O Fuzzy rthmetc, Possblty Theory ad Theory of Evdece suco P. Cucala, Jose Vllar Isttute of Research Techology Uversdad Potfca Comllas C/ Sata Cruz de Marceado 6 8 Madrd. Spa bstract Ths paper explores the exstg relatoshp betwee Possblty Theory ad Theory of Evdece, whe they are both appled to fuzzy arthmetc. Possblty Theory arthmetc s based o the exteso prcple (proecto of the ot possblty dstrbuto), whle Theory of Evdece, the cosoat bodes of evdece obtaed from each operad are combed to a ew ot body of evdece, whch ca geeral be o cosoat. Idetcal behavour s foud whe the ot possblty dstrbuto s calculated usg the m operator, whle Possblty Theory gves more specfc results whe others T-orms are used. Ths has bee cosdered by some authors as a Theory of Evdece drawbac (Dubos & Prade 989). Ths paper shows that Theory of Evdece may be a more realstc ucertaty model whe put data are obtaed from radom expermets wth mprecse outcomes. INTRODUCTION There s a straght forward relatoshp betwee Possblty Theory ad Theory of Evdece, whe cosoat bodes of evdece are volved. I ths case possblty ad plausblty measures cocde. Iterpretg basc assgmets as desty fuctos, where the radom varables are the focal elemets, smulatos ca be performed from gve possblty dstrbutos. Ths paper shows how the sum of two fuzzy umbers ad ca be calculated, applyg both the exteso prcple ad the theory of evdece, ad compares the results. Secto revews some basc deftos. Secto shows how the ot possblty/plausblty dstrbuto ca be obtaed. Secto compares the possblty/plausblty dstrbuto of the uo of two pots. Secto compares the results of summg two fuzzy umbers usg both approaches, ad fally some coclusos are preseted Secto 6. DEFINITIONS REVIEW Plausblty ad elef measures are fuzzy measures defed by (see (Klr 988) (Shafer 987)): such that: [ ] Pl : P( X), [ ] el : P( X), Pl(... ) Pl( ) Pl( ) < + ( ) Pl(... ) < el(... ) el( ) Pl( ) + ( ) el(... ) where P( X) s the power set of crsp subsets of X. Plausblty/elef measures ca also be defed, gve a body of evdece (F,m), as: Pl( ) = m( ) el( ) = m( ) where are the focal elemets ad m s the basc probablty assgmet (lvarez 99). Whe the body of evdece s cosoat, that s, ts focal elemets are ested, the the plausblty (resp belef) measure s called possblty (resp ecessty) measure, ad the followg propertes hold: [ ] π [ π π ] Pl( ) = max Pl( ), Pl( ) ( ) = max ( ), ( ) [ ] N [ N N ] el( ) = m el( ), el( ) ( ) = m ( ), ( ) Tag to accout the body of evdece, a fuzzy set ca be gve a probablstc terpretato. The basc assgmet s vew as a probablty desty fucto whose radom varable s the set of focal elemets. The possblty/plausblty dstrbuto fucto s defed from the plausblty measure defto by: µ( x) = Pl( { x} ) = m( ) = m( ) { x} x µ( ) x ca the be terpreted as a probablty dstrbuto fucto of the focal elemets, ad Motercarlo method ca be used to obta a realsato of the radom varable, that s, to obta a set.
