Fuzzy Probability Approximation Space and Its Information Measures

Transcription

1 Fuzzy Pobablty Appomato Space ad Its Ifomato Measues Qghua Hu, Dae Yu Hab Isttute of Techology, Cha Abstact ough set theoy has attacted much atteto modelg wth mpecse ad complete fomato A geealzed appomato space, called fuzzy pobablty appomato space has bee poposed by toducg pobablty to fuzzy appomato space The ovel defto combes thee types of ucetaty to a model Ifomato o kowledge s cosdeed as a patto of the uvese ough set famewok We toduce Yage s etopy to measue kowledge quatty mpled fuzzy pobablty appomato space It s show that the fomato measue fo fuzzy pobablty appomato space s a atoal eteso of the Shao s oe ad t degades to Shao s etopy case whee attbutes ae omal ad objects ae equalty-pobable The a ufom fomato measue fo Pawlak s ough set model, fuzzy ough set model ad fuzzy pobablty ough set model s fomed based oe Yage s etopy Itoducto ough set methodology has bee wtessed geat success modelg mpecse ad complete fomato ough set methodology pesets a ovel paadgm to deal wth ucetaty ad has bee appled to featue selecto [, 2], kowledge educto [3], ule etacto [4, 5, 6], ucetaty easog [7, 8] ad gaulaty computg [9, 0, 39]The Pawlak s ough set model does t cosde ucetaty duced by fuzzess ad pobablty applcatos Some geealzatos of Pawlak s model wee poposed whee fuzzy sets ad fuzzy elatos est ough set theoy ad fuzzy set theoy wee put togethe, ough-fuzzy sets ad fuzzy-ough sets wee defed [, 2] The popetes ad aomatzato of fuzzy ough set theoy [3-7] wee aalyzed detal Ad the geealzed methods wee appled to mg stock pce [8], vocabulay fo fomato eteval [9] ad fuzzy decso ules [20, 2] The omal ough set models, both Pawlak s ough set model ad fuzzy ough set model, mplctly take a assumpto that the objects ae equalty-pobable Howeve, pactce t s ot ecessay that the objects ae ufomly dstbuted A pobablty dstbuto may be defed ove U A theoy o pobablty appomato space o a pobablty ough set model s desable ths case Gve a uvese U, a pobablty dstbuto o U, ad some omal, eal-valued o fuzzy attbutes, t s teestg costuctg a measue to compute the dsceblty powe of a famly of attbutes o equvalece elatos, whch ca lead to lkelhood to compae the kowledge quatty geeated by dffeet attbutes o elatos It wll help us fd the mpotat attbute set ad edudacy of fomato system Shao [22] defed a fomato measue of a adom vaable wth the fame of commucato theoy Fote ad Kampe [23, 24] gave the aomatc fomato measue, whee the wod fomato was assocated both to measues of evets ad measues of pattos ad suggested that the ucetaty measue s assocated to a famly of pattos of a gve efeetal space I [26, 27]a measue, sutable to opeate o

2 domas ove whch fuzzy equvalece elatos have bee defed, was toduced, whee the sematcs of fuzzy evets was take to accout Ucetaty measue o fuzzy pattos geeated by fuzzy equvalece elatos was aalyzed documets [28, 29] I ough set famewok, attbutes ae called kowledge whch s used to classfy the elemets to dsceble clustes Kowledge toduced by a attbute set mples the pattos of a efeetal uvese Moe kowledge wll lead to a fe patto, ad the we ca get a moe pefect appomato of a subset uvese Theefoe kowledge deceases ucetaty chaactezg the cocepts Dmshmet of ucetaty ca be cosdeed as a cease of kowledge I ths pape we wll ufy the epesetato ad use the tem kowledge, stead of ucetaty Fst we use Shao s etopy to compute the kowledge quatty toduced by omal attbutes o csp equvalece elatos, the a eteso fomato measue wll be peseted, whch s sutable fo the case whee fuzzy attbutes o fuzzy elatos ae defed ased o the eteso, the poblem of measug the fomato fuzzy appomato spaces s solved The est of the pape s ogazed as follows: we wll evew some deftos about fuzzy ough set model ad gve fuzzy pobablty ough set model secto 2 Secto 3 toduced a eteded fomato measue fo fuzzy equvalece elato ad fuzzy patto The we apply the poposed fomato measues to fuzzy pobablty appomato space secto 4 The cocluso s gve secto 5 2 Fuzzy pobablty appomato space I ths secto we wll tegate thee types of ucetaty pobablty, fuzzess ad oughess togethe, ad peset the defto of fuzzy pobablty appomato space Defto Gve a o-empty fte set, s a elato defed o, deoted by a elato mat M () : 2 M ( ) = whee j [0, ] s the elato value of ad s a fuzzy equvalece elato, f satsfes ) eflectvty: ( ) =, U 2) Symmety: ( = ( y, ), y U j y, z, 3) Tastvty: ( Z) m{ (, ( y, z)} Gve abtay set, s a fuzzy equvalece elato defed o y, some opeatos o elato matces ae defed as ) = = (, y 2 ( 2 2 ( ma{ (, 2 ( y 2 ( m{ (, 2 ( y 2 ( 2 ( y 2) = = )} 3) = = )} 4) ) y A csp equvalece elato wll geeate a csp patto ad a fuzzy equvalece elato geeates a fuzzy patto Defto 2 The fuzzy equvalece classes geeated by a fuzzy equvalece elato s defed as whee =, U / {[ ] } = 2 [ ] = ample Gve a object set = {, 2, 3}, s fuzzy equvalece elato o as follows: 09 0 = The the fuzzy equvalece classes ae 09 0 [ ] = + + 2, [ 2 ] = + + 2, 3

