DealingwithuncertaintyandimprecisionbymeansoffuzzynumbersA.Gonzalez,O. - PDF document

DealingwithuncertaintyandimprecisionbymeansoffuzzynumbersA.Gonzalez,O.Pons,M.A.VilaDepto.CienciasdelaComputacineI.A.,E.T.S.deIngenieriaInformatica,UniversidaddeGranada,18071Granada,SpainReceived1July1998;accepted1February1999Theproblemofthecombinationofimprecisionanduncertaintycombinationfromtheapproximatereasoningpointofviewisaddressed.Animpreciseanduncertainin-formationcanberepresentedasafuzzyquantitytogetherwithacertaintyvalue.Inordertosimplifytheuseofsuchinformation,itisnecessarytocombinetheimprecisionanduncertaintyofthefuzzynumber.Inthispaperweproposeamethodforcombiningthembasedontheuseofinformationmeasures.The®rststepconsistsintruncatingthefuzzynumberbythecertaintyvalue.Sincenon-normalizedfuzzynumbersarediculttouse,wetransformthetruncatedfuzzynumberintoanormalizedfuzzynumberwhichcontainsthesameamountofinformation.Toformalizethisprocess,wedevelopatheoreticalcontextfortheinformationmeasuresonfuzzyvalues.Westudythefuzzynumberstransformationanditsproperties,andgiveanapproximatereasoninginter-pretationtotheapproach.1999ElsevierScienceInc.Allrightsreserved.Uncertainty;Imprecision;Fuzzynumber;Implicationfunction;InformationInternationalJournalofApproximateReasoning21(1999)233±256 *Correspondingauthor.Tel.:+34-58-243199;fax:+34-58-243317;e-mail:opc@decsai.ugr.es1E-mail:A.Gonzalez@decsai.ugr.es2E-mail:vila@decsai.ugr.es0888-613X/99/$±seefrontmatterÓ1999ElsevierScienceInc.Allrightsreserved.PII:S0888-613X(99)00024-9 IntroductionTheco-existenceofimprecisionanduncertaintywithinaconcretedatumappearsinmanyapplications.Forexample,inthestudyofoptimizationmethodsinfuzzygraphs[4]orintheframeworkofuncertainfuzzydatabases[1].Inthesetwocases,thegivensolutionsgiverisetoaseriesofinconveniencesderived,mainly,fromtheuseofnon-normalizedornon-trapezoidalfuzzysets,respectively.Inthispaperweinvestigatetheproblemsassociatedwiththecombina-tionofimprecisionanduncertaintyfromtheapproximatereasoningpointofview.Todothat,weusethegeneraltransformationfunctionTintro-ducedin[14]thatwillallowustorelateourresultstootherapproachesoftheliteratureandthatwillopennewwaysforthetreatmentofthisprob-lem.Thistypeofinformationcanbeexpressed,ingeneral,byanimprecisevalueA(represented,forexample,byatrapezoidalfuzzynumber)togetherwithacertaintylevelaassociatedwithsuchvalue.Thesituationcanbeformulatedasaconditionalexpressioninthefollowingterms.IfthedatumistotallytruethenitsvalueisA.Sincewehaveacertaintylevela1,thegeneralizedmodusponenscouldbeformulatedas:Ifthecertaintylevelis1,thenthevalueisA.Ifthecertaintylevelisa1,thenthevalueisA0:Thissituationisequivalenttothegenericcase: Therefore,anaturalwaytosolvetheproblemistoconsiderthatthedatumwearehandlingisA0de®nedas:lA0…x†ˆI…a;lA…x††whereIisamaterialim-plicationfunctionwhichre¯ectstheinterpretationgiventothecompatibilitydegree.Thereexistintheliteraturetwomainwaysofdealingwithimpreciseanduncertaindataandcanbeinterpretedasfollows:1.ToTruncate:Ifthedatumis…a;A†,thenA0isde®nedbythemembershipfunctionlA0…x†ˆmin…a;lA…x††whichdirectlyimpliesthatweareusingMamdani'simplicationinourreasoning.2.ToExpand:Ifweassumethataisanecessity,thenA0isgivenbythemem-bershipfunctionlA0…x†ˆmax…1ÿa;lA…x††,whichcorrespondstoKleene±Dienes'implicationasfoundationofourreasoning.Thesetwoapproachescorrespondtothedisjunctiveandconjunctiverepresentationoftheinferencerule,respectively.234A.Gonzalezetal./Internat.J.Approx.Reason.21(1999)233±256 ourpointofview,weunderstandthattheuseoftheseimplicationfunctions[2,3,15]forinformationrepresentation(whichisour®nalobjective)couldinduceanerror,sincedatumA0weareusingwillbeevaluatedintermsofcompatibilitywithotherdataand,inthesecases:1.Mamdani'simplicationresultsinadecreaseofthecompatibilitytolevelainanycase.Thisresultseemstobereasonableasaisthecertaintydegreebutitobligesustoworkwithnon-normalizedfuzzynumbers.2.Kleene±Dienes'implicationimposesthatanydatumwouldbecompatiblewithA0atleastatlevel1ÿa.Thisresultmaynotbesuitableforsomeapplications,sinceitassignsthesamepossibilitytoallthepointsoftheunderlyingdomainindependentlyfromthedistancetothesupportsetofA.Letusthinkthat,forexample,ifAisveryheavywithcertaintya,theval-uescloseto0willhavethesamepossibilitythanthoseclosetothesupportofveryheavy.Ourmainobjectiveisto®ndatransformationfunctionthat,basedondi€erentcriteria,ensuresusasuitablechange.Theintuitiveideasusedto®ndsuchatransformationfunctionare:·TotruncateAatlevela(weobtainAa).