Type-2 Fuzzy Sets: Properties and Applications

Transcription

1 vailable olie at Iteratioal Joural of Idustrial Egieerig ad Maagemet Sciece Type-2 Fuzzy Sets: Properties ad pplicatios Jorge Forcé Departmet of Fiace ad Operatios Maagemet, Iseberg School of Maagemet, Uiversity of Paris, Paris, Frace bstract The cocept of typc-2 fuzzy set was itroduced by Zadeh as a extesio of the cocept of a ordiary fuzzy set. Such sets arc fuzzy sets whose membership grades themselves are type-l fuzzy sets; they are very useful i circumstaces, where it is difficult to determie a exact membership fuctio for a fuzzy set; hece, they are useful for icorporate liguistic ucertaities. I this paper, we will discuss about set operatios, algebraic operatios ad properties of membership grades of type-2 fuzzy sets. lso we will propose some applicatios of type-2 fuzzy sets such as i type-2 fuzzy eural etwork systems ad typc-2 fuzzy logic systems. Keywords: type-2 fuzzy sets, fuzzy operatio, fuzzy eural etwork systems, fuzzy logic systems Itroductio The cocept of a type-2 fuzzy set was itroduced by Zadeh [] as a extesio of the cocept of a ordiary fuzzy set that called type-l fuzzy set. Membership grades of these sets are type-l fuzzy sets. They are very useful i circumstaces where it is

2 vailable olie at Iteratioal Joural of Idustrial Egieerig ad Maagemet Sciece difficult to determie a exact membership fuctio for a fuzzy set; hece, they are useful for icorporatig liguistic ucertaities. e.g., the words that are used i liguistic kowledge ca mea differet thigs to differet people [2], a fuzzy relatio of higher type, (for example type-2) has bee regarded as oe way to icrease the fuzziess of a relatio. I fact Type-2 fuzzy sets allow for liguistic grades of membership ad also offer improvemet o iferece with type-l fuzzy sets [3]. Type-2 sets ca be used to covey the ucertaities i membership fuctios of type-l sets, due to the depedece of the membership fuctios o available liguistic ad umerical iformatio. Liguistic iformatio does ot give ay iformatio about the shapes of the membership fuctios. Whe membership fuctios arc determied or tued based o umerical data, the ucertaity i the umerical data traslates ito ucertaity i the membership fuctios (MF). I all such cases iformatio about the liguistic/umerical ucertaity ca be icorporated i the type-2 framework. Fig.l (a) shows a type-2 fuzzy set i which the primary membership grade for each poit i the set is a Gaussia typc- fuzzy set. Ucertaity i the primary membership grades of the typc-2 MF cosists of a shaded regio the uio of all primary membership grades ad we call the footprit of ucertaity of the type-2 MF. Fig. (b) shows the secodary membership i x=4(e.g., μ(4)). (see fig.l.) Mizumoto ad Taaka [4] have studied Ihe set-theoretic Operatios of type-2 sets, properties of membership grades of such sets, ad, have examied the operatios of algebraic product ad algebraic sum for them [5]. Niemie [6] has provided more details

3 vailable olie at Iteratioal Joural of Idustrial Egieerig ad Maagemet Sciece Figure. Type-2 Fuzzy Set about algebraic structure of type-2 sets. Dubois ad prade [7] ad Kaufma ad Gupta [8] have discussed the joit ad meet operatios betwee fyzzy umbers uder miimum t- orm. Set operatios, algebraic operatios, properties of membership grades ad some applicatio of type-2 fuzzy sets are the ext sectios of this paper. Set Operatios o Type-2 Sets ad Cosider two fuzzy sets of type-2, ad, i a uiverse X. Let ( ) x ( ) be the membership grades (fuzzy sets i J [0,] ) for each x as x ( x ) f x( u) / u ad ( x ) g x( w) / w u, respectively, where u, w J x, idicate w the primary membership of x ad f x (u), g x ( w) [0, ] idicate the secodary memberships of x. Usig Zadeh's Extesio Priciple [,7], the membership x