3 0 0 [ 3 ] = Theoem Gve abtay set, s a fuzzy equvalece elato defed o The fuzzy quotet set of by elato s deoted by y, we have ) ( =0 [ ] [ y] = 0 2) [ ] = 3) [ ] = [ y] ( = Defto 3 A thee-tuple < U, P, > s a fuzzy pobablty appomato space (shotly, FPAS) o a fuzzy pobablty fomato system (FPIS), whee U s a oempty ad fte set, called the uvese, P s the pobablty dstbute ove U, s a famly of fuzzy equvalece elatos defed o U Defto 4 Gve a fuzzy pobablty appomato space < U, P, >, s a fuzzy subset of U The lowe appomato ad uppe appomato, deoted by ad, ae defed as ( ) = { ( ( ( ) : y U}, U ( ) = {( ( ( : y U}, U These deftos ae the atoal eteso of some models Let s deve the othe model fom these deftos Case s a csp subset ad s a csp equvalece elato o U: ( ) = y U, y U : y y [ ] [ ] ( ) = y U : [ ] ( ( ( ) = y ( φ ( =, ( = I ths case these deftos ae cosstet wth Pawlak, ough set model Case 2 s a fuzzy subset of U ad s a csp equvalece elato o U: ( ) = { = { = { ( ) = { = { = { ( ( ( ) : y U} ( : ( = } ( : y [ ] } ( ( : y U} ( : ( = } ( : y [ ] I ths case, the ough sets ae called ough fuzzy sets Case 3 s a subset of U ad s a fuzzy equvalece elato o U: ( ) = m{ = m{- ( } y ( ) = ma{ = ma ( y } ( ( ( ) : y U} ( ( ( ) : y U} Fom the above aalyss we ca coclude that the deftos of lowe ad uppe appomatos of fuzzy set fuzzy fomato system ae the atoal geealzatos of classc model Fuzzy pobablty fomato system (FPIS) s the geeal case of the othe ough set model FPIS wll degeeate to the omal fuzzy fomato system f pobablty dstbuto s ufom ad fuzzy fomato system wll degeeates to Pawlak s ough set model f equvalece elato s csp ad s the csp subset of U The membeshp of a object the fuzzy postve ego s defed as U POS sup = ( ) ( d ) U / d, belogg to Defto 5 Gve a fuzzy pobablty fomato system <U, P, A>, ad d ae two subset of attbute set A, the depedecy degee of d to s defed as γ ( ) ( d) = ) POS ( d ) U The dffeece betwee fuzzy appomato space ad fuzzy pobablty appomato space s toducg pobablty dstbute ove U Ths leads to a moe geeal geealzato of Pawlak s ough set model I classc ough set model take the equalty-pobablty assumpto

