Fuzzy Sets and Their Extensions: Representation, Aggregation and Models PDF
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Fuzzy Sets and Their Extensions: Representation, Aggregation and Models Studiesin Fuzziness andSoft Computing,Volume 220 Editor-in-chief Prof.JanuszKacprzyk SystemsResearchInstitute PolishAcademyofSciences ul.Newelska6 01-447Warsaw Poland E-mail:[email protected] Furthervolumesofthisseriescanbe Models,2007 foundonourhomepage:springer.com ISBN978-3-540-68994-2 Vol.206.A.Sengupta(Ed.) Vol.214.IrinaGeorgescu Chaos,Nonlinearity,Complexity,2006 FuzzyChoiceFunctions,2007 ISBN978-3-540-31756-2 ISBN978-3-540-68997-3 Vol.207.IsabelleGuyon,SteveGunn, Vol.215.PaulP.Wang,DaRuan, MasoudNikravesh,LotfiA.Zadeh(Eds.) EtienneE.Kerre(Eds.) FeatureExtraction,2006 FuzzyLogic,2007 ISBN978-3-540-35487-1 ISBN978-3-540-71257-2 Vol.208.OscarCastillo,PatriciaMelin, Vol.216.RudolfSeising JanuszKacprzyk,WitoldPedrycz(Eds.) TheFuzzificationofSystems,2007 HybridIntelligentSystems,2007 ISBN978-3-540-71794-2 ISBN978-3-540-37419-0 Vol.217.MasoudNikravesh,Janusz Vol.209.AlexanderMehler, Kacprzyk,LoftiA.Zadeh(Eds.) ReinhardKo¨hler ForgingtheNewFrontiers:Fuzzy AspectsofAutomaticTextAnalysis,2007 PioneersI,2007 ISBN978-3-540-37520-3 ISBN978-3-540-73181-8 Vol.218.MasoudNikravesh,Janusz Vol.210.MikeNachtegael,Dietrich Kacprzyk,LoftiA.Zadeh(Eds.) VanderWeken,EtienneE.Kerre, ForgingtheNewFrontiers:Fuzzy WilfriedPhilips(Eds.) PioneersII,2007 SoftComputinginImageProcessing,2007 ISBN978-3-540-73184-9 ISBN978-3-540-38232-4 Vol.219.RolandR.Yager,LipingLiu(Eds.) Vol.211.AlexanderGegov ClassicWorksoftheDempster-ShaferTheory ComplexityManagementinFuzzy ofBeliefFunctions,2007 Systems, 2007 ISBN978-3-540-38883-8 ISBN978-3-540-25381-5 Vol.220.HumbertoBustince, Vol.212.ElisabethRakus-Andersson FranciscoHerrera,JavierMontero(Eds.) FuzzyandRoughTechniquesinMedical FuzzySetsandTheirExtensions: DiagnosisandMedication,2007 Representation,AggregationandModels. ISBN978-3-540-49707-3 IntelligentSystemsfromDecisionMakingto Vol.213.PeterLucas,Jose´A.Ga´mez, DataMining,WebIntelligenceandComputer AntonioSalmero´n(Eds.) Vision,2008 AdvancesinProbabilisticGraphical ISBN978-3-540-73722-3 · · Humberto Bustince Francisco Herrera Javier Montero Editors Fuzzy Sets and Their Extensions: Representation, Aggregation and Models Intelligent Systems from Decision Making to Data Mining, Web Intelligence and Computer Vision With126Figuresand44Tables Foreword by Didier Dubois HumbertoBustince FranciscoHerrera DepartmentofAutomaticsandComputation DepartmentofComputerScience UniversidadPu´blicadeNavarra andArtificialIntelligence(DECSAI) CampusArrosadias/n UniversityofGranada 31006Pamplona PeriodistaDanielSaucedoArandas/n SPAIN 18071Granada Email:[email protected] SPAIN Email:[email protected] JavierMontero FacultyofMathematics UniversidadComplutense PlazadelasCiencias3 28040Madrid SPAIN Email:[email protected] LibraryofCongressControlNumber:2007932776 ISSNprintedition:1434-9922 ISSNelectronicedition:1860-0808 ISBN 978-3-540-73722-3 SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violations areliableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com (cid:2)c Springer-VerlagBerlinHeidelberg2008 Theuseofgeneral descriptive names,registered names,trademarks, etc. inthis publication does not imply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotective lawsandregulationsandthereforefreeforgeneraluse. Typesetting:IntegraSoftwareServicesPvt.Ltd.,India Coverdesign:WMXDesign,Heidelberg Printedonacid-freepaper SPIN:11679158 89/3180/Integra 5 4 3 2 1 0 Foreword Fuzzy Sets and Their Extensions: Representation, Aggregation and Models IntelligentSystems fromDecisionMakingtoDataMining, Web Intelligenceand ComputerVision Fuzzy sets are now more than 40 years old, and have come of age. However, the developmentoffuzzysettheoryatthetheoreticallevel,anditssuccessfulapplica- tionstoscienceandtechnologyhaveoftenruninisolation.Onlyalittlepartofthe