·TonormalizeAa(weobtainAT).Ifweassumethetranslationofuncertaintyintoimprecision,thenimpreci-sionofATmustbelargerthanAimprecisionbutATwillneverbede®nedonthewholedomain.TheideaistoincreasesuchimprecisionaroundthesupportsetofvalueA.ThetransformationusedisaccordingtoequitativedistributionofimprecisiononthesupportofAwhichisvalidwhennomoreinformationisprovided,i.e.,imprecisionisdistributedaccordingtoametricwhichtakesintoaccountthenearnesstotheoriginalinformation.Followingtheseideas,whenwehavetheinformationthatXisblackwithcertaintya,wewillnevergiveapositivepossibilitytocolourwhitebuttocoloursnearenoughtoblackde-pendingonvaluea.ThiswayofreasoninghasnotbeenusedyetandwillpermitustoensurethattheinformationamountprovidedbyanuncertainimprecisevalueAisthesameastheinformationprovidedbyitstransformationAT,whichisfullytrueandnormalized.Thepaperisorganizedasfollows.InSection2,thepreliminaryconceptsandthenotationusedareintroduced.InSection3,theaxiomaticde®nitionofaninformationfunctiononfuzzynumbersanditspropertiesaregiven.Basedonthisinformationfunction,atransformationforfuzzynumbersisintroducedinSection4.Thistransformationensuresthattheamountofinformationbeforeandafteritsapplicationremainsequal.InSection5,wearegoingtoprovethatthetransformationfunctionde®nedisanimplicationfunction.Todoit,wearecheckingthatalltheconditionsanimplicationfunctionmusthold,arealsoheldbyourtransformationfunction.Finally,inSection6,themainconclusionsofthisworkaresummarized.A.Gonzalezetal./Internat.J.Approx.Reason.21(1999)233±256235 PreliminaryconceptsAfuzzyvalueisafuzzyrepresentationabouttherealvalueofaproperty(attribute)whenitisnotpreciselyknown.Inthispaper,accordingtoGoguen'sFuzzi®cationPrinciple[10],wewillcalleveryfuzzysetofthereallinefuzzyquantity.Afuzzynumberisaparticularcaseofafuzzyquantitywiththefollowingproperties.De®nition1.ThefuzzyquantityAwithmembershipfunctionlA…x†isafuzzynumberi€:1.8a2‰0;1Š;Aaˆfx2RjlA…x†Pag(a-cutsofA)isaconvexset.2.lA…x†isanupper-semicontinuousfunction.3.ThesupportsetofASupp…A†ˆfx2RjlA…x†�0gisaboundedsetofR,whereRisthesetofrealnumbers.Thegivende®nitionisbasedonthede®nitiongivenbyDuboisandPrade[7]butwedonotrequireeithernormalizationorthatthemodalintervalisasingleton.Wewilluse~Rtodenotethesetoffuzzynumbers,andh…A†todenotetheheightofthefuzzynumberA.Forthesakeofsimplicity,wewillusecapitallettersatthebeginningofthealphabettorepresentfuzzynumbers.Theinterval‰aa;baŠ(seeFig.1)iscalledthea-cutofA.Sothen,fuzzynumbersarefuzzyquantitieswhosea-cutsareclosedandboundedintervals:Aaˆ‰aa;baŠwitha2…0;1Š.ThesetSupp…A†ˆfx2RjlA…x†�0giscalledthesupportsetofA.Ifthereis,atleast,onepointxverifyinglA…x†ˆ1wesaythatAisanor-malizedfuzzynumber.Sometimes,atrapezoidalshapeisusedtorepresentfuzzynumbers.Thisrepresentationisveryusefulasthefuzzynumberiscompletelycharacterizedby Fig.1.Fuzzynumber.236A.Gonzalezetal./Internat.J.Approx.Reason.21(1999)233±256 parameters…m1;m2;a;b†andtheheighth…A†asshowninFig.2.Otherparametricalrepresentationforfuzzynumberscanbefoundin[5].Wewillcallmodalsetallvaluesintheinterval‰m1;m2Š,i.e.,thesetfx2Supp…A†j8y2R;lA…x†PlA…y†g.Thevaluesaandbarecalledleftandrightspreads,respectively.Whenafuzzynumberisnotnormalized,thissituationcanbeinterpretedasalackofcon®denceintheinformationprovidedbysuchnumbers[6,11].Infact,theheightofthefuzzynumbercouldbeconsideredasacertaintydegreeoftherepresentedvalue.Ontheotherhand,ifweassumetheseconsiderations,normalizedfuzzynumbersrepresentimprecisequantitiesonwhichwehavecompletecertainty.Aswewillseealongthispaper,thisuncertaintycanbetranslated,usingsomesuitabletransformations,intoimprecision,takingintoaccountthatthelesstheuncertainty(orthemorethecertainty)aboutafuzzynumber,themoreistheimprecisionofsuchanumber.Thistransformationwillbedoneinsuchawaythattheamountofinformationofthefuzzynumberwillbeconstantbeforeandafterthemodi®cation.3.AninformationmeasureonfuzzyvaluesAspointedoutinSection2,wearegoingtotranslateuncertaintyintoim-precisionandviceversaundercertainconditions.Themostimportantoftheseconditionsisthattheamountofinformationprovidedbythefuzzynumberremainsequalbeforeandafterthetransformation.Sothen,the®rststepistode®neaninformationfunctionforfuzzynumbers.Weproposeanaxiomaticde®nitionofinformation,partiallyinspiredinthetheoryofgeneralizedinformationgivenbyKampedeFeriet[13]andthatcanberelatedtotheprecisionindexes[8]andthespeci®cityconcept,introducedbyYagerin[16]. Fig.2.Trapezoidalfuzzynumber.A.Gonzalezetal./Internat.J.Approx.Reason.21(1999)233±256237 on2.LetD~RjRD;wesaythattheapplicationIde®nedasI:D!