4 So ) = /, =, 2,, The γ ( d) = U = = U ) U POS ( d) POS ( d ) U POS ( d ) ( ) ( ) ( ) Ths fomula s the same as that fuzzy ough set model [30], whch shows that the fuzzy pobablty appomato space wll degade to a fuzzy appomato space whe the equalty-pobablty assumpto s take Defto 6 Gve a fuzzy fomato system <U, A, V, f>, A, a, f U / = U /( a), we say kowledge a s edudat o supefluous othewse, we say kowledge a s dspesable If ay a belogg to s dspesable,we say s depedet If attbute subset A s depedet ad U/ = U/A, we say s a educt of A Defto 7 Gve a fuzzy fomato system <U, A, V, f>, A = C d s a subset of C a, a s edudat elatve to d f γ ( d) = γ ( d) othewse a s dspesable s depedet f a, a s dspesable, othewse s depedet s a subset of C s a educt of C f satsfes: ) γ d) = γ ( d) ( C 2) a : γ ( d) < γ ( d) -a Compag the fuzzy pobablty appomato space wth fuzzy appomato space we fd that the cetal dffeece s the fucto of depedecy I fuzzy appomato space, we assume the objects ae ufomly dstbuted ad ) = / U I the fuzzy pobablty appomato space the pobablty of s p ) Whe the pobablty ) = / U, the fuzzy ( pobablty appomato space degades to a fuzzy appomato space, ad f the equvalece elato ad the object subset to be appomated ae both csp, we get a Pawlak s appomato space I applcatos the pobablty ca be cosdeed as a weght of the object Pobablty s oly oe of the weghtg methods Weghtg gves us a ovel dmeso to ject fomato out of data to pocessg, whch ca tegate the po fomato wth data 3 Ifomato o fuzzy equvalece elatos Shao s fomato measue just woks the case whee a csp equvalece elato o a csp patto s defed, whch s sutable fo Pawlak s ough set model I ths secto we wll gve a ovel fomula to compute Shao s etopy fo csp elato mat epesetato, ad the a geealzato of the etopy s poposed fo fuzzy elato matces Futhemoe, we wll peset aothe geealzato fo pobablty fuzzy fomato systems ad use the poposed etopes to measue the fomato fuzzy pobablty appomato spaces 3 Shao s etopy measues elato mat fom fo csp equvalece elatos Gve a fomato system <U, A, V, f>, Abtay elato U U {0, } ca be deoted by a elato mat M(): whee 2 M ( ) = j s the elato value betwee elemet ad j If s a equvalece elato we say M() s a equvalece elato mat A equvalece elato mat satsfes: ) eflectvty: ( ) =, U 2) Symmety: ( = ( y, ), y U 3) Tastvty: ( =, ( y, z) = ( z) Gve a abtay set,, y, some opeatos o elato mat ae defed as ) = = (, y 2 ( 2

5 2) = = ma{ (, ( )} 2 ( 2 y 2 ( m{ (, 2 ( y 2 ( 2 ( y 3) = = )} 4) ) Thee ae some popetes betwee csp attbute set ad elatos duced by the coespodg attbutes: ) A = A = 2) A A 3) C = A C = A The equvalece class cotaed elato s deoted by [ ] = wth espect to whee j = 0 o The cadalty of [ ] s defed as [ ] = j Defto 8 Gve a fomato system <U, A, V, f>, abtay equvalece elato o U, deoted by a elato mat M(), the we defe the fomato measue fo elato as whee [ ] = H ( ) = log, = Theoem 2 Gve a fomato system <U, A, V, f>, A, s a equvalece elato geeated by attbutes o U H () s computed as Shao s oe ad H ) s computed as defto 8 The ( H ) ) ( Poof Staghtfowad ample 3 Assumed thee ae a fomato system wth thee objects, A equvalece elato mat defed o the uvese s 0 M () = The equvalece classes ae, } ad } The the fomato quatty s { 2 { H ( ) = log log log = log log The computato s the same as Shao oe ths case Theoem 3 Gve a fomato system <U, A, V, f>,, A,, s two equvalece elato geeated by attbutes ad [ ] ad [ ] s the equvalece classes duced by ad The jot etopy of ad s H ( ) = H ( ) = = log [ ] [ ] Theoem 4 Gve a fomato system <U, A, V, f>,, A,, s two equvalece elato geeated by attbutes ad [ ] ad [ ] s the equvalece classes duced by ad The codtoal etopy codtoed to H ( ) s H( ) ) = [ ] log = [ ] [ ] Hee the ovel computatoal fomulae of Shao s fomato wll bg geat advatage to geealze them to fuzzy cases 32 Ifomato measue o fuzzy equvalece elatos As we kow, fuzzess ests may eal-wold applcatos Pawlak s ough set model just woks the csp case D Dubos etc geealzed the model to the fuzzy case I ths secto we wll peset a geealzato of Shao s etopy The ovel measue has a same fom as Shao s oe ad ca wok the case whee a fuzzy equvalece elato s defed Gve a fte set U, A s a fuzzy o eal-valued

