Fuzzy Set Theory in Modeling Uncertainty Data. via Interpolation Rational Bezier Surface Function

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1 Appled Mahemacal Scences, Vol. 7, 013, no. 45, 9 38 HIKARI Ld, Fuzzy Se Theory n Modelng Uncerany Daa va Inerpolaon Raonal Bezer Surface Funcon Rozam Zakara Deparmen of Mahemacs, Faculy of Scence and Technology, Unvers Malaysa Terengganu, Malaysa. rozam_z@yahoo.com Abd. Faah Wahab Deparmen of Mahemacs, Faculy of Scence and Technology, Unvers Malaysa Terengganu, Malaysa. faah@um.edu.my Copyrgh 013 Rozam Zakara and Abd. Faah Wahab. Ths s an open access arcle dsrbued under he Creave Commons Arbuon Lcense, whch perms unresrced use, dsrbuon, and reproducon n any medum, provded he orgnal work s properly ced. Absrac Laely, he research of raonal splne funcon wdely epandng because he desgnng of curve and surface, whch s he addonal parameer known as wegh ha had. Ths wegh use o changng he curve and surface shape o f he real daa form. Then, he nerpolaon mehod adaped n raonal splne funcon as nerpolaon raonal splne funcon, whch can be consruced by usng pecewse raonal splne funcon. Ths mehod s beng used o model he collecon of daa se o vsualze he daa for more he undersandng he meanng of he daa hrough he curve especally surface forms. However, he problem arses when he daa become unceran, whch s unable o model hem. Therefore, n hs paper, we nroduced fuzzy nerpolaon raonal bcubc Bezer surface (laer known as FIRBBS) whch modelng he fuzzy daa afer had been defned by he fuzzy ses defnon. The consrucon of FIRBBS s based on he defnon of fuzzy se heory and nerpolaon raonal bcubc Bezer surface. Then, he llusraon gven n fgures for more undersandng he mehod. Keywords: Fuzzy se heory, raonal bcubc Bezer surface, nerpolaon, fuzzy daa, alpha-cu, defuzzfcaon

2 30 Rozam Zakara and Abd. Faah Wahab 1 Inroducon In Compuer Aded Geomerc Desgn (CAGD) feld, raonal splne funcon s a useful ool n modelng daa whch s advanage n weghs whch hey had. The weghs also named as addonal parameers gve full freedom o a user o conrol he shape of he desgn curve and surface whch meanng ha he curve changng when he wegh s changng [5-7,13,15]. For raonal splne curves, Sarfraz e. al [14] nroduced a pecewse raonal cubc splne curves where possesses parameers (weghs) n each nerval, whch can be used o conrol he shape of he curves. Ths pecewse raonal cubc splne also known as nerpolaon raonal cubc splne. The eended of pecewse raonal cubc funcon s he raonal bcubc funcon whch nroduced by Hussan and Hussan [8]. The raonal nerpolaon as he pecewse raonal mehod, one of he mehods n CAGD feld fro desgnng. Ths nerpolaon mehod used n modelng daa, especally n curves desgn and convered o surfaces form. The modelng daa hrough he nerpolaon surface face he dffculy when he daa become he uncerany whch s he daa no precsely obaned. Ths uncerany daa canno be modeled wh he varous mehod o consruc a surface. Therefore, he fuzzy se heory whch nroduced by Zadeh [16] used as he soluon of he uncerany daa. The eenson of fuzzy se heory s fuzzy number concep [4,11,1,17] used o defne he uncerany daa. The srucure of hs paper begns wh he basc defnons whch hese defnons are beng used o consruced he proposed mehod hrough Secon. Then, Secon 3 dscuss abou he consrucng model for modelng he uncerany daa by usng nerpolaon raonal Bezer surface funcon. Also, n hs secon, we wll llusrae he fuzzfcaon and defuzzfcaon processes ogeher wh her llusraon. Secon 4 and 5 are boh he dscusson and concluson of he consruced model respecvely. Prelmnares In hs Secon, we gve some defnon whch ncluded he defnon of fuzzy se heory, fuzzy number and fuzzy daa pon(defned from uncerany daa pons usng fuzzy number conceps). In addon, he fuzzfcaon process(alpha-cu operaon) and defuzzfcaon process ye sll defned n hs secon. Defnon 1. Le X be a unversal se and A X Se A called fuzzy se denoed by A f for every X here ess μa : X [ 0,1] a form of membershp funcon ha characerzng he membershp grade for every elemen of A n X s defned by