theoreticalapparatuswaseffectivelyusedinpastapplications.Themostprominent ones,namelyfuzzyrule-basedmodelingandcontrolengineering,weredirectlyin- spiredfromaseminalpaperbyLotfiZadehin1973,suggestinghowtouseexpert knowledgeforsynthetizingcontrollaws, andfromthe firstexperimentspublished by Abe Mamdani. Later in the eighties, when spectacular applications were blos- soming in Japan, fuzzy rule-based systems were systematized and simplified by MichioSugenoandcolleagues,andbecameabasicapproachtonon-linearsystem modelingand control,soon hybridizedwith neuralnetworksin the nineties. Thus fuzzy systems significantly contributed to the raise of computational intelligence, and a lot of learning techniquesfor the constructionof (supposedly interpretable) fuzzymodelsfromdataweredevelopedundertheflagofsoftcomputing. Evenifthisareawasquitesuccessful,itispatentthattherole,inthesuccessof fuzzylogic,ofnewfuzzyset-relatedconceptsdevelopedquiteatthesametimein themathematicalnicheofthefuzzysetcommunitywaslimited.Towit,thenotionof fuzzyruleisnowmuchbetterunderstood,butfuzzyextensionsofmaterialimplica- tionshavenotbeenpopularinsystemsengineeringsofar.Othernotions,likeaggre- gationoperations,fuzzyintervals,fuzzypreferencerelations,possibility measures havealsoacquiredstrongtheoreticalunderpinningsinthemeantime.Theendofthe ninetieshavealsowitnessedtremendousprogressinthefoundationsofformalfuzzy logic, due to the impulse given by reputed logicians like Petr Hajek and Daniele Mundici.Yettheyonlyhad a limitedimpactondomainslike databasesandinfor- mation retrieval, risk analysis, artificial intelligence, mathematical programming, andmultiplecriteriadecision-making,wheretheiruselooksnatural.Onereasonis thelackofcommunicationbetweenmathematicsandengineeringinmanycases. From scanning its contents, this book looks like an outline of what could be the nextgenerationof fuzzylogic applications.Strikinglyenoughit doesnotdeal withsystemsengineeringnorsoftcomputing.Instead,newexcitingtopicsarepro- posed like data miningand web intelligence,as well as more traditionaloneslike decision-making and computer vision, for which fuzzy set methods are available for a long time, but the impact of fuzzy sets was clearly marginal so far. With the tremendous gain in maturity of fuzzy set mathematics, observed in the last v vi Foreword twenty years, no doubtsthat new generationsof researchers,trained with the new results,willproposenewtechniquesforsuchapplicationareasasappearinginthis book. The extensive series of chapters in this book, devoted to aggregation oper- ators, their axiomatization, their construction and their identification from data is verysymptomaticoftherenewalofresearchtopicsinfuzzysettheory. The other part of the theoretical section of this book raises some questions. Severalchaptersaretypicalofthenewtrendstowardshybridconstructions,namely fuzzy sets with rough sets, and fuzzy sets with random sets. The other two top- ics, that is fuzzy sets of higher order and Atanassov’s intuitionistic fuzzy sets are more problematic at this stage of their development.Type-two fuzzy sets as well asinterval-valuedfuzzysetsareanoldconstructsuggestedbyZadehalreadyinthe seventies.Theyarenaturalconceptsaddressingtheapparentparadox,facedbystan- dardfuzzysets,ofmodelingimpreciseconceptsusingprecisemembershipgrades. Of course, they fall under the usual objection of regression to infinity (why then should the imprecise membershipgrades be modeled by type 1 fuzzy sets ?). But thisisnotatallavalidpracticalobjection.Afirstmoreactualdifficultyisthattype-2 fuzzysystemsmayinvolvemanymoretuningparametersthantype-1fuzzysystems. Sotheymaynotbeaveryconcisetool.Anothermorebasicdifficultyliesintheex- tensionoffuzzyconnectivesviatheextensionprinciple,studiedintheseventiesby MizumotoandTanaka(andDuboisandPrade!),thathidesdependencyphenomena. If an interval-valuedor type-2 fuzzy set represents uncertaintyaboutmembership