6 attbute set, whch geeates a fuzzy equvalece elato o U The fuzzy elato mat M ) s deoted A by M ( A ) 2 = ( A 2 whee j [0, ] s the elato value of ad j Defto 9 The fuzzy quotet set geeated by the fuzzy equvalece elato s defed as whee U / = {[ ] } = 2 [ ] = Defto 0 The cadalty [ ] of [ ] s + s the o-egatve eal-umbe set Ths map bulds a foudato o that we ca compae the dsceblty powe, patto powe o appomatg powe of multple fuzzy equvalece elatos topy value ceases mootoously wth the dsceblty powe o the kowledge s feess So the fe patto s, the geate etopy s, ad the moe sgfcat attbute set s Defto 2 Gve a fuzzy fomato system <U, A V, f>, A s the fuzzy attbute set, subsets of A classes cotag The jot etopy of H ( ) ] ad [ ] ae two [ ae fuzzy equvalece geeated by ad s defed as ) = = log,, espectvely [ ] [ ] defed as [ ] = j Defto 3 Gve a fuzzy fomato system <U, A V, f>, A s the fuzzy attbute set, ae two As show eample, the [ ] = = 9 [ 09 0 = ] 3 Defto Ifomato quatty of the fuzzy attbute set o the fuzzy equvalece elato s defed as whee H( A ) A) = log, [ ] = = Ths measue has the same fom as the Shao s oe defed as defto 8 ut t has bee geealzed to the fuzzy case The fomula of fomato measue foms a map: + H :, whee s a equvalece elato mat subsets of A classes cotag [ ae fuzzy equvalece ] ad [ ] geeated by The codtoal etopy of defed as H ( ) = = log [,, espectvely codtoed to ] [ ] Theoem 5 H( ) ) H( ) [ ] Theoem 6 ) H ( A ) 0, = holds f ad oly f j =,, j 2) H( A ) ma{ H( A ), H( )} 3) A H( A ) A ) 4) H( ) = 0 A A s

7 Poof Staghtfowad 33 Ifomato measues o fuzzy pobablty equvalece elato Shao s etopy ad the poposed measue wok o the assumpto that all the objects ae equalty-pobable I ths secto we wll gve a geealzato whee a pobablty dstbuto s defed o U Gve a fuzzy pobablty fomato system <U, A V, f, P>, A s the fuzzy attbute set, whch geeates a famly of fuzzy equvalece elatos o U, P s the pobablty dstbuto ove U, p ( ) s the pobablty of object A abtay fuzzy equvalece elato U U geeated by attbutes s deoted by a elato mat M ) : ( M ( ) = 2 whee j [0, ] s the elato value of ad j The fuzzy quotet set by the fuzzy equvalece elato s = {[ ]}, whee U/ = 2 [ ] = Defto 4 The epected cadalty equvalece class [ s defed as ] = p ( ) j j of a fuzzy Defto 5 The fomato quatty of fuzzy attbute set o fuzzy equvalece elato H (, = = ) log s defed as Ths measue s detcal wth Yage s etopy [26] the fom, but dffeet the goal The fomato measue we gve s to compute the dsceblty powe of a fuzzy attbute set o a fuzzy equvalece elato whee a pobablty dstbute s defed o U whle Yage s etopy s to measue the sematcs of a fuzzy smlaty elato Defto 6 Gve a fuzzy fomato system <U, A V, f, P>, A s the fuzzy attbute set, P s the pobablty dstbuto o U, ae two subsets of A [ ] ad [ ] ae fuzzy equvalece classes cotag geeated by,, espectvely The fuzzy equvalece elatos duced by, ae deoted by ad S The jot etopy of ad s defed as H (, = H ( S, = ) log, whee = p ( )( s ) j j j = Defto 7 The codtoal etopy of to s defed as whee H (, = j j = ) log, codtoed = p ( ) ad = p ( )( s ) Theoem 7 H(,, H(, Poof H(, H(, = = ) log = ) log = = H (, ( = ) log The foms of the poposed fomato measues ae detcal wth that of Shao s oes, ad they ca be used to measue the fomato geeated by a fuzzy attbute set, a fuzzy equvalece elato o a fuzzy patto I the follows, the poposed fomato measues wll be appled to fuzzy pobablty appomato Space 4 Ifomato measues o fuzzy pobablty j ) j j