4 vailable olie at Iteratioal Joural of Idustrial Egieerig ad Maagemet Sciece grades for uio, itersectio ad complemet of type-2 fuzzy sets bee defied as follows [4]: Uio: ad have () Itersectio: ( x) ( x) ( x) ( fx( u)* gx( w) /( u w (2) Complemet: ( x) ( x) ( x) ( f x( u)* gx( w) /( u* w ), ), ( x) ( x) f ( u) /( ) (3) x u Where represets die max t-coorm ad * represets a t-orm. The itegrals idicate logical uio. I the sequel, we adhere to these defiitios, ad, as i [4], we refer to the operatios, ad as joi, meet ad egatio, respectively. If ( ) ad ( ) have discrete domais, the itegrals i (l)-(3) are replaced by x x summatios. Usig the defiitios i (l)-(3), we examie the operatios of joi, meet ad egatio i more detail, uder miimum t-orm (). We cosider oly covex, ormal membership grades. Our goal is to obtai algorithms that let us compute the joi, meet ad egatio. The type- sets, F i this Sectio ca be thought of as membership grades of some type-2 sets,, for some arbitrary iput x 0. Major result for joi ad meet uder miimum t-orm is give i: Theorem. Suppose that we have covex, ormal, type- real fuzzy sets F,..., F characterized by membership fuctios f,..., f respectively. Let u, u2,..., u be

5 vailable olie at Iteratioal Joural of Idustrial Egieerig ad Maagemet Sciece real umbers such that u u... u 2 ad The, usig max t- coorm ad mi t-orm, f ( u ) f 2( u 2)... f ( u ). i fi( ), u, F ( ) f ( ), u u, k i i k i k k i f ( ), u. i i (4) d i fi ( ), u, Fi ( ) i fi ( ), uk uk, k i f ( ), u. i i I, (5) Dubois ad Prade preset the result i Theorem just for the case =2, i the cotext of fuzzificatio of max ad mi operatios. s a cosequece of Theorem, we have the followig iterestig result [8]: Corollary : If we have a covex, ormal, type- fuzzy sets membership fuctios f,..., f respectively, such that F,..., F characterized by f( ) f( ki), ad k k... k ; the, usig max t-coorm ad mi t-orm, i F F ad F 0 2 F i. The proof follows by a direct applicatio of Theorem to the type- sets described i the statemet of the corollary. lgebraic Operatios Just as the t-coorm ad t-orm operatios have bee exteded to the membership grades of type-2 sets usig the Extesio Priciple, algebraic operatios betwee type- sets have bee defied usig Extesio Priciple [7,8]. biary operatio

6 vailable olie at Iteratioal Joural of Idustrial Egieerig ad Maagemet Sciece "" defied for crisp umbers, ca be exteded to two type- sets, ad F f ( w) / w 2, as w 2 F f ( u) / u u follows: F * F [ f ( u)* f ( w)] /( u w) 2 2 (6) u w Where * idicates the t-orm used. We will maily be iterested i multiplicatio ad additio of type- sets. Observe that multiplicatio of two type- sets uder product t-om is the same as the meet of the two type- sets uder product t-orm; therefore, all our earlier discussio about the meet uder product t-orm applies of the multiplicatio of type- sets uder product t-orm. Theorem 4. Give iterval type- umbers i F,..., F with meas ml,...,m ad spreads s, s2,...,s, their affie combiatio if i, where i ( i,..., ) ad β are crips costats, is also a iterval type- umber with mea im i i, ad spread i. i s i Theorem 5. Give type- Gaussia fuzzy umbers F,..., F, with meas ml,...,m ad stadard deviatio, 2,...,, their affie combiatio if i, where ( i,..., ) ad β are crisp costat, is also a Gaussia fuzzy umber with mea i i im i i, ad stadard deviatio, where if product t- 2 2 i i i orm is used ad if miimum t-orm is used. i i