grades,theyareuncertaintype-1fuzzysets,notfuzzysetswithmembershipgrades in a more complex scale than the unit interval. So for instance even if the mem- bershipgradeμ (x)isill-known,weneverthelessknowthatμ (x)+μc(x) = 1, A A A whereAcisthecomplementofA,hencemin(μ (x),μc(x))≤0.5.But,ifμ (x)∈ A A A [a,b],computingtherangeofthemembershipgradeofx in A∪ Ac as min([a,b],1−[a,b])=[min(a,1−b),min(b,1−a)] usingtheextensionprincipleonfuzzyconnectives,cannotretrievethisupperbound, as we may have min(b,1-a)> 0.5. This is one more example of the lack of truth- functionalityinthepresenceofuncertainty.Thisisclearlyalimitationofthetruth- functionalcalculusof interval-valuedfuzzysets (and a fortiori, type-2fuzzy sets) oneshouldbeawareof.Interestingly,theinterpretationofafuzzysetasanill-known (random)crispsetleadstothesamelossoftruth-functionality,asexplainedinthe chapterbyLawry. The case of intuitionistic fuzzy sets, advocatedby Atanassov,in this book,has beendiscussedelsewhere.Theysufferfromalackofconnectionwithcurrentintu- itionisticlogic(despitethefactthatsomeideaswereborrowedbytheirfounderfrom early intuitionism). Moreover, the algebraic operations proposed for them make themformallyequivalenttointerval-valuedfuzzysets(hencetheyalsosufferfrom the difficulty pointed out in the previous paragraph). Yet, the idea of Atanassov, namelyevaluatingdegreesofmembershipanddegreesofnon-membershipinasep- arate way, is very important, as it refers to the phenomenonof bipolarity.Indeed, it has been observedby cognitivescientists that the humanbrain tends to process positive and negative aspects of information separately. More work is needed to Foreword vii clarifythespecificroleofAtanassovmembership/non-membershippairs,asdistinct fromtheolderinterval-valuedfuzzysets,andfromwhatisnowadaysreferredtoas intuitionism. The second part of the book will be very useful to charter the most promis- ing future applications areas for fuzzy set theory. In the case of decision-making, preferencemodelingwillcertainlybenefitfromtheprogressmadeinthetheoryof fuzzyrelations,becausethenaturalconceptofgradualpreferenceisnotreallyen- counteredinclassicaldecisiontheorybutformereutilityfunctions.Thesuccessof fuzzyreferencerelationswilldependonthedevelopmentofmeasurement-theoretic foundationsforthem,andtheexistenceoftoolsthatdemonstratetheirsimplicityof useandefficiencyinsolvingissuescrisprelationscannotaddress(likeinvotingthe- ories).Anotherchallengeinfuzzydecisiontheoryisitscapabilitytofoundlinguis- tic approaches.Here,thedifficultyisthesame asinqualitativereasoningatlarge: findingacompromisebetweenmathematicallywell-behavedbutpoorlyexpressive frameworks(aslikethesigncalculus),andmoreexpressivemathematically-illbe- havedsettings(likeabsoluteordersofmagnitude).Inparticular,itisnotclearthat linguisticvariablesgroundedonnon-measurablenumericalscales(likeusing[0,1] toevaluateabstractnotionslikebeauty)arethewaytogo.Severalchaptersofferthe stateoftheartinlinguisticdecision-making. Dataminingandweb intelligenceareclearlynewtopicswherefuzzysetshave a roleto play.Thefirst data-miningtoolspresupposeda Booleanworld.Adapting them to numerical attributes is problematic if crisp partitions of attribute ranges areused.Fuzzysetsmayaddresstheissueofsensitivitytothresholds,evenifthis advantageneedstobeproperlyassessed.Neverthelesstheconceptoffuzzyassoci- ation ruleand moregenerallyfuzzyrule appearsto be more flexiblethanits crisp counterpart.Thewebisalsoanexcellentopportunitytocombineseveralfuzzyset methods developed in the recent years: formal fuzzy logic for the description of fuzzyontologies,fuzzypreferencemodelingandaggregationtechniquesforsearch engines,linguisticinformationprocessinginrecommendersystemsandotherkinds ofe-services.Thesetopicsarewelldocumentedinthebook. Lastly,theselectionofpapersincomputervisionalsowitnesseshowfuzzytech- niqueshaveslowlybutstronglyenteredthevarioustool-boxesofthefield.Thatthis