8 appomato space The above secto pesets a fomato measue fo fuzzy equvalece elatos whe a pobablty dstbuto s defed Hee we wll apply t to the fuzzy pobablty appomato space Theoem 8 Gve a fuzzy pobablty fomato system <U, A V, f, P>, A s the fuzzy attbute set, P s the pobablty dstbuto o U, ae two subsets of A [ ] ad [ ] ae fuzzy equvalece classes cotag geeated by,, espectvely The fuzzy equvalece elatos duced by, ae deoted by ad S, espectvely The we have: ) A : H(, 0 2) H(, ma{ H(,, H(, } 3) o : H(, H(, = 4) o : H(, = 0 Theoem 9 Gve a fuzzy pobablty fomato system <U, A V, f, P>, A a, H(, a, f a s edudat H(, > H( a, f s depedet s a educt f satsfes: ) H (, A, 2) a : H(, > H( a, Defto 8 The sgfcace of a attbute a s defed as SIG( a, ), H( a, Theoem 0 Gve a fuzzy pobablty fomato system <U, A V, f, P>, A = C d s a subset of C a, H ( d a, d, f a s edudat elatve to d H ( d a, > H( d, f s depedet s a educt of C elatve to d f satsfes: ) H ( d, C d, 2) a : H( d a, > H( d, Defto 9 The elatve sgfcace of a attbute a s defed as SIG( a,, d) d a, H( d, 5 Coclusos The cotbuto of the pape s two-fold O oe sde, we geealze the fuzzy appomato space to fuzzy pobablty appomato space by toducg a pobablty dstbuto o U Futhemoe, we popose ovel fomato measues o fuzzy equvalece elatos to compute the fomato quatty fuzzy pobablty appomato space The poposed fuzzy pobablty appomato space combes thee types of ucetaty: adomcty, fuzzess ad oughess togethe It s show that the fuzzy pobablty appomato space wll degade to fuzzy appomato space whe the equalty-pobablty assumpto holds If equvalece elatos ad the subset to be appomated both ae csp, the appomato space s Pawlak s oe The poposed measues tegate fuzzess, pobablty wth oughess, whch s showed a atoal geealzato of othe cases The methods to measue fomato Pawlak s appomato space, fuzzy appomato space ad fuzzy pobablty appomato space ae peseted ufom foms based o the geealzatos efeece [] W Swask, oma Hags, Lay ough sets as a fot ed of eual-etwoks tetue classfes Neuocomputg Vol 36, No -4, 200, pp [2] W Swask A Skowo ough set methods featue selecto ad ecogto Patte ecog Lettes Vol 24, No6, 2003, pp [3] M, Ju-Sheg Wu, We-Zh Zhag, We-u Appoaches to kowledge educto based o vaable pecso ough set model Ifomato Sceces Volume: 59, Issue: 3-4, Febuay 5, 2004, pp [4] Tsumoto, Shusaku Automated etacto of heachcal decso ules fom clcal databases