possblty dstrbuto ca be also be represeted terms of ts alpha-cuts (Dubos & Prade 989) (Dubos & Prade 986 a): { F α α ( ] }, where F = { w ( w) } µ F( w) = sup { α (, ] w Fα } α µ F α I the followg, oly possblty ad plausblty measures wll be cosdered. JOINT POSSIILITY / PLUSIILITY DISTRIUTION. JOINT DISTRIUTION IN POSSIILITY THEORY Havg two fuzzy umbers ad wth possblty dstrbutos π ( a) ad π ( b) defed over U ad U, the ot possblty dstrbuto π x ca be obtaed combg them wth a T-orm: where a π { } = π π ( ( a, b ) ) T( ( a ), ( b )) x U ad b U. Fgure shows ad possblty dstrbutos ad the ot dstrbuto π x for the mmum, product or Lucasevcz T-orms. s t ca be see, the m T-orm gves the least specfc result. Whe both umercal varables a ad b correspod the same physcal varable, a possblty dstrbuto ca be obtaed cuttg the prevous surfaces wth a = b, as show Fgure (Zadeh 977). It s supposed that the sources are completely relable, as couctve cosesus has bee used (Dubos & Prade 988).. JOINT DISTRIUTION IN THEORY OF EVIDENCE If we cosder the cosoat body of evdece of each fuzzy set ( F, m ) ( F, m ) as the radom varables desty fuctos, the relatoshp betwee both radom varables ca be used to calculate the ot basc assgmet m x. The plausblty measure s the gve by: Pl(( a, b )) = m ( C ) C {( a, b )} x where C are the focal elemets of the ot body of evdece, defed, or a subset, depedg o the relatoshp betwee the radom varables..8.8.8.6.6.6....... a.. a.. a. b b b Fgure : Jot possblty dstrbuto, wth dfferet T-orms: a) mmum, b) product, c) Luasewcz.8.8.8.6.6.6............ b b b a a a Fgure : ad possblty dstrbuto whe a = b, wth dfferet T-orms: a) mmum, b) product, c) Luasewcz
If the ot body of evdece s cosoat, ths plausblty measure s also a possblty measure. I the followg, three ds of relatoshps betwee the radom varables wll be aalysed: α = α, depedece, ad α = α. s explaed (Ta 99), whe probablty s cocetrated ad uformly dstrbuted o the ma dagoal of the ot doma, α ad α are perfect postve correlato. It ca also be terpreted as cocordace betwee ad sources of owledge (for example, the same strumet has bee used to measure ad tervals). Whe the whole probablty s cocetrated ad uformly dstrbuted o the at-dagoal, α ad α are perfect egatve correlato. It ca be terpreted as a dscrepacy betwee both sources of owledge, or betwee precso measuremets. Idepedece remas wth ts usual terpretato... α = α relatoshp I ths case the ot basc assgmet s oly defed the le show Fgure. ad basc assgmets are dsplayed X ad Y axes, ad the ot basc probablty assgmet the correspodg subset of x. Each pot of the basc assgmet represets a terval that s a ( F, m ) body of evdece focal elemet. Every terval s represeted by ts lower lmt, ad thus the doma of the basc assgmet s the doma of the tervals lower lmts. Ths graphcal represetato s smlar to (Ta 99), where focal elemets are represeted by ther correspodg alpha value. ut lower lmt represetato allows to show plausblty measures the basc assgmet graph..8 m() x basc probablty assgmet m() m(x) To calculate the possblty of a ad b (see Fgure ), every ot focal elemet cotag the par ( a, b ) has to be cosdered. That s, π (( a, b )) m ( C ) = = x or, expressed for cotuous varables: C π (( a, b )) = m ( C ) dc C = C x whch s equal to the area dcated Fgure. b.... C C (a,b) (a,b) C...6.8...6.8 a Fgure : Jot body of evdece focal elemets whe α = α Sce ths pot belogs to the le correspodg to α α (( a, b )) = ( a ) = ( b ). =, t s π π π To calculate the possblty of the par ( a, b ) show Fgure, t should be oted that the ot focal elemets cotag ths pot are the same ot focal elemets that cota the prevous oe ( a, b ). That s, π = π = π = π π (( a, b )) (( a, b )) ( b ) m( ( a ), ( b )).6.. m(x).8...6.8...6.8.6 Pos(a.ND.b) Fgure : Jot basc assgmet whe α = α Every par ( a, b ) located the doma of the ot basc assgmet m x represets a focal elemet of the ot body of evdece ( F, m ) x x. Fgure shows that ths case the ot focal elemets are ested, ad thus plausblty measures wll be possblty measures.... (a,b).......6.7.8.9 Fgure : Possblty of ( a, b ) whe α = α