‰0;1ŠisaninformationonDifitveri®es:1.I…A†ˆ18A2R,2.8A;B2Djh…A†ˆh…B†andAB)I…B†6I…A†:The®rstconditionmeansthatrealnumbersaretotallyinformativeand,thesecondone,thatconsideringtwofuzzynumberswiththesameheight,ifoneofthemiscontainedintheotherone,thenitisobviousthatthe®rstone,whichismoreprecise,isalsomoreinformative.Thegivende®nitionofinformationisverysimilartothede®nitionoftheprecisionindex,infact,whenappliedtonormalizedfuzzynumbers,bothofthemcoincide.Thiscoincidenceisveryreasonablebecause,whenthereisnouncertainty,informationisequivalenttoprecision.Inthisway,theinforma-tionfunctionisageneralizationoftheprecisionindexes.De®nition3.LetA2Dbeafuzzynumber.WesaythatAhasthemaximuminformationorthatAistotallyinformativewithrespecttoIi€I…A†ˆ1.Obviously,realnumbersaretotallyinformativewithrespecttoanyinfor-mationmeasureI,thatis,8r2R;I…r†ˆ1.Theinformationaboutfuzzynumbersdependsondi€erentfactors,inparticular,onimprecisionandcertainty.Wefocusongeneraltypesofinfor-mationrelatedonlytothistwofactors.Tocomputeameasureoftheimprecisioncontainedinafuzzynumber,wewillconsiderameasureoftheimprecisionofitsa-cuts,whichareclosedin-tervalsonwhichthefollowingfunctionisde®ned:8A2~R;fA…a†ˆbaÿaaifa6h…A†;0otherwise:(Fromthisimprecisionfunctiononthea-cuts,wede®nethetotalimprecisionofafuzzyvalueasacombinationoftheimprecisionineverylevela.Whenaˆ0,wewillconsiderthatfA…0†isthelengthofthesupportset.De®nition4.Theimprecisionofafuzzynumberisde®nedasfollows:f:~R!R‡0;8A2~R;f…A†ˆZh0…A†fA…a†da:238A.Gonzalezetal./Internat.J.Approx.Reason.21(1999)233±256 imprecisionfunctionfcoincideswiththeareabelowthemembershipfunctionofthefuzzyvalue,asshowninFig.3.Obviously,itisheldthat8A;B2~RjAB)fA…a†6fB…a†8a2‰0;hŠwhenh…A†ˆh…B†ˆhand,thereforef…A†6f…B†:Relatedtotheheight(certainty)andtheimprecisionofafuzzyvalue,wede®nethefollowinggeneraltypeoffunctionon~R:IF:~R!‰0;1Š;IF…A†ˆF…h…A†;f…A††:Thefollowingresultguaranteesthat,forcertaintypesofFfunctions,IFisainformationfunctiononR.Proposition1.LetF:…0;1ŠR‡0!‰0;1Šsuchthat1.F…1;0†ˆ1;2.8y;z2R‡0jy6z)F…x;z†6F…x;y†8x2…0;1Š:then,IFisaninformationfunctionon~R.Proof.LetA2R.Thenobviouslyh…A†ˆ1andf…A†ˆ0.Then,IF…A†ˆF…1;0†ˆ1.LetA;B2~RjABandh…A†ˆh…B†.Thenf…A†6f…B†anditisveri®edthatF…h…B†;f…B††ˆIF…B†6F…h…A†;f…A††ˆIF…A†andtherefore,IFisaninformationon~R.WhenFveri®esthepreviousconditions,wewillcallfunctionIFanF-information.Inthisway,associatedwithaclassoffunctions,wecanbuildsomeparticulartypesofinformationon~Rwiththepropertyofnotdepending Fig.3.A.Gonzalezetal./Internat.J.Approx.Reason.21(1999)233±256239 thepositionthefuzzyvaluehasonR,asshowninthefollowingproposi-tion.Proposition2.LetFbeafunctionverifyingconditionsestablishedinProposi-tion1,A2~Randt2R.Then,IF…A†ˆIF…At†:Proof.IfA2~Randt2RthenAt2~RandlAt…z†ˆlA…zÿt†8z2R.Be-sides,h…At†ˆsupz2RflAt…z†gˆsupz2RflA…zÿt†gˆsupz2RflA…z†gˆh…A†and…At†aˆAa‡t,resultingthatfA…a†ˆfAt…a†andf…A†ˆf…At†;thereforetheresultisimmediate.Therearemanywaystobuildinformationfunctionsbut,forourpurpose,wearede®ninginformationassociatedwithaparticularfunction.ThisF-informationwillpermit,subsequently,thede®nitionoftransformationsthatkeepconstanttheamountofinformationafuzzynumberprovides.LetusconsiderthefunctionF:…0;1ŠR‡0!‰0;1Š;F…x;y†ˆ xk
y‡1;k2R‡;thattriviallyveri®estheconditionsestablishedinProposition1.Hence,wecande®nethefollowingF-information.De®nition5.Wede®nethefunctionIF:~R!‰0;1Š;8A2~R;IF…A†ˆ h…A†k
f…A†‡1;whereh…A†isthefuzzynumberheight,f…A†istheimprecisionassociatedwithAandk6ˆ0aparameterwhichdependsonthedomainscale(inSection5,thisparameteriswidelyexplained).Evidently,byProposition1,IFisaninformationfunctionand,ittriviallyfollowsthat8A2~R;06IF…A†6h…A†61:Ascanbeimmediatelydeducedfromitsde®nition,informationIFisalwaysboundedbythefuzzynumberheight.Therefore,fuzzynumberswithmaximuminformationwithrespecttoIFmustalsohavemaximumheight(h…A†ˆ1)and,consequently,minimumimprecision(f…A†ˆ0).240A.Gonzalezetal./Internat.J.Approx.Reason.21(1999)233±256 fuzzynumbersshowninFigs.4(a)and(b)providethesameinforma-tionas 0:51‡1ˆ 13‡1ˆ0:25assumingkˆ1.ThefuzzynumbersshowninFigs.4(b)and(d)arethesamefuzzynumbersexpressedindi€erentdomainscales.Astheinformationprovidedbybothnumbersshouldbethesame,thekparametermustbeadaptedtothescalechangesconsideringabaseorreferencescalewherekissetto1(inthiscasekilometers) 13‡1ˆ 1 11000