7 vailable olie at Iteratioal Joural of Idustrial Egieerig ad Maagemet Sciece See [4] for the proof of theorems 4,5. Properties of Membership Grades Mizumoto ad Taaka [4] discuss properties of membership grades of type -2 sets i detail, uder the miimum t-orm ad maximum t-coorm. These properties have examied uder product t-orm ad maximum t-coorm for covex, ormal membership grades, the fidigs are summarized i table l. See Chapter 3 of [5] for details. I provig the idetity law, the cocepts of 0 ad membership grades type-2 sets are used [4]; they are represeted as /0 ad /, respectively, which is i accordace with our earlier discussio about type- sets beig a special case of type- 2 sets. elemet is said to have a zero membership i a type-2 set if it has a secodary membership equal to correspodig to the primary membership of 0, ad if it has all other secodary membership equal to 0. Similarity a elemet is said to have a membership grade equal to i a type-2 set, if it has a secodary membership equal to correspodig to the primary membership of ad if all other secodary membership are zero. ll operatios o type-2 sets collapse to their type- couterparts, i.e., if all the type-2 sets are replaced by their pricipal membership fuctios (assumig pricipal membership fuctio ca be defied for all of them), all results remai valid. We, therefore, coclude that if there are ay set-theoretic laws that are ot satisfied by type- fuzzy sets, we ca safely say that type-2 sets will ot satisfy those laws either; however, the coverse of this statemet may ot be true. If ay coditios is satisfied by type- sets, it may or may ot be satisfied by type-2 sets. Observe, from Table, that the (max, product) t-coom/t-orm pair, for type-2 sets, does ot satisfy idempotet, absorptio, distributive, DE Morga s ad complemet laws, hece, if the desig of a type-2 fuzzy logic system (cotroller) ivolves the use of these laws, it will be i error. Observe also that some laws that are satisfied i the type- case uder {max, product} t-coorm/t-orm pair are ot satisfied i the type- 2 case.

8 vailable olie at Iteratioal Joural of Idustrial Egieerig ad Maagemet Sciece Table. Properties of Membership Grades Set-theoretic laws Miimum t-orm Product t-orm Reflexive ti-symmetric Trasitive Idempotet Commutative Type- Type-2 Type- Type-2, C, C Yes Yes Yes No * Yes Yes No No * * ssociative bsorptio Distributive Ivolutio De Morga's Laws Idetity ( ) C ( C ) ( C C * )* *( * ) *( ) Yes Yes No No ) ( * Yes Yes Yes No *( C ) ( * C ) ( * C ) Yes Yes Yes No ( * C ) ( C )*( C ) Yes Yes No No Yes Yes No No * * Yes Yes No No 0

9 vailable olie at Iteratioal Joural of Idustrial Egieerig ad Maagemet Sciece * * 0 0 Comlemet (Failure) * 0 pplicatios of Type-2 Fuzzy Sets Type-2 fuzzy sets have bee used i decisio makig [9,0], solvig fuzzy relatio equatios [], survey processig [2], pre-processig of data [3], ad modelig ucertai chacl states [4].The most importat applicatio of type-2 fuzzy sets is i fuzzy logic systems ad fuzzy eural etwork systems that is proposed i follow. Type-2 Fuzzy Logic Systems Quite ofte, the kowledge user to costruct rules i a fuzzy logic system (FLS) is ucertai. This ucertaily leads to ruler havig ucertai atecedets ad/or cosequets, which i tur traslates ito ucertai atecedet ad/or cosequet membership fuctios.fig.2 shows the structure of a type-2 FLS. It is very similar to the structure of a type- FLS [6]. For a type- FLS, the output processig block oly cotais the defuzzifier. We assume that the reader is familiar with type- FLSs, so that here we focus oly o the similarities ad differeces betwee the two FLSs. The fuzzifier maps the crisp iput ito a fuzzy set this fuzzy set ca, i geeral, be a type-2 set. I the type- case, we geerally have "IF-THEN" rules, where the th rule has the from "R': IF x is iputs; F s F ad x 2 is F2 ad x p is F p THEN y is G, where: x s are