3 Fuzzy se heory n modelng uncerany daa 31 1 f A (full membershp) μa( ) = c ( 0,1) f A (non-full membershp) (1) 0 f A (non-membershp) So, fuzzy se A can be wren as A = {(, μa( ) )} whch s A n X s a se of order par denoed genercally by n X wh grade of membershp μ A( ) n [0,1] [,3]. Defnon. Le R be a unversal se whch R s a real number and A s subse o R. Fuzzy se A n R (number around A n R) called fuzzy number whch eplaned hrough he α-levelse (srong α-cu ) ha s f for every α ( 0,1], here es se A α n R unl Aα = { R: μa ( ) > α}. If alpha value are α α > α >,..., > α, hen A,..., A [1-3]. k A α α α k Defnon 3. Fuzzy se P n a space of S sad se of fuzzy conrol pons f for every α level se was chosen, here ess poned nerval ha s P=< P, P, P > n S wh every P s crsp pon and membershp funcon μp : S [0,1] whch s defned as μ P( P ) = 1, 0 f P S 0 f P S μ ( P ) = c (0,1) f P S and μ ( P ) k (0,1) f P S P = P 1 f P S 1 f P S wh μp( P ) and μp( P ) are lef membershp grade value and rgh membershp grade value respecvely and generally wren as P= { P : = 0,1,,..., n} () for every, P =< P, P, P > wh P, P and P are lef fuzzy conrol pon, crsp conrol pon and rgh fuzzy conrol pon respecvely. Ths s also same as appled o wo and hree dmenson space [,3]. Defnon 4. A fuzzy number A s a rangular fuzzy number whch s cenred a p wh lef wdh, δ > 0 and rgh wdh, β > 0 f s membershp funcon forms as a 1 δ f a δ a a A ( ) = 1 β f a a+ β (3) 0 oherwse whch can be wren as A= ( a, δ, β ). Therefore, f he crsp nerval s obaned

4 3 Rozam Zakara and Abd. Faah Wahab by usng he alpha cu operaon, hen he nerval of A α [17]. α (0,1], from ( α) ( α) ( a δ) ( a δ) ( a+ β) ( a+ β) = α, = α a ( a δ) ( a+ β) a ( α ) ( a δ) = ( a ( a δ)) α + ( a δ) ( α ) ( a+ β) = (( a+ β) a) α + ( a+ β) Then, ( α) ( α) Aα = [( a δ),( a+ β) ] = [( a ( a δ )) α + ( a δ), (( a+ β) a) α + ( a+ β)]. surely can be obaned (4) Defnon 5. Le he α -cu of fuzzy daa pons, P has been appled by usng Def. 4. Then P named as defuzzfcaon of P f for every P P, P = P = n (5) { } for 0,1,..., α α < P + P + P > = 0 where for every P = where P α and P α are he h 3 lef and rgh values of fuzzy daa pons afer he alpha-cu process respecvely. 3 Fuzzy Daa Modelng va Inerpolaon Raonal Bezer Surface Funcon In hs Secon 3, we wll dscuss of how he fuzzy daa pons had been defned by usng fuzzy number conceps and beng modeled n surface form by nerpolae hem by usng Bezer surface funcon. Ths modelng of fuzzy daa pons, hen called as fuzzy nerpolaon raonal bcubc Bezer surface (FIRBBS). Here, we choose bcubc surface form as we wan o llusrae he surface as he recangular pach. Defnon 6. Le π : a= 0 < 1... < m = b be paron of [ ab, ] and λ : c= y0 < y1 <... < yn = d be paron of [ cd, ]. The fuzzy raonal bcubc Bezer funcon s defned over each recangular pach [, + 1] [ y, y+ 1], where = 0,1,,..., m 1; = 0,1,,..., n 1 as: Where Cy (, ) = C ( y, ) = AuFA ( ) (, ) ( v), T, (6)