Calculatg the possblty of every par ( a, b ), the ot possblty dstrbuto s the oe obtaed possblty theory whe the m T-orm s used. Numercal smulato ca be appled to obta the same result R usg both approaches, whe ad refer to the same physcal varable. (Fgure.a). To perform the smulato, radom varables are obtaed by Motecarlo method: a value of alpha s geerated as a uform dstrbuto betwee ad. Wth ths value, alpha-cuts of ad are obtaed, whch are realsatos of the radom varables. The tersecto of both tervals s calculated, ad the result s a focal elemet of ( FR, mr ). These focal elemets are ested ad the possblty measure assocated to the resultg body of evdece ca be obtaed by the formula π ( r ) m ( R ) = =.. Idepedece relatoshp If ad radom varables are depedet, the ot basc assgmet doma s the whole cartesa product x, where the basc assgmet s uformly dstrbuted (see Fgure 6)..8.6.. m()...6.8...6.8 R x basc probablty assgmet m() m(x) Fgure 6: Jot basc assgmet whe ad are depedet b.... C C...6.8...6.8 a Fgure 7: Jot body of evdece focal elemets whe ad are depedet Plausblty of ( a ad b ) s calculated cosderg every ot focal elemet that cotas the pot ( a, b ). Every focal elemet, represeted by a pot located the volume base Fgure 8, cotas ( a, b ). pplyg the plausblty formula for cotuous varable, the plausblty s the volume show Fgure 8. Pl(( a, b )) = m ( C ) dc = x C ( a, b ) = m ( ) m ( ) d d = = = = m ( ) d m ( ) d = ( a ) ( b ) π π = = Furthermore, ( a, b ) plausblty s equal to the product of possbltes π ( a ), π ( b ), ad thus the plausblty dstrbuto obtaed from the theory of evdece s the same as the possblty dstrbuto obtaed from the possblty theory. s t wll be explaed later, the dfferece s that the uderlyg body of evdece possblty theory s cosoat, whle the theory of evdece t s ot. ga, every pot the doma represets a ot focal elemet, buld from a focal elemet of ( F, m ) ad a focal elemet of ( F, m ) (represeted both by ther lower lmts). Two of these ot focal elemets are show Fgure 7. s the doma s the whole cartesa product, the focal elemets are ot ested, ad the plausblty measures assocated to the ot body of evdece are ot possblty measures. m(x) Pl(a.ND.b).8.6. (a,b).......6.7.8.9.. Fgure 8: Plausblty of ( a, b ) whe ad are depedet
The same cocluso was reached usg umercal smulato, whe ad refer to the same varable. Numercal smulato wth depedet radom varables gves the same result as the possblty theory approach, whe the product t-orm s used. Numercal smulato has bee performed as before, but two dfferet values of alpha are obtaed depedetly, oe for a the other oe for. Every ot focal elemet from C to C has to be cosdered, because t cotas the pot ( a, b ) gvg: C Pl(( a, b )) = mx( C ) dc C = C whch ca be terpreted as the area show Fgure... α = α relatoshp Whe the relatoshp betwee ad s somehow cotradctory, α = α, the ot focal elemets are oly defed the le show fgure 9, where the ot basc assgmet s uformly dstrbuted (Ta 99)..8.6. m(x) Pl(a.ND.b) x basc probablty assgmet. (a,b). q.......6.7.8.9 p m() m().8.6 m(x) Fgure : Plausblty of ( a, b ) whe α = α.....6.8...6.8 Fgure 9: Jot basc assgmet whe α = I ths case, as show Fgure, focal elemets are aga ot ested, gvg a o cosoat body of evdece, ad thus, plausblty measures are ot possblty measures. b.... (a,b) C (a,b) C...6.8...6.8 a Fgure : Jot body of evdece focal elemets whe α = α α Plausblty measure ca be expressed terms of the tal bodes of evdece: C Pl(( a, b )) = m ( C ) dc = C C = C = m ( C ) dc m ( C ) dc = = m ( ) d m ( ) d = = m ( ) d m ( ) d = = = = ( a ) + ( b ) x C x x C = C C = C q = p = π p π d geeral, for ay par( a, b ), t s: ( π π ) Pl(( a, b )) = max, ( a ) + ( b ) that s, Luasewcz T-orm. The ot plausblty dstrbuto ths case s the same as the possblty dstrbuto obtaed by possblty theory, the dfferece beg aga the uderlyg body of evdece. Numercal smulato gves the same result. value of alpha s obtaed (uformly dstrbuted betwee ad ) ad the other oe s calculated accordg to α = α. q Let us cosder the pot ( a, b ) Fgure. Sce o ot focal elemet cotas that pot, ts plausblty s ull. O the other had, ( a, b ) plausblty s ot ull.