9 usg ough set model pet Systems wth Applcatos Volume: 24, Issue: 2, 2003, pp [5] N Zhog J Dog S Ohsuga ule dscovey by soft ducto techques Neuocomputg Volume: 36, Issue: -4, Febuay, 200, pp [6] T P Hog L Tseg S Wag Leag ules fom complete tag eamples by ough sets pet Systems wth Applcatos Vol 22, No 4, 2002, pp [7] Polkowsk, L Skowo, A ough Meeology: A New Paadgm fo Appomate easog Ite Joual of Appomate easog Vol 5,No 4, 996, [8] Pawlak, Zdzslaw ough sets, decso algothms ad ayes' theoem uopea Joual of Opeatoal eseach Volume: 36, Issue:, 2002, pp 8-89 [9] Pawlak, Z Gaulaty of kowledge, dsceblty ad ough sets Poceedgs of 998 I te Cof o fuzzy systems, 06-0, 998 [0] Zadeh, LA Towads a theoy of fuzzy fomato gaulato ad ts cetalty huma easog ad fuzzy logc fuzzy sets ad systems, 9, -27, 997 [] D Dubos, H Pade ough fuzzy sets ad fuzzy ough sets Iteatoal Joual of geeal systems 7 (2-3),990, [2] D Dubos, H Pade Puttg fuzzy sets ad ough sets togethe, : Slowsk (d), Ittellget Decso suppot, Kluwe Academc, Dodecht, 992, [3] Mos, Nehad N Yakout, MM Aomatcs fo fuzzy ough sets Fuzzy Sets ad Systems Volume: 00, Issue: -3, Novembe 6, 998, pp [4] Aa Maa Kee, tee A compaatve study of fuzzy ough sets Fuzzy Sets ad Systems Volume: 26, Issue: 2, Mach, 2002, pp [5] Wu, We-Zh M, Ju-Sheg Zhag, We-u Geealzed fuzzy ough sets Ifomato Sceces Volume: 5, May, 2003, pp [6] W Wu W Zhag Costuctve ad aomatc appoaches of fuzzy appomato opeatos Ifomato Sceces Vol59, No3-4, 2004, [7] M, Ju-Sheg Zhag, We-u A aomatc chaactezato of a fuzzy geealzato of ough sets Ifomato Sceces Vol60, No-4, 2004, [8] Wag, Y-Fa Mg stock pce usg fuzzy ough set system pet Systems wth Applcatos Volume: 24, Issue:, 2003, pp 3-23 [9] S Padm Mguel et al Vocabulay mg fo fomato eteval: ough sets ad fuzzy sets Ifomato Pocessg ad Maagemet Vol 37, No, 200, pp 5-38 [20] Feádez Saldo, JM Muakam, S ough set aalyss of a geeal type of fuzzy data usg tastve aggegatos of fuzzy smlaty elatos Fuzzy Sets ad Systems Vol 39, No 3, 2003, pp [2] Q She ad A Chouchoulas A ough-fuzzy appoach fo geeatg classfcato ules Patte ecogto, 35(): , 2002 [22] C Shao, W Weave The mathematcal theoy of commucato, uvesty of Illos pess, Champag, IL, 964 [23] Fote Measue of fomato: the geeal aomatc theoy, IO 2 3 (969) [24] JKampe de Feet, Fote Ifomato etc pobablte CAS Pas, Se A 265(967) 0-4,43-46 [25] L Zadeh, Pobablty measues of fuzzy evets, JMath Aal Appl 23 (968) [26] Yage topy measues ude smlaty elatos Iteat J Geeal systems 20 (992) [27] Headez, J ecases A efomulato of etopy the pesece of dstgushablty opeatos Fuzzy sets ad systems [28] adko Mesa, Ja ybak topy of fuzzy

10 pattos: a geeal model Fuzzy sets ad systems 99, 998, [29] Calo etoluzza, Vvaa Dold, Gloa Naval Ucetaty measue o fuzzy pattos Fuzzy sets ad systems 42(2004) 05-6 [30] chad Jese, Qag She Fuzzy-ough attbute educto wth applcato to web categozato Fuzzy sets ad systems 4 (2004) [3] Petes, JF Pawlak, Z Skowo, A A ough set appoach to measug fomato gaules 2002 Compute Softwae ad Applcatos Cofeece, Poceedgspp35 39 [32] Slowsk, Vadepoote, D A geealzed defto of ough appomatos based o smlaty Kowledge ad Data geeg, I Tasactos o, Vol 2, No2, 2000, [33] Q She, A Chouchoulas A Modula Appoach to Geeatg Fuzzy ules wth educed attbutes fo the motog of Comple Systems geeg applcatos of atfcal Itellgece, vol3, pp ,2000 [34] Z Pawlak ough classfcato Ite J huma-compute studes 999,5, [35] Yage O the topy of Fuzzy Measues I Tas o fuzzy systems Vol 8, No 4, 2000, [36] Yage Ucetaty epesetato Usg Fuzzy Measues I Tasacto o systems, ma ad cybeetcs Cybeetcs, Vol32, No, 3-20, 2002 [37] Geco, S, Mataazzo,, Słowńsk, : ough appomato by domace elatos, Iteatoal Joual of Itellget Systems, 7 (2002) o 2, 53-7 [38] Geco, S, Mataazzo,, Słowńsk, : ough sets methodology fo sotg poblems pesece of multple attbutes ad ctea uopea J of Opeatoal eseach, 38 (2002), o 2, [39] Y T Yao ad YY Yao, Iducto of Classfcato ules by Gaula Computg, SCTC 2002 pp [40] W Pedycz, Shadowed sets: bdgg fuzzy ad ough set, : S K Pal, A Skowo (ds), ough Fuzzy Hybdzato: A New ted Decso Makg, Spge, el, 999 [4] G Y Wag, H Yu, D C Yag Decso table educto based o codtoal fomato etopy Chese J computes Vol 25, No 7, 2002, pp -9