3000‡1;sothen,ifthebasescaleiskilometersandthecurrentscaleinmeters,kpa-rametermustbesetto1=1000. Fig.4.Fuzzyvaluesexamples.A.Gonzalezetal./Internat.J.Approx.Reason.21(1999)233±256241 caseofFig.4(c)isnotreallyafuzzynumberrepresentationifwestrictlyfollowDe®nition1,butitisveryillustrativetoseehowthesetypesoffuzzyquantitieswithin®nitesupport,provideinformation0,asf…A†ˆ1.Wecouldalsowonderwhichintervalhasthesameinformationamountasaconcretenormalizedtrapezoidalfuzzynumber.Theanswertothisquestionisthefollowing±LetussupposeournormalizedtrapezoidalfuzzynumberisthegeneralonerepresentedinFig.5(a).Oneofthepossibleintervals(theonecenteredinthemodalset)withthesameinformationasthefuzzynumberisrepresentedinFig.5(b)expressedbyBˆ‰m1ÿ a2;m2‡ b2Š.ItcanbeeasilyprovedthatI…A†ˆI…B†andthatBˆE…A†,i.e.themeanvalueofAinthesenseofDuboisandPrade[9,12].Oncewehaveaninformationfunctiononfuzzynumbers,wewanttouseittode®netransformationswhichpreservesuchinformationfunctionvalue.Theideaisto®ndanequivalentrepresentationoftheconsideredfuzzynumberinsuchawaythatwechangeuncertaintybyimprecisionkeepingconstanttherelationshipbetweenthemde®nedbytheinformationfunction.4.Fuzzynumberstransformations4.1.BasicmodelTheaimofthetransformationsweareproposinginthissectionis,basically,tobeabletomodifytheheightofafuzzynumberbutkeepingtheinformationcontainedinit.Thereasonfordoingthisisthat,inmostapplications,itisveryconvenientthatfuzzynumbersarenormalized(simplicity,betterunderstand-ingforusers,etc...).Givenafuzzynumber,atransformationonitwillgiveanotherfuzzynumberwiththesameinformationamountbutdi€erentheight.Sothen,tode®netransformations,wewillrequestthattheinformation Fig.5.Normalizedtrapezoidalfuzzynumberandthecorrespondinginterval.242A.Gonzalezetal./Internat.J.Approx.Reason.21(1999)233±256 remains®xed,i.e.,wewillmodifycertaintyandimprecisionbutkeepingconstanttherelationbetweenbothnumbers,whichisde®nedbytheinformationfunction.Thede®nitionoftransformationwillbeobtainedfromtheconditionofequalityintheinformationbut,asa®rststep,wemustestablishwhatweunderstandfortransformationofafuzzynumberonasubsetof~R.De®nition6.Letusconsidera2…0;1ŠandtheclassoffuzzynumbersD~R.WesaythatTa:D!~RisatransformationforaninformationfunctionIonD,ifitveri®esthat:1.Ta…A†2D;2.h…Ta…A††ˆa;3.I…Ta…A††ˆI…A†8A2D:Inthisway,foraheightlevela,Ta…A†neednotexistbut,ifitdoes,itmustverifytheconditionsabove.De®nition7.GiventhetransformationTa,wesaythatA2Distransformableforaˆ…0;1ŠifthereexistsTa…A†.WewilldenoteH…A†ˆfa2…0;1Šj9Ta…A†gthesetoflevels,whereAistransformable.Thoughmostoftheresultsobtainedherecanbegeneralizedforanytypeoffuzzynumber,wewillfocusontrapezoidalonesforthesakeofsimplicityinthetransformationfunction.WewillnotebystheclassoftrapezoidalfuzzynumbersonR.GivenafuzzynumberA2s,wearelookingfortheconditionsthatanotherfuzzynumberB,with®xedheighta2…0;1Š,mustholdtohavethesamein-formationamountasA.Proposition3.LetA;B2sbetwofuzzynumberswithheightsh…A†ˆaAandh…B†ˆaB,respectively.Then,IF…A†ˆIF…B†()fB…0†‡fB…aB†ˆfA…0†‡fA…aA†‡ 2kD…aA;aB†;whereD…aA;aB†ˆ aBÿaAaA
aB:Proof.Itisimmediatefortrapezoidalfuzzynumbers,takingintoaccountthatf…A†ˆ……fA…0†‡fA…aA††=2†
aA:A.Gonzalezetal./Internat.J.Approx.Reason.21(1999)233±256243 is,thesumofbaseimprecisionandmodalimprecisionmustbemod-i®edbythevalue…2=k†
D…aA;aB†forAcanbetransformedintoafuzzynumberBof®xedheight.Besides,ifwepretendtoputuptheheightofA…aAaB†,thenD…aA;aB†ispositiveandthesumofbaseimprecisionandmodalimprecisionofBmustaugment;ontheotherhand,toputdowntheheight…aBaA†,sinceD…aA;aB†isnegative,imprecisionmustbedecreased.Whentheheightis®xed,itisobviousthatimprecisionremainsequal.Sothen,therelationbetweenuncertaintyandimprecisionisthefollowing:·Anincreaseofcertaintymeansanincreaseofimprecision.·Adecreaseofimprecisionmeansadecreaseofcertainty.Proposition3permitsustode®neatransformationassumingthat:1.Modalimprecisionispreserved.2.Theincrease/decreaseofimprecisionisequallydistributedintherightandleftsidesofthefuzzynumberindependentlyfromitsshape.De®nition8.LetA2sbeafuzzynumbersuchthatAˆf…m1;m2;a;b†;aAg;wherem1;m2;aandbareshowninFig.2andaAistheheightofA.Leta2…0;1Š.WewilldenoteD…aA;a†ˆDandde®neTa…A†ˆm1;m2;a‡ Dk;b‡ Dk;aforthoseainwhichthetransformationmakessense.Proposition4.Taisatransformationfortrapezoidalfuzzynumbers.Proof.LetusassumethatthereexistsTa…A†fora2…0;1Š.Then,obviouslyTa…A†2sandh…Ta…A††ˆa.Ontheotherhand,fTa…A†…0†‡fTa…A†…a†ˆfA…0†‡fA…aA†‡ 2kD:ByProposition3,IF…A†ˆIF…Ta…A††andusingDe®nition6,Taisanin-formationons:De®nition9.LetAˆf…m1;m2;a;b†;aAgbeatrapezoidalfuzzynumber.Wede®nethelowestlimitofthetransformationasl…A†ˆmax aAk
a
aA‡1; aAk
b