PLUSIILITY OF ( a, b ) ( a, b ). POSSIILITY THEORY The possblty of a set s the maxmum possblty of every pot belogg to t, that s: π (( a, b ) ( a, b )) = max( π ( a, b ), π ( a, b )) where the ot possblty s obtaed wth a T-orm.. THEORY OF EVIDENCE Gve a body of evdece ( F, m ), the plausblty x x measure of a two pots set s gve by: ( ) Pl ( a, b ) ( a, b ) = m ( C ) x C ( a, b ) or C ( a, b ) that s, every ot focal elemet cotag at least oe of the two pots must be cosdered. Fgure.a shows the ot focal elemets that cota (a,b ) or (a,b ), whe the relatoshp betwee the radom varables s α = α. I ths case, the plausblty of the uo s equal to the plausblty of ( a, b ), whch s the maxmum plausblty of both pots (max operator s obtaed whe focal elemets are ested, (Klr 988)). The result s the same as possblty theory because the uderlyg body of evdece s also the same. Fgure.b shows the ot focal elemets whe the radom varables are depedet. ll the ot focal elemets located the mared area cota at least oe of the two pots. Expressg the formula for cotuous varables, the plausblty s the tegral of the uform dstrbuto m x over the dcated area (that s, the volume whose base s the dcated area). Ths volume s geeral greater or equal tha the volumes obtaed for Pl( ( a, b )), or Pl( ( a, b )). The t s: ((, ) (, )) ( (, ), (, )) Pl a b a b max Pl a b Pl a b Ths dscrepacy betwee both approaches s due to the fact that the body of evdece obtaed from the theory of evdece s ot cosoat, whle possblty theory always cosders, amog all the dfferet bodes of evdece wth the same possblty/plausblty dstrbuto, the uderlyg cosoat body of evdece. The use of max operator the exteso prcple meas that the cosdered body of evdece s the cosoat oe. Fgure.c shows the ot focal elemets that cota at least oe of the pots, whe the relatoshp betwee the radom varables s α = α. ga, expressg the plausblty for cotuous varables, ts value s gve by the area whose base s mared, ad geeral t s greater or equal tha the dvdual plausblty measures. s the prevous case, the body of evdece cosdered by the theory of evdece s ot cosoat, whle the body of evdece uderlyg possblty theory calculus s the cosoat oe. SUM OF ND I ths secto the sum of two fuzzy umbers ad wll be dscussed usg the prevous results. Gve ad ther sum R=+ wll be obtaed computg the possblty/plausblty of the uo of pots located the le defed by a + b = r, where r U R.. POSSIILITY THEORY pplyg the exteso prcple, the possblty of each r s obtaed as the maxmum of the possbltes of the pars (a,b ) verfyg a + b = r, whch defes a secto the ot possblty dstrbuto. Fgure shows the secto obtaed for a partcular r..8.8.8 (a,b).6.6.6 (a,b)... (a,b)....8.8.8 (a,b) (a,b).6.6.6 (a,b)............6.7.8.9......6.7.8.9......6.7.8.9 Fgure : Jot focal elemets cotag ( a, b ) or ( a, b ). a) α = α, b) depedece, c) α = α
.8.8.8.6.6.6...... b a+b=. a. b a+b=. a. b a+b=. a. Fgure : a + b = costat cut wth dfferet T-orms: a) mmum, b) product, c) Luasewcz. THEORY OF EVIDENCE Gve a le defed by a + b = costat = r, the plausblty of r s calculated summg the basc probablty assgmet of every ot focal elemet cotag ay of the pots of the le. Fgure.a shows these focal elemets whe the radom varables are related by α = α. For example the possblty of r = s gve by the possblty of the pot where the dagoal ad the le a + b = tersect. Fgure.b shows the ot focal elemets where the radom varables are depedet. The plausblty measure s the volume whose base s the mared area. Whe the radom varables are related by α = α, see fgure.c, there are o ot focal elemets located α = α le cotag ay pot defed by a + b =. Ths meas that the plausblty of, ths case, s