10 vailable olie at Iteratioal Joural of Idustrial Egieerig ad Maagemet Sciece re atecedet sets (i=l,...,p); y is the output; ad Gs are cosequet sets. The distictio betwee type- ad type-2 is associated with the ature of the membership fuctios, which is ot importat while formig rules; hece, The structure of the rules remais exactly the same i the type-2 case, the oly differece beig that some or all of the sets ivolved are of type-2; so, the th rule i a type-2 FLS has the form "R': IF x is F ad x2 is F 2 ad... ad xp is F p THEN y is G. It is ot ecessary that all the atecedets ad the cosequet be type-2 fuzzy sets. s log as oe atecedet or the cosequet set is type-2, we will have a type-2 FLS. I a type- FLS, the iferece egie combies rules ad gives a mappig from iput type- fuzzy sets to output type- fuzzy sets. Multiple atecedets i rules are coected by the t-orm (correspodig to itersectio of sets). The membership grades i the iput sets are Figure 2. The Structure of a Type-2 FLS Combied with those i the output sets usig the sup-star compositio. Multiple rules may be combied usig the t-coorm operatio (correspodig to uio of

11 vailable olie at Iteratioal Joural of Idustrial Egieerig ad Maagemet Sciece sets) or durig defuzzificatio by weighted summatio. I the type-2 case, the iferece process is very similar. The iferece egie combies rules ad gives a mappig from iput type-2 fuzzy sets to output type-2 fuzzy sets. To do this oe eeds to fid uios ad itersectios of type-2 sets, as well as compositios of type-2 relatios. I a type- FLS. The defuzzifier produces a crisp output from the fuzzy set that is the output of the iferece egie, i.e. a type-0 (crisp) output is obtaied from a type- set. I the typc-2 case, the output of the iferece egie is a type-2 set; so we use "exteded versios" (usig Zadeh's extesio priciple [] of type- defuzzificatio methods). This exteded defuzzificatio gives a type- fuzzy set. Sice this operatio takes us from the type-2 output sets of the FLS to a type- set, we call this operatio "type reductio" ad the type-reduced set so obtaied a "typereduced set". To obtai a crisp output from a typc-2 FLS, we ca defuzzify the type-reduced set. The most atural way of doig this seems to be by fidig the cetroid of the typereduced set, however, there exist other possibilities like choosig the highest membership poit i the type-reduced set. Type-2 Fuzzy Neural Network Systems I this subsectio we cosider a typc-2 PLS system with a rule base of R rules i type-2 Fuzzy Neural Network System, e.g., -iput ad m-output with R rules. The jth cotrol rule id described as the followig form: R': IF x is ad x is rule umber, the ad THEN y is ad... ad y m is 's arc type-2 MFs of the atecedet part, ad m where j is a p 's are type- fuzzy sets of the cosequet part. Herei, the atecedet part MFs arc represeted as a upper MF ad a lower MF, Deote (x) ad (x) (see Fig. 3). The cosequet

12 vailable olie at Iteratioal Joural of Idustrial Egieerig ad Maagemet Sciece part is a iterval set [, ]. The rules let us simultaeously accout for ucertaity about atecedet membership fuctios ad cosequet parameters values. Whe the iput arc give, the firig stregth of the jth rule is m ( x ) ( x )... (7) ( x m m m 2 2 ) Where is the meet operatio [7]. Here i, the atecedet operatio is product-torm. That is, equatio (7) ca be calculated by m ( x ). m ( x )... ( x) (8) m ad 2 m m m ( x ). m ( x)... ( x) (9) 2 m Fially, the type reductio ad defuzzificatio should be cosidered. Similar to the FNN, here i the ceter of sets (COS)-type reductio method is used to fid M M ad y 2 2 i i y (0)

13 vailable olie at Iteratioal Joural of Idustrial Egieerig ad Maagemet Sciece Where Figure 3. Fuzzy Iferece of Typc-2 FNN deotes the firig stregth membership grad Hece, the defuzzificd output of a iterval type-2 FLS is y yr y () 2 Note that, if rule umber R is eve M R ad 2 M R 2. O the other had R is odd, y y 2 ( ).( y r M M M M 4 ) (2) Figure 3 summarizes above discussio ad show a fuzzy iferece system (jth rule) of type-2 FNN system. It is obvious that the type-2 FNN system is a geeralizatio of the FNN system. That is, the type-2 FNN system ca be reduce to a type- oe if