aA‡1:Itcanbeprovedimmediatelythatl…A†isanumberintheinterval…0;1ŠanditislessorequalthantheheightofA.244A.Gonzalezetal./Internat.J.Approx.Reason.21(1999)233±256 tion5.A2sistransformable()aPl…A†:Proof.A2sistransformable()9Ta…A†andtheexistenceofTa…A†meansthatthespreadsofAarepositiveornull,asitistheonlypossiblerestrictiontobuildit.Therefore,a‡ DkP0b‡ DkP09=;()aPl…A†:Followingthisresult,thetransformationdomainisH…A†ˆ‰l…A†;1Š,whereA2s.SincethelowestlimitofthetransformationisalwayslessorequalthantheheightofA,itisalwayspossibletomakeatransformationforputtinguptheheightofafuzzynumberbut,onthecontrary,thereisaminimumlevelfromwhichtransformationsarenotpossible.InFig.6wehaverepresentedgraphicallythebehaviorofTawhentheheightisdecreasedand,therefore,imprecisionisalsodecreased.Ontheotherhand,inFig.7itisshownhowanincrementofheightproducesanincrementofimprecision.Thisresultagreeswiththefollowingassertion:``Imprecisionanduncertaintycanbeconsideredastwoantagonisticpointsofviewaboutthesamereality,whichishumanimper-fection...andifthecontentsofapropositionismademoreprecise,thenuncer- Fig.6.Transformationthatdecreasesimprecision. Fig.7.Transformationthataugmentsthecertainty.A.Gonzalezetal./Internat.J.Approx.Reason.21(1999)233±256245 willhavetobeaugmented''[6],whichisawaytoenunciatetheprincipleofincompatibilitybetweencertaintyandprecision,establishedbyZadehin[17].Consideringthatf…A†isameasurefortheimprecisionofthefuzzynumberAandthat1ÿh…A†isameasureofitsuncertainty,thisprinciplecanbeenunciatedas:·Iff…A†decreases,thenh…A†decreases.·Ifh…A†increases,thenf…A†increases.FunctionIF®xestheconstantrelationshipbetweenimprecisionandun-certaintyandisassociatedwiththeconceptwerepresentusingafuzzynumber.Ontheotherhand,astransformationstoputuptheheightarealwayspossible,wecanalwaysnormalize(aˆ1)thefuzzynumbersweareworkingwith.Normalizationmeansalossofuncertainty,i.e.,thesecurityonthevalidityofthefuzzyrepresentation.InFig.8itisshownhowthefuzzynumberxkilo-meterswithcertaintydegreelessthan1istransformedintoabiggerfuzzynumberwithcertainty1.Note.Wecanseehow,contrarytothemodelofexpandingimprecisionoverthewholedomain,ourmodelassumesimplicitlythatimprecisionmustbedistributeddependingonthenearnesstotheoriginalconcept.Proposition6(TaProperties).LetA2sanda;b2H…A†:Thenthefollowingpropertiesareverified:1.Th…A†…A†ˆA;2.Ta…Tb…A††ˆTa…A†;3.Th…A†…Ta…A††ˆA:Proof.LetusconsiderAˆf…m1;m2;a;b†;h…A†g,then1.SinceD…h…A†;h…A††ˆ0,thenTh…A†…A†ˆA;2.Tb…A†ˆf…m1;m2;a‡D…h…A†;b†;b‡D…h…A†;b††;bg; Fig.8.Anincreaseofcertaintyproducesanincreaseofimprecision.246A.Gonzalezetal./Internat.J.Approx.Reason.21(1999)233±256 a…Tb…A††ˆm1;m2;a‡ D…h…A†;b†‡D…b;a†k;b‡ D…h…A†;b†‡D…b;a†k;aand,sinceD…h…A†;b†‡D…b;a†ˆD…h…A†;a†thenTa…Tb…A††ˆTa…A†;3.Ittriviallyfollowsfrom(1)and(2).4.2.ThekparameteranditsexperimentalcomputationAswepointedoutattheendofSection3,kparameterisadjusteddependingonthedomainscaletakingintoaccountthatthereisapre-®xedbasescale,forwhichkparameterissetto1.Theideaisthatidenticalfuzzynumbers,thoughexpressedinadi€erentscale,mustprovideexactlythesameinformationamountandthatthisinformationmustbethesamebeforeandafteratrans-formationisapplied.Inthenextsub-sectionswearegoingtoillustratewithsomeexamplestheuseofkparameterinthecasethatthebasescaleisused(kˆ1)andinthecasewhenitisnot.4.2.1.Fuzzyvaluesinthesamedomainscale:kˆ1LetussupposewearegiventhefuzzynumberAˆf…3;4;1;1†;0:5gfortheconcept`Ibelieveitisfewkilometersfaraway'wherethebelievehasbeenquanti®edby0:5,andwewantAtobenormalized,thatis,aˆ1forthepropositionbecome'Itisfewkilometersfaraway'.Sincetheinformationamountbeforeandafterthetransformationmustbethesame,IF…A†ˆIF…T…A††and,byProposition3,D…0:5;1†ˆ…1ÿ0:5†=…10:5†.Therefore,thetrans-formationofAisT…A†ˆ3;4;1‡ 1k
…1ÿ0:5†…10:5†;1‡ 1k
…1ÿ0:5†…10:5†;1ˆf…3;4;2;2†;1gconsideringkˆ1,sincethetransformationisfromkilometersintokilometers.4.2.2.Fuzzyvaluesindi€erentdomainscaleInthepreviouscase,kˆ1aswewereconsideringa®xedscale,butwhatwouldhappenifweweregiventhesameinformationintwodi€erentscales?Inthiscase,kissettothenumberofunitsofthebasescalecontainedinaunitofthescaleweareusing.Forexample,ifthebasescaleiskilometersandthescaleinuseismeters,kˆ1=1000,i.e.thenumberofkilometerscontainedinameter.Ifthebasescaleiscentimetersandthescaleinuseismeters,thenkˆ100.A.Gonzalezetal./Internat.J.Approx.Reason.21(1999)233±256247 