zero. The fgure also shows the ot focal elemets that must be tae to accout to calculate the plausblty of.. ga t ca be checed that the plausblty of the uo s greater or equal to the maxmum of all of them: ( ) Pl( r ) max Pl( a, b ) a + b = r The values of R uder these three assumptos were also computed usg umercal smulatos. ga Motecarlo method was used to obta the focal elemets of ad that had to be added. part from the case where α = α that gves the same result usg both approaches, geeral theory of evdece leads to less specfc dstrbutos tha possblty theory..8.8.8 alfa()= alfa().6.6.6 alfa()=alfa().......8.8.8 a+b=..6.6.6 a+b=. a+b=. a+b=..........6.7.8.9......6.7.8.9......6.7.8.9 Fgure : Jot focal elemets cotag ay pot of a + b = costat : a) α = α, b) depedece, c) α = α
.8.8.8.6.6.6.......8.8.8.6.6.6.................. Fgure : Jot focal elemets wth dfferet T-orms: a) mmum, b) product, c) Luasewcz 6 CONCLUSIONS Ths paper aalyses two dfferet approaches to aggregate possblty dstrbutos ad to operate fuzzy umbers, usg possblty theory ad fuzzy arthmetc oe had, ad theory of evdece o the other had. Gve the focal elemets of each of the operads (terpreted as radom varables), the couctve ot possblty/plausblty dstrbutos obtaed from both methods are detcal whe: the radom varables are related by α = α ad the ad operator used possblty theory s the m t-orm. the radom varables are depedet ad the ad operator used possblty theory s the product T-orm. the radom varables are related by α = α ad the ad operator used possblty theory s the Luasewcz t-orm. Gve a subset of the couctve ot possblty dstrbuto, ts possblty s calculated usg the max operator, ad thus mplctly assumg a cosoat uderlyg body of evdece. However theory of evdece, the plausblty dstrbuto must be obtaed from a explctly calculated body of evdece, whch s geeral ot cosoat, leadg to dfferet ad less specfc results. If we suppose (see (Dubos & Prade 986 b)) that fuzzy umbers ad are obtaed from radom expermets wth mprecse outcomes, that s, each measure s a terval where o dstctos ca be made, the theory of evdece seems a more realstc model. ddtoally ths meas that Motecarlo smulato ca be used to perform the computatos. O the cotrary the ested uderlyg focal elemets obtaed from the ot possblty dstrbutos (usg product ad Luasewcz t-orms) ca ot be terpreted the same way (see fgure ), sce they ca ot be obtaed combg ad focal elemets. s see prevously, the combato of ad focal elemets oly produce rectagles parallel to the axes. Theory of evdece approach leads to less specfc results tha possblty theory, although t could be cosdered a more realstc ucertaty model uder the above assumptos, ot ust a possblty theory approxmato (see (Dubos & Prade 989)). Refereces E. lvarez, E. Castllo (996). Ucertaty Measures Expert Systems. Mcrocomputers Cvl Egeerg 9, 9-66 D. Dubos, H. Prade (986 a). Set Theoretc Vew of eleve Fuctos. Iteratoal Joural of Geeral Systems, 9-6. D. Dubos, H. Prade (986 b). Fuzzy Sets ad Statstcal Data. Europea Joural of Operatoal Research, - 6. D. Dubos, H. Prade (988). Represetato ad Combato of Ucertaty wth elef Fuctos ad Possblty Measures. Computatoal Itellgece, - 6. D. Dubos, H. Prade (989). Fuzzy Sets, Probablty ad Measuremet. Europea Joural of Operatoal Research, - G.J. Klr, T.. Folger (988). Fuzzy Sets, Ucertaty, ad Iformato. Pretce-Hall Iteratoal Edtos G. Shafer (987). elef Fuctos ad Possblty Measures. alyss fuzzy formato, vol., -8 S.K. Ta, P.Z. Wag, X.Z. Zhag (99). Fuzzy Iferece Relato based o the Theory of Fallg Shadows. Fuzzy Sets ad Systems, 79-88 L.. Zadeh (977). Fuzzy Sets as a bass for a Theory of Possblty. Fuzzy Sets ad Systems, -8