5 Fuzzy se heory n modelng uncerany daa 33 y y F, F, 1 F, F +, + 1 y y F+ 1, F+ 1, + 1 F+ 1, F+ 1, + 1 F (, ) =, y y F, F, 1 F, F +, + 1 y y F+ 1, F+ 1, + 1 F+ 1, F + 1, + 1 A( u) = [ a ( u) a ( u) a ( u) a ( u)], A () v = [ a0() v a1() v a() v a3()] v, wh 3 (1 u) + 3 uα (1 u) (1 v) + 3 vˆ α (1 v) a ( u) =, a ( v) =, ( ) q ( v) q u 3 3 β (1 ) v + 3 v ˆ β (1 v) a ( u) =, a ( v) =, ( ) q ( v) 3 u + u u 1 1 q u u(1 u) v(1 v) a ( u) =, a ( v) =, ( ) q ( v) q u (1 ) (1 ) a ( u), ( ), u u v v 3 = a3 u = q( u) q( v) 3 3 q ( u) = (1 u) + 3 uα(1 u) + 3 u β(1 u) + u 3 ˆ 3 q ( v) = (1 v) + 3 vˆ α (1 v) + 3 v β (1 v) + v. Subsung he values A, F and A n Eq. 6, hen he fuzzy raonal bcubc Bezer funcon Cy (, ) can be epressed as: 3 3 (1 u) σ, + 3 u(1 u) τ, + 3 u (1 u) ε, + u κ, Cy (, ) = (7) (1 u) + 3 uα (1 u) + 3 u β (1 u) + u 3 3 where

6 34 Rozam Zakara and Abd. Faah Wahab 3 y ˆ y 3 σ = (1 v) F + 3 v(1 v) ( ˆ α F + F ) + 3 v (1 v)( β F F ) + v F q ( v),,,,,, + 1, + 1, y y τ = (1 v) ( α F F ) 3 v(1 v),,, ( ˆ α ( α F F ) α F F ,,,, ) + 3 v (1 v) ˆ y y 3 ( β ( α F + F ) α F F, + 1, 1, 1, 1 ) + v ( α F + F ) q ( v), + + +, + 1, y y ε = (1 ) ( ) 3 (1 ), 1, 1, ( ˆ ( ) v β F F + v v α β F F + β F F , + 1, + 1, + 1, ) 3 (1 ) ˆ y y 3 + v v ( β ( β F F ) β F + F + 1, , + 1 1, 1 1, 1) + v ( β F + F ) q ( v), , , y (1 ) 3 (1 ) ( ˆ ) 3 (1 )( ˆ y κ = v F + v v α F + F + v v β F F ), + 1, + 1, + 1, + 1, , vf q(). v + 1, + 1 Unforunaely, hese fuzzy raonal funcons are no very useful for fuzzy surface desgn as any one of he free fuzzy parameer α,, ˆ and ˆ β α β apples o he enre nework of fuzzy curves. Thus, here s no local conrol on he fuzzy surface. Ths ambguy s overcome by nroducng varable fuzzy weghs and desred local conrol has been acheved. For hs purpose new free fuzzy parameers α, β, ˆ α and ˆ β are nroduced such ha:,,,, α ( y ) = α, β ( y ) = β, ˆ α ( ) = ˆ α, ˆ β ( ) = ˆ β,,,, = 0,1,,..., m 1; = 0,1,,..., n 1. The shape of he fuzzy surface can be modfed by assgnng dfferen values o hese fuzzy parameers. Ths propery of free fuzzy parameers wll mpose dfferen consran on α ˆ ˆ,, β,, α, and β,. 3.1 Choces of Dervaves y y In mos applcaons, he dervave parameers d, F,, F, and F, are no gven and hence mus be deermned eher from gven fuzzy daa or by some oher means. These mehod are he appromaon based on varous mahemacal heores. An obvous s menoned here [8-10]: Fuzzy Arhmec Mean Mehod Fuzzy arhmec mean mehod s he hree-fuzzy pon dfference appromaon based on arhmec manpulaon. Ths mehod s defned as:

7 Fuzzy se heory n modelng uncerany daa 35 F0, =Δ 0, + ( Δ0, Δ1, ) Δ, +Δ 1, F, =, = 1,,3,..., m 1; = 0,1,,..., n, Fm, =Δ m 1, + ( Δm 1, Δm, ). y F,0 =Δ,0 + ( Δ,0 Δ,1), Δ y, +Δ, 1 F, =, = 0,1,,..., m; = 1,,3,..., n 1, y Fn, =Δ, n 1 + ( Δ, n 1 Δ, n ). y y ( F, 1 F, 1) ( F 1, F 1, ) y F, =, = 1,,..., m 1; = 1,,..., n 1, where Δ, = F+ 1, F, and Δ, = F, + 1 F,. Thus, for llusraed of FIRBBS hrough Eq. 7 whch he llusraons are used fuzzy daa pons s gven n Fg. 1. (a) Fgure 1. Unform FIRBBS (a) wh meshes (b) whou meshes whch nerpolae all fuzzy pons daa. In Fg. 1a and 1b, FIRBBS was consruced hrough Eq. 7, whch nerpolaed all he 16 unform fuzzy daa pons. The unform fuzzy wegh values of he surface can change he fuzzy surface shape as shows n Fg.. (b) (a) Fgure. Unform fuzzy wegh changng he unform FIRBBS due o fuzzy wegh values changng wh (a) all fuzzy weghs values equal o 6.5 and (b) all fuzzy wegh values equal o 0.5. (b)

8 36 Rozam Zakara and Abd. Faah Wahab Fg. shows ha how he unform fuzzy wegh values of unform FIRBBS can change he unform fuzzy surface shape. These unform fuzzy wegh values, ˆ ˆ α,, β,, α, and β, where α ˆ ˆ, = β, = α, = β, = 6.5 shows unform fuzzy surface shape as n Fg. a and α ˆ ˆ, = β, = α, = β, = 0.5 shows surface shape as n Fg. b. In Fg. 1, unform FIRBBS was consruced hrough Eq. 7 nerpolae all 16 unform fuzzy pons daa whch also descrbe he unform fuzzy daa a z-as (fuzzy hegh). For he operaon of alpha-cu oward FIRBBS based on Def. 4, he vsualzaon can be shown hrough Fg. 3. Fgure 3. The unform alpha-cu operaon agans unform FIRBBS. Fg. 3 llusraed he operaon of unform alpha-cu was appled on unform fuzzy daa whch a z-as. The alpha value s 0.5, whch means ha f he alpha value s ended o 1, hen he FIRBBS approached o sngle surface (crsp surface).. Ne s he defuzzfcaon process of unform FIRBBS afer fuzzfcaon process has beng appled. Ths defuzzfcaon process s appled o oban crsp fuzzy soluon of FIRBBS. Based on Def. 5, hen we obaned he defuzzfcaon surface of FIRBBS whch llusraed va Fg. 4. Fgure 4. The crsp fuzzy soluon surface of FIRBBS afer defuzzfcaon process. Fg. 4 shows ha he modelng of crsp fuzzy soluon of FIRBBS afer he defuzzfcaon process has been appled. The defuzzfcaon surface s eacly he same wh he crsp surface because he fuzzy nerval of fuzzy daa pons are symmery, meanng ha he lengh beween crsp daa pon and lef fuzzy daa