usseetheinformationfunctionbehaviorwithanexample.Inthiscase,wemustestablishabasescaleasareference,forexample,kilometers.Inthissituation,letussupposewewanttonormalizethefollowingfuzzynumbergiveninkilometersandmeters,Akmˆf…3;4;1;1†;0:5ginkilometers;Amˆf…3000;4000;1000;1000†;0:5ginmeters:Akmtransformationgivestheresultobtainedabove:T…Akm†ˆf…3;4;2;2†;1g.ForAmtransformation,kparametermustbesetto1=1000(therelationbetweenthescaleweareusingandthebasescale)andtheresultisT…Am†ˆf…3000;4000;2000;2000†;1gwhichisthesameresultobtainedforT…Akm†butrepresentedinthecorrespondingscale.IfweconsidernowthatAkmˆf…3000;4000;1000;1000†;0:5ginkilometers,kparameterwillbe1andthetransformationwillbeT…Akm†ˆf…3000;4000;1001;1001†;1g:Aswecansee,thetransformationfunctionTiscorrectwithrespecttothechangetodi€erentdomainscales.4.2.3.ExperimentalcomputationofthebasescaleInSection4.2.2,wehaveseenhow,thankstokparameter,wecanusedi€erentdomainscalesforthefuzzynumberswearehandling.Butthereisanotherkeypointwhenusingtransformationsandthefollowingquestionarises.Shouldtheincrease/decreasebethesameandnotdependonthemeaningofthefuzzynumber?or,inotherwords,shouldtheincreasebethesameirrespectiveofthefactthatwearedealingwithagesorwithdistance?Uptohere,wehaveconsideredthattheusercouldchangesuchincrease/decreasethroughthescalefactork.Inthissectionwearegoingtosee,inanexperi-mentalway,howwecanadjustthetransformationmodeltoeachproblemdomain.Letusthink,forexample,thatwhendealingwithages,theuserispreparedtoadmitthatapproximately40yearsisanyageintheinterval‰38;42Š(spreadis2)butwhentalkingaboutdistanceinkilometers,approximately7kilometersisanydistanceintheinterval‰6;8Š(spread1).Takingthesecommentsintoaccount,itseemstobereasonablethatkpa-rameterisnotonlydependentonthedomainscalebutalsoontheconceptthefuzzynumberisrepresenting.Inthissense,kshouldhavetheformkˆk0
k;wherekistherelationbetweenthescaleweareusingandthebasescale(asintheprevioussection)andk0isthecorrectionfactorthatdependsonthemeaningandthatallowsustodeterminethereferencescale.Letusseenowhowk0couldbeexperimentallycalculatedconsideringthatweareworkinginabasescale,i.e.kˆ1.248A.Gonzalezetal./Internat.J.Approx.Reason.21(1999)233±256 ussupposewehavetherealvaluef…x;x;0;0†;agwitha1(seeFig.9(a)).Wecouldasktheusertowhatpointheispreparedtorelaxxastoacceptitascompletelytrue.Letussupposetheusersaysthathewillacceptitascompletelytrueifweenlargexatbothsideswithaspreadc,asshowninFig.9(b),becomingtheinitialrealvaluethefuzzynumberf…x;x;c;c†;1g.Ifitisso,cˆ0‡ Dk0andDˆ 1ÿa1
aand,subsequently,valuek0isk0ˆ 1ÿac
awithc6ˆ0:Forexample,ifwehavetherealvaluef…5;5;0;0†;0:9gandtheuseradmitsitascompletelytrueifwetransformitintof…5;5;0:5;0:5†;1g,asshowninFig.10,thenthecomputedvaluefork0isk0ˆ 1ÿ0:90:50:9ˆ 0:10:45ˆ 14:5ˆ0:22: Fig.9.Relaxationofxaccordingtouser'scredibility. Fig.10.Relaxationofvalue5forbeingtrue.A.Gonzalezetal./Internat.J.Approx.Reason.21(1999)233±256249 then,whereaskdependsonlyonthescalewearedoingthetransfor-mationon,k0permitstheusertospecifythechangesmagnitudedependingdirectlyonthedomainoftheproblemwearetackling.Asaconclusionoftheexperimentalcomputationofk,wecansaythat:1.Thetranslationofcertaintyintoimprecisionisvalidfortheconcreteprob-lemwearetackling,asithasbeenelicitatedexperimentallybytheuser.2.Wecanadjustthetransformationtoparticularproblemsanddomains.3.Itisobviousthattheexperimentforobtainingk0shouldberepeatedmanytimesusingdi€erentandseparatedomainvaluesanddi€erentcertaintylev-els.Theideaistoobtainanaveragek0,thatis,ifwedonexperimentsandki0isthevalueobtainedinexperimenti,thenk0ˆXniˆ1ki0
1n:5.ManagementofuncertainfuzzydataAspointedoutintheintroduction,therearetwomainapproachestodealwithuncertainfuzzynumbers.Fromasemanticalpointofviewandforourpurposes,itseemstobemorereasonabletotruncateatlevelathantoextendthesupportsettothewholedomain.Nevertheless,truncatinghastheincon-veniencethatnon-normalizedfuzzynumbersmustbehandled.Tobeabletoworkalwayswithnormalizedfuzzynumbers,weusethetransformationfunctionintroducedinthispapersettingaˆ1.Thisfunctionwillconvertanuncertainimprecisevalue…A;a†toanA0whichisnormalizedandprovidethesameinformationastheoriginalAvaluetruncatedtoanalevel.Alltheseprocessescanbesummarizedasfollows:1.WestartfromafuzzynumberAandacertaintylevelaattachedtoit.2.WetruncateAatlevelaassumingthatAheightisitscertaintyvalue.WebuildAa.3.Totakeadvantageofnormalizedfuzzynumberproperties,wetransformAaintoAT(normalizedversionofAausingourtransformationfunction).Wearegoingtoillustratethisprocesswithanexample.LetAbeatrapezoidalnumberexpressedasAˆf…m1;m2;a;b†;1gwithanassociatedcertaintylevela.AtruncatedtolevelaisAaˆm1…fÿa
…1ÿa†;m2‡b
…1ÿa†;a
a;b
a†;agandthetransformationofAaisATˆm1ÿa
…1ÿa†;m2‡b
…1ÿa†;a
a‡ 1ÿak
a;b
a‡ 1ÿak
