measure s fte The set α( M ) = α( ) M 2 If M ad B are fuzzy sets such that for α α( ) α > we have α = B α + the B M 3 If M ad B are fuzzy sets such that for α α( ) α > we have α = B + α the B M Theorem 32 Let M be a comlete fuzzy set σ algebra o U The the set M = { U M α α( M ) ; = α } s a set σ algebra Theorem 33 Let M be the fuzzy set σ algebra Theorem 3 Let M be a set σ algebra o U { } ( M ) = = < < < ( ) α α α α α ; a fte set of real umbers wth the followg roerty: Let be a sequece α α( M ) ( α) α( M ) of sets = () M () ( ; ) { } { } The the famly of fuzzy sets M = { α = (); () } geerated by the system of sequeces s a fuzzy set σ algebra We say that the fuzzy set σ algebra M s geerated by the set σ algebra M Now we ca ask f ad whe we are able to geerate a crs set σ algebra from a fuzzy set σ algebra Oe terestg class of fuzzy set σ algebras that are able to geerate a crs σ algebra s gve by the followg defto geerated by a set σ algebra M robablty measure o M The fuzzy set fucto : M where ( ) ( ) = su { α α [ ;] } ( α ) = s a fuzzy robablty measure We say that s geerated by Defto 38 Let ( UM ) be a robablty sace M the fuzzy set σ algebra geerated by the set σ algebra M the fuzzy robablty measure o M geerated by the robablty measure The UM s a fuzzy robablty sace ad trlet ( ) ( ) UM s geerated by the robablty sace ( ) UM fuzzy set M s called a fuzzy radom evet fuzzy umber ( ) s called a fuzzy robablty of the fuzzy radom evet Defto 39 Let be a fuzzy set o a uversal set U [ ] ( x ) = :; U the membersh fucto fucto (f t exsts) where ( α) = x ff α s called a quas-verse membersh Defto 36 Let F ( U) be a fuzzy set s fucto of the fuzzy set called a ste fuzzy set f the set Theorem 34 For B M α ( ) = { α [ ; ] x U; ( x) = α } B α [ ;] we have ( )( α ) ( B)( α) s fte The set α ( ) s called a set of membersh degrees of the fuzzy set Theorem 35 For M α [;] we have: Defto 37 famly of ste fuzzy sets M s a) ( ) ( α) = ( α ) called comlete f has the followg roertes:

4 b) ( ) = suα ( ) ( ) ( ) = α ω j [ ] Defto 3 Fuzzy radom evets B M are deedet f radom evets B are deedet for αβ ; Fuzzy radom evets M = are mutually deedet f radom evets α are mutually deedet for α [ ;] = Theorem 36 If fuzzy radom evets B M where M s a comlete fuzzy set σ algebra are deedet the the fuzzy radom evets B ; B ; B are also deedet Theorem 37 For ay two deedet fuzzy radom evets B M ad for ay set of mutually deedet fuzzy radom evets M = we have: a) ( B) = ( ) ( B) b) = ( ) = = α β k The calculated fuzzy robabltes of fuzzy evets are: ( ) ( ) ( ) ( ) lot of the membersh fucto of fuzzy robablty ( ) s show Fg Examles Examle 4 (Zadeh-tye fuzzy robablty) We have a fuzzy robablty (fuzzy buch) o the uversal set Π= wth the membersh fucto: { } ( ) where robablty measures for = 234 o the { 23 45} basc sace Ω= are gve: ω j Let fuzzy evets k for k = 23 have the member- sh fuctos: Fg 4 lot of the membersh fucto of fuzzy robablty ( ) s show Fg Fg 42

5 lot of the membersh fucto of fuzzy robablty ( ) s show Fg Fg 43 Examle 42 (Yager-tye fuzzy robablty) Suose a far co s fled te tmes Let be the radom evet "head faces u -tmes" = The radom evet has the robablty ( ) 2 = from the bomal dstrbuto B(;/2): (2 - ) 45(2 - ) 2(2 - ) (2 - ) 252(2 - ) 2(2 - ) 2(2 - ) (2 - ) (2 - ) 2 - B C D B C D The calculated fuzzy robabltes ( B) C ( ) ad ( D) have the membersh fuctos: (2 - ) 56(2 - ) 76(2 - ) 386(2 - ) B ( ) (2 - ) 76(2 - ) 386(2 - ) 638(2 - ) C ( ) (2 - ) 56(2 - ) 76(2 - ) D ( ) 8 3 The membersh fuctos of the fuzzy robabltes ( B) C ( ) ad ( D) are lotted Fg 44 Let fuzzy radom evets: B : head faces u seldom C : head faces u sometmes D : head faces u may tmes have the membersh fuctos o the uversal set U = : { } B C D Fg 44 B C D

6 5 Cocluso We have reseted two dfferet fuzzy robablty models s comared to the secod model the frst oe ca also deal wth ossble ucertaty recarous formato about the observed robablty dstrbuto The frst fuzzfcato s based o the exected value of the membersh fucto of a fuzzy evet wth resect to the fuzzy buch of robablty measures [2] The secod fuzzfcato o the cotrary assumes oly oe robablty measure ad ts cocet s earer the classcal theory of robablty The frst model s relatvely flexble ad has already bee mlemeted o a C to calculate relablty [3 4] The secod model s tesvely examed [9] ad we exect ts major alcato to relablty calculato as well [8] G J Klr B Yua (995) Fuzzy Sets ad Fuzzy Logc st ed retce Hall New Jersey 995 ISBN [9] K Slavíček (22) Fuzzy ravděodobostí míra (Fuzzy robablty Measure) hd Thess FME BUT Bro [] R R Yager (979) Note o robabltes o Fuzzy Evets I Iformato Scece (8) Elsever [] L Zadeh (968) robablty Measures ad Fuzzy Evets J of Math alyss ad lcatos 23(2) ckowledgemets The aer was suorted by research desg CEZ: J22/98: 269 No-tradtoal methods for vestgatg comlex ad vague systems Refereces [] U Höhle (976) Maße auf uscharfe Mege Z Wahrsch Verw Geb [2] Z Karíšek (2) Fuzzy robablty ad ts roertes I Medel 2 6 th Iteratoal Coferece o Soft Comutg Bro ISBN [3] Z Karíšek (2) Fuzzy robablty Dstrbuto Characterstcs ad Models I roceedgs East West Fuzzy Colloquum 2 9 th Zttau Fuzzy Colloquum Zttau ISBN [4] Z Karíšek (22) The Fuzzy Relablty wth Webull Fuzzy Dstrbuto I roceedgs East West Fuzzy Colloquum 22 th Zttau Fuzzy Colloquum Zttau ISBN [5] E Klemet (98) Fuzzy σ-algebras ad Fuzzy Measurable Fuctos Fuzzy Sets ad Systems 4 (98) [6] E Klemet (982a) Theory of Fuzzy Measures: Survey I: Guta M M / Sachez E Fuzzy formato ad decso rocesses msterdam / New York: North- Hollad [7] E Klemet (982b) Some Remarks o a aer by R R Yager Iformato Sceces