14 vailable olie at Iteratioal Joural of Idustrial Egieerig ad Maagemet Sciece the fuzzy sets is type-. We ca fid that details computatio of those systems are the same. Coclusio I this paper we have reviewed type-2 fuzzy sets. Set operatios, algebraic operatios ad properties of membership grades of type-2 fuzzy sets are proposed that are very useful i some applicatios of type-2 fuzzy sets especially i fuzzy logic systems ad eural etwork systems. Suggest that for more iformatio see [4,8]. Refereces ) Zadeh L, (975), "The Cocept of a Liguistic Variable ad Its pplicatio to pproximatio Reasoig-", Iformatio Sciece, 8, ) Medel J.M, (999), "Computig With Words Whe Word Ca Mea Differet Thigs to Differet People", 3 rd ual Symposium o Fuzzy Logic ad pplicatios, Rochester, New York, Jue, ) Hisdal F. (98), 'The IF THEN ELSE Statemet ad Iterval-Valued Fuzzy Sets of Higher Type", Iteratioal Joural of Ma-Machie Study,, ) Mizumoto M. ad Taaka K, (976), "Some Properties of Fuzzy Sets of Type-2", Iformatio ad Cotrol, 3, ) Mizumoto M. ad Taaka k., (98)," Fuzzy Sets of Type-2 uder lgebraic Product ad lgebraic Sum". Fuzzy Sets ad Systems, 5, ) Neimie J., (977), "O the lgebraic Structure of Fuzzy Sets of Type-2", Kyberetica, 3(4). 7) Dubois D. ad Prade H., (980), "Fuzzy Sets ad Systems: Theory ad pplicatios", cademic Press, Ic., New York.

15 vailable olie at Iteratioal Joural of Idustrial Egieerig ad Maagemet Sciece 8) Kaufma. ad Gupta M.M., (988), "Itroductio to Fuzzy rithmetic: Theory ad pplicatio", Va Nostrad Rcihold, New York. 9) Chaegua J.L. ad Guarrte M. ad ltschaeffl.g., (987), " pplicatio of Type-2 Fuzzy Sets to Decisio Makig i Egieerig", CRC Press, Ic., oca Rato, F.L. 0) Yager R R., (980), "Fuzzy Subsets of Type-2 i Decisios", J.Cyberetic, 0, ) Wagekecht M. ad Hartma K., (988), "pplicatio of Fuzzy Sets of Type-2 to the Solutio of Fuzzy Equatio Systems", Fuzzy Sets ad Systems, 25, ) Karik N.N. ad Medel J.M., (999), "pplicatio of Ttype-2 Fuzzy Logic Systems: Hadlig the Ucertaity ssociated With Surveys", Proceedig of FUZZ-IEEE '99, Seoul, Korea, ugust. 3) Joh R.I. Iocet P.R. ad ares M.R., (9998), "Type-2 Fuzzy Sets ad Neuro-Fuzzy Clustrig of Rad-ographic Tibia Images", IEEE Iteratioal Coferece o Fuzzy Systems, chorage, k, US, May, ) Karik N.N. ad Medel J.M.,(200), "Operatios o Type-2 Fuzzy Sets", Fuzzy Sets ad Systems, 22, ) Karik N.N. ad Medel J.M, (998), " Itroductio to Type-2 Fuzzy Logic Systems, USC Report, October. 6) Medel J.M, (995),"Fuzzy Logic Systems for Egieerig: Tutorial", IEEE, 83, ) Medel J.M, (200), "Ucertai Rule-ased Fuzzy Logic Systems: Itroductio ad New Directios", Pretice-Hall:NJ.

16 vailable olie at Iteratioal Joural of Idustrial Egieerig ad Maagemet Sciece 8) Liag Q. ad Medel J.M., (2000), "Iterval Type-2 Fuzzy Logic Systems: Theory ad Desig, IEEE Trasactio o Fuzzy Systems.