a;1:250A.Gonzalezetal./Internat.J.Approx.Reason.21(1999)233±256 canbeprovedthatthedirectexpressionforcomputingATfromAandaisATˆ 1ÿaa Sa A 1ÿaa Supp…A†;whereand arethefuzzyextensionsofsumandproductoperators,SthefuzzynumberexpressedbySˆf…0;0;1=k;1=k†;1gandSupp…A†thesupportsetofAexpressedbythefuzzynumberSupp…A†ˆf…m1ÿa;m2‡b;0;0†;1g.Theadvantageofthisprocessisobvious.Uncertaintyandimprecisionareincludedinthefuzzynumberitselfandthereisnoneedtodeveloporusedi€erentmechanismsfromthosealreadyintroducedfornormalizedfuzzynumbers.5.1.AnapproximatereasoninginterpretationAswementionedintheintroduction,theproblemoftheco-existenceofbothuncertaintyandimprecisioncanbeformulatedbymeansofthecompo-sitionalruleofinference.Infact,someapproachesthatsolvethisproblemmakeuseofwell-knownimplicationfunctionsonthecertaintydegreeandtheruleconsequent Thestartingpointofourapproachisquitedi€erentsincewehavetrans-formedAinsuchawaythattheinformationprovidedbyAtruncatedatthecertaintylevelispreserved.Anyway,wearegoingtoprovethatthisapproachisveryclosetotheapproximatereasoningone.Infact,weareprovingthatforthecaseoftrapezoidalfuzzynumbers,thewholeprocessoftransformationappliedisanimplicationfunctioninthesenseofTrillasandValverde[15].Todothat,wearegoingto®ndtheexpressionwhichsummarizesthewholeprocessoftransformationofAwithcertaintyaintoAT.Thisexpressioniscalculatedinthefollowingproperty.Property1.LetAbeatrapezoidalfuzzynumberexpressedasAˆ…m1;m2;a;b†andletussupposethattheuncertaintylevelofAisa.LetAtruncatedtolevelbeAaandthenormalizedversionofAabeAT.Intheseconditions,itisverifiedthatA.Gonzalezetal./Internat.J.Approx.Reason.21(1999)233±256251 v2RjlA…v†�0;lAT…v†ˆ1a6lA…v†;min aa
a‡ Dk
lA…v†‡ Da
a‡D‡ Dk; bb
a‡ Dk
lA…v†‡ Db
a‡D‡ Dk();8�����:whereDˆD…a;1†ˆ…1ÿa†=aandkisthescalefactor.Proof.Provingthispropertyisquitedirecttakingintoaccountthatthea-cutofAhasthefollowingmembershipfunctionlAa…v†ˆaiflA…v†Pa;lA…v†otherwise:Sothen,ATˆf…m1a;m2a;a
a‡…D=k†;b
a‡…D=k††;1g,wherem1aˆm1ÿa:…1ÿa†ˆm1ÿa‡a
aandm2aˆm2‡b
…1ÿa†ˆm2‡bÿb
a.ThegraphicalrepresentationofthesetrapezoidalfuzzynumbersisshowninFig.11.FromFig.11wecandirectlyobtainthatlAT…v†ˆ1ifa6lA…v†.Letusseenowthecasewherethisconditionisnotheld.Inthissituation,a�lA…v†�0andv2…m1ÿa;m1aŠorv2…m2a;m2a‡bŠ.Besides, Fig.11.GraphicalrepresentationofA,AaandAT.252A.Gonzalezetal./Internat.J.Approx.Reason.21(1999)233±256 A…v†ˆ vÿ…m1ÿa†aifv2m1ÿa;m1a…Š; …m2‡b†ÿvbifv2m2a;m2‡b…Š:8����Finally,lAT…v†ˆ vÿ…m1ÿ…aÿD=k††a
a‡D=kifv2m1ÿ…aÿD=k†;m1a…Š; …m2‡b‡D=k†ÿvb:a‡D=kifv2m2a;m2‡b‡D=k…Š:8����:Makingoperations,wehavelAT…v†ˆ vÿ…m1ÿa†
aa
…a
a‡D=k†‡ D=ka
a‡D=kˆ aa
a‡D=k
lA…v†‡ D=ka
a‡D=kifv2m1ÿ…aÿD=k†;m1a…Š; …m2‡bÿv†
bb
…b
a‡D=k†‡ D=kb
a‡D=kˆ bb
a‡D=k
lA…v†‡ D=kb
a‡D=kifv2m2a;m2‡b‡D=k…Š:8���������Whenv2…m1ÿ…aÿD=k†;m1aŠthen, aa
a‡D=k
lA…v†‡ D=ka
a‡D=k6 bb
a‡D=k
lA…v†‡ D=kb
a‡D=kandwhenv2…m2a;m2‡b‡D=kŠthen, bb
a‡D=k
lA…v†‡ D=kb
a‡D=k6 aa
a‡D=k
lA…v†‡ D=ka
a‡D=k:MakingthesuitableoperationsweobtaintheexpressionofthegiventransformationfunctionlAT…v†ˆ1ifa6lA…v†;min a
a
k
lA…v†ÿa‡1a
a2
kÿa‡1; b
a
k
lA…v†ÿa‡1b
a2
kÿa‡1:8��:Thisresultledustode®neastransformationfunctionforthefuzzynumberAthefunction8x;y2…0;1Š;I…x;y†ˆ1ifx6y;min a
x
yÿx‡1a
x2ÿx‡1; b
x
yÿx‡1b
x2ÿx‡1:8:Therelationshipofthisresultwiththeapproximatereasoningisshowninthefollowingproperty.A.Gonzalezetal./Internat.J.Approx.Reason.21(1999)233±256253 y2.ThefunctionI…x;y†previouslydefinedisanimplicationfunctioninthesenseofTrillasandValverde[15].Proof.1.Iisadecreasingfunctioninx.Thatis8x;x0jx6x0;I…x0;y†6I…x;y†.Infact,±ifx;x06ytheresultisimmediate.±ifx6yandx0�yitisalsoimmediatesinceI…x0;y†61.±ifx;x0�ytheexpressionis…a
y
xÿx‡1†=…a
x2ÿx‡1†giventhatx�y.Inthiscase,thenumeratorincreaseswithxmoreslowlythanthedenominatorand,therefore,thequotientisadecreasingfunction.2.Iisanincreasingfunctioniny.Thatis,8y;y0jy6y0;I…x;y†6I…x;y0†.Thisresultisobvioussincebothfunctions(numeratoranddenominator)arelin-earwithrespecttoywithpositivecoecients.Therefore,thefunctionwillbeincreasinginy.3.I…0;y†ˆ18y;06yandI…0;y†ˆ1:4.I…1;y†ˆy:Consideringthefollowingexpressionforxˆ1,theresultisdi-rectlyobtained.I…x;y†ˆmax0;min a
yÿ1‡1aÿ1‡1; b