9 Fuzzy se heory n modelng uncerany daa 37 pon s same wh he lengh beween crsp daa pon and rgh fuzzy daa pon. 4. Dscusson The consrucon of FIRBBS became one of he mehods n modelng fuzzy daa n surface form. Ths modelng surface used o nerpolae he daa o gve he llusraon of ha surface where hs mehod s eended from fuzzy nerpolaon bcubc Bezer surface. The advanage of hs mehod compared wh he oher mehod such as fuzzy nerpolaon bcubc Bezer surface s he addonal fuzzy parameer known as fuzzy weghs, whch can change he shape of he fuzzy surface due o fuzzy wegh values. 5. Concluson As he concluson, he FIRBBS was nroduced, whch can be used n modelng fuzzy daa (afer defnng he uncerany daa by usng he fuzzy number) whch represen he fuzzy surface. Ths mehod also can be used for modelng he non-fuzzy daa or eac daa. Furhermore, he fuzzy weghs as he addonal fuzzy parameers whch FIRBBS had, brng he advanage o change he fuzzy surface wh changng he values of fuzzy weghs. Therefore, wh hs mehod, he vsualzaons of hese fuzzy daa are suable and hs mehod also can be reduced o fuzzy nerpolaon bcubc Bezer surface when all fuzzy wegh's values are se o 1. Acknowledgemen The auhors would lke o hank Research Managemen and Innovaon Cenre (RMIC) of Unvers Malaysa Terengganu and Mnsry of Hgher Educaon (MOHE) Malaysa for fundng(frgs, vo5944) and provdng he facles o carry ou hs research. References [1] W. Abd. Faah, Pemodelan Geomer Menggunakan Teor Se Kabur, Unvers Sans Malaysa, 008. [] W. Abd. Faah, M.A. Jamaluddn, A.M. Ahmad, Fuzzy Geomerc Modelng, Proceedngs. Inernaonal Conference on Compuer Graphcs, Imagng and Vsualzaon, 009. CGIV 009., 009, pp [3] W. Abd. Faah, M.A. Jamaluddn, A.M. Ahmad, M.T. Abu Osman, Fuzzy Se In Geomerc Modelng, Proceedngs. Inernaonal Conference on

10 38 Rozam Zakara and Abd. Faah Wahab Compuer Graphcs, Imagng and Vsualzaon, 004. CGIV 004., 004, pp [4] D. Dubos, H. Prade, Fuzzy Ses and Sysems: Theory and Applcaons, Academc Press, New York, [5] G. Farn, NURBS for Curve and Surface Desgn: from Proecve Geomery o Praccal Use, nd ed., AK Peers, Ld, [6] G. Farn, Curves and Surfaces for CAGD: A Praccal Gude, 5h ed., Academc Press, USA, 00. [7] G. Farn, J. Hoschek, M.-S. Km, Handbook of Compuer Aded Geomerc Desgn, Elsever Scence B.V., The Neherlands, 00. [8] M.Z. Hussan, M. Hussan, Vsualzaon of Surface Daa Usng Raonal Bcubc Splne, Journal of Mahemacs, 38 (006), [9] M.Z. Hussan, M. Hussan, Vsualzaon of Daa Subec o Posve Consrans, Journal of Informaon and Compung Scene, 1- (006), [10] M.Z. Hussan, M. Hussan, Vsualzaon of 3D daa preservng convey, Journal of Appled Mahemacs & Compung, 3 (007), [11] G.J. Klr, B.Yuan, Fuzzy Ses and Fuzzy Logc: Theory and Applcaon, Prence Hall, New York, [1] G.J. Klr, U.S. Clar, B. Yuan, Fuzzy Se Theory: Foundaon and Applcaon, Prence Hall, New Jersey, [13] D.F. Rogers, An Inroducon o NURBS: Wh Hsorcal Perspecve, Academc Press, USA, 001. [14] M. Sarfraz, Z. Habb, M. Hussan, Pecewse nerpolaon for desgnng of p aramerc curves, Proceedngs of an IEEE Conference on Informaon Vsualzaon, London, 1998, pp [15] F. Yamaguch, Curves and Surfaces n Compuer Aded Geomerc Desgn, Sprnger-Verlag, Germany, [16] L. Zadeh, Fuzzy Ses, Informaon and Conrol, 8 (1965), [17] H.-J. Zmmermann, Fuzzy Se Theory and Is Applcaons, Kluwer Academc, USA, Receved: February 1, 013