yÿ1‡1bÿ1‡1ˆy:5.I…x;I…y;z††ˆI…y;I…x;z††(propertyofinterchangeability).Therearetwocases:·y6z.InthiscasetherightsideofthepropertybecomesI…x;1†ˆ1byap-plyingproperties1and4,sinceI…x;z†PI…1;z†ˆz.Consequently,y6z6I…x;z†.Ontheotherhand,theleftsideisalsoequalto1,sincewehavethat:y6z,I…y;z†ˆ1andI…x;1†ˆ1,whichprovestheequality.·zy.Therearetwopossibilities:±xPy�z.Wearegoingtoprovethat:x6I…y;z†andy6I…x;z†,andsubsequentlythatI…x;I…y;z††ˆI…y;I…x;z††ˆ1.Infact,thefollowingex-pressionsareheld:xˆI…1;x†6I…y;x†6I…y;z†andyˆI…1;y†6I…x;y†6I…x;z†.±y�zPx.AccordingtoIproperties,I…x;z†ˆ1)I…y;I…x;z††ˆ1.WearegoingtoprovethatI…x;I…y;z††ˆ1.Todothat,wemustprovethatx6I…y;z†.Letussupposethecontrary:x�I…y;z†,thenI…1;x†�I…y;z†PI…y;x†andthisleadustotheexpressionI…1;x†�I…y;x†;y61whichisincontradictionwiththedecreasingcharacterofI:Sothen,Iisanimplicationfunction.6.ConclusionsTheproblemofimprecisionanduncertaintymanagementthroughfuzzynumbershasbeenaddressed.Fuzzynumbersareausefultoolforrepresenting254A.Gonzalezetal./Internat.J.Approx.Reason.21(1999)233±256 informationbut,inmanycases,thisimpreciseinformationcanbegivenwithanuncertaintydegreeInthesecases,thefuzzynumberincludesadditionalinformationaboutthecon®denceofthisinformation.Inthispaperwehaveproposedamethodthatallowsustotransformthewholeinformation(imprecision‡uncertainty)intoanewfuzzynumberinsuchawaythat:·Itmaintainsaprincipleofdistributionoftheimprecisionbasedonametricthattakesintoaccountthedistancetotheoriginalconcept,thatis,thecloseranelementistotheconceptthemoreistheincreaseofitsmembershiptothementionedconcept(basedonZadeh'sprinciple).·Itpermitstoadjusttheresultstotheusers'pointofviewbyusingthescalefactor,i.e.wecanadjustthetransformationtoparticularproblemsanddo-mains.·Itisinterpretedfromtheapproximatereasoningpointofviewand,subse-quently,itguaranteessoundresults.·Itiseasytoimplementand,therefore,tobeincludedasadatapreprocessingmodule.·Themethod,notonlynormalizesbutequalizes.Thisisanimportantchar-acteristicforproblemswhereanacceptedlevelofuncertaintyexists(notnec-essarily1).·Ifbissetto1,allthesoftwaredevelopedfornormalizedfuzzysetsisre-us-ableandnonewversionsarenecessarytotreatuncertainty.Asfutureavenuesforresearchwecanmention:1.Togiveageneralexpressionforfuzzynumbers.Inthispaperwehaveonlyconsideredthecaseoftrapezoidalfuzzynumbersbuttheresultscouldbegeneralizedforanykindoffuzzynumber.2.Touselinguisticuncertaintyinsteadofuncertaintylevels.3.Tostudyhowtransformationsa€ecttheresultsobtainedfromarithmeticoperations,matchingorrankingoffuzzyvalues.References[1]G.Bordogna,G.Pasi,Managementoflinguisticquali®cationofuncertaintyinafuzzydatabase,in:ProceedingsofIPMU'98,SeventhInternationalConferenceonInformationProcessingandManagementofUncertaintyinKnowledge-basedSystems,Paris,France,1998,pp.435±442.[2]Z.Cao,D.Park,A.Kandel,Investigationsontheapplicabilityoffuzzyinference,FuzzySetsandSystems49(1992)151±169.[3]J.L.Castro,J.J.Castro,J.M.Zurita,Thein¯uenceofimplicationfunctionsinfuzzylogiccontrollerswithdefuzzi®cation®rstandaggregationafter,in:SeventhIFSAWorldCongress,Prague,1997,pp.237±242.[4]M.Delgado,J.L.Verdegay,M.A.Vila,Onvaluationandoptimizationproblemsinfuzzygraphs:Ageneralapproachandsomeparticularcases,ORSAJournalonComputing2(1990)74±83.A.Gonzalezetal./Internat.J.Approx.Reason.21(1999)233±256255 M.Delgado,M.A.Vila,W.Voxman,Onacanonicalrepresentationoffuzzynumbers,FuzzySetsandSystems93(1998)125±135.[6]D.Dubois,Modelesmathematiquesdel'imprecisetdel'incertainenvued'applicationsauxtechniquesd'aidealadecision,Ph.Dthesis,UniversiteScienti®queetMedicaledeGrenoble,1983.[7]D.Dubois,H.Prade,Fuzzynumbers.anoverview,in:Bezdek(Ed.),TheAnalysisofFuzzyInformation,CRSPress,BocaRaton,1985.[8]D.Dubois,H.Prade,PossibilityTheory,PlenumPress,NewYork,1985.[9]D.Dubois,H.Prade,Themeanvalueofafuzzynumber,FuzzySetsandSystems24(1987)279±300.[10]J.A.Goguen,L-fuzzysets,JournalofMathematicalAnalalysisandApplications18(1967)145±174.[11]A.Gonzalez,MetodosSubjetivosparalaComparaciondeNumerosDifusos,Ph.Dthesis,GranadaUniversity,1987.[12]A.Gonzalez,Astudyoftherankingfunctionapproachthroughmeanvalues,FuzzySetsandSystems35(1990)29±41.[13]KampedeFeriet,J.Latheoriegeneraliseedel'informationetlemesuresubjetivedel'information,in:KampedeFeriet,C.F.Picard(Eds.),LecturenotesinMath:Theoriesdel'information,vol.398,Springer,Berlin,1973,pp.1±28.[14]O.Pons,J.C.Cubero,A.Gonzalez,M.A.Vila,Uncertainfuzzyvaluesstillintheframeworkof®rstorderlogic,in:ProceedingsofSecondInternationalWorkshoponLogicProgrammingandSoftComputing:TheoryandApplications,Manchester,UK,1989.[15]E.Trillas,L.Valverde,Onimplicationandindistinguishabilityinthesettingoffuzzylogic,J.Kacpryzk,R.R.Yager(Eds.),ManagementDecisionSupportSytemsusingFuzzySetsandPossibilityTheory,VerlagTUVRheinlan,1985,pp.189±212.[16]R.R.Yager,Measurementofpropertiesonfuzzysetsandpossibilitydistributions,E.P.Klement(Ed.),ProceedingsoftheThirdInternationalSeminaronFuzzySetTheory,JohannesKeplasUniversityLiuz,1981,pp.211±222.[17]L.A.Zadeh,Outlineofanewapproachtotheanalysisofcomplexsystemsanddecisionprocesses,IEEETransactionsSystems,ManandCybernetics3(1973)28±44.256A.Gonzalezetal./Internat.J.Approx.Reason.21(1999)233±256