areacouldbenefitfromfuzzysetswaspointedoutquiteearlybythelateA.Rosen- feld in the seventies, with continued efforts until recently. Filtering, thresholding and segmentation techniques, colour processing and pixel classification methods, amongothersubproblemsincludefuzzyingredientsaswitnessedbyvarioussurvey chaptersinthisvolume. Systemsandcontrolengineersusedtobe themainadvertisersoffuzzysets till the endof the twentieth century.Thisbooksuggeststhatthe new centurywill see fuzzyset theorybecomingan importantmethodologyfor informationscience and engineeringatlarge. March1,2007 DidierDubois Preface Fuzzy Sets and Their Extensions: Representation, Aggregation and Models IntelligentSystems fromDecisionMakingtoDataMining, Web Intelligenceand ComputerVision Thisbookhasitsoriginsintheinvitedtalkspresentedatthe“SecondInternational Workshop of Artificial Intelligence, Fuzzy Logic and Computer Vision”, held in Pamplona,Spain(November30toDecember2,2005).Theseinvitedtalkscovered thedifferenttasksthatwemusttakeintoconsiderationfor“fuzzylogicbasedreal applications”,fromfuzzylogicfoundations(representationandaggregationopera- tors)toinformationfusionandspecificmodelconstructions.Althoughmodelcon- structiondependsonthekindofproblemandpresentsagreatvarietyofsituations, the Workshop paid special attention to computer vision applications, and it also includedsomeinvitedtalksondecision-making,webintelligenceanddatamining. Duringtheconferencewerealizedthattherewasscientificdemandforabookof- feringagoodstate-of-the-artcollectionofpaperstogetherwithawide-rangingview ofapplications,insuchawaythatreaderscouldfind,inasinglevolume,thethree mostimportanttaskstotakeintoconsiderationforfuzzylogicrealapplications:rep- resentation,aggregationproceduresandavarietyofmodelsindifferentapplication areas,consideringthedifferentsemanticsforfuzzymembershipfunctionsthatexist intheliterature(similarity,preferenceanduncertainty). Thebookhasbeenconceivedaccordingtoafixedscheme,coveringawideview ofpast,presentandfutureresearchrelatedtothisfield,togetherwithastrictselec- tionofprestigiousauthorsasaguaranteeforqualitypapers,butstillmaintaininga standardanonymouspeer-reviewforeverypaper(otherresearchers,alsoprestigious in each topic, kindly accepted to collaborate in this project). We paid attention to non-standardrepresentationsthatextendfuzzysets,aggregationprocedures,andthe wholeprocessofintelligentinformationmanagementusingfuzzylogic,focusingon four important application areas: decision-making, data mining, web intelligence andcomputervision. Thepresentbookbringstogethermanyofthementionedinvitedtalksatthe“Sec- ond InternationalWorkshop of Artificial Intelligence, Fuzzy Logic and Computer Vision”,plusacollectionofwellrecognizedresearchercontributions,withtheaim ofpresentinganextensivebackgroundoneachtopic.Itcollectsasetofpapersand gives a triple perspective: papers for revision, papers prospecting these areas and paperspresentinginterestingnovelapproaches. Onceallthe papershadbeenrevisedandcorrected,Prof.D. Duboiskindlyac- ceptedtowrite the Preface.Thebookcontains34chaptersdividedinto twoparts: ix x Preface Part I devoted to foundation issues (Sect. 1 and Sect. 2) and Part II to the four applicationareaspreviouslymentioned(Sects.3,4,5and6). More specifically, the first part is divided into two sections. Section 1 contains fourreviewpapersintroducingsomenon-standardrepresentationsthatextendfuzzy sets(type-2fuzzysets,Atanassov’sIFS,fuzzyroughsetsandcomputingwithwords under the fuzzy sets perspective). Section 2 contains six review and prospect pa- persthatrevisedifferentaggregationissuesfromatheoreticalandpracticalpointof view.Thesecondpartisdividedintofoursections.Section3isdevotedtodecision- making,containingsevenpapersthatshowhowfuzzysetsandtheirextensionsare animportanttoolformodelingchoiceproblems(e.g.,sensoryevaluation,preference representation, group decision making, consensus and voting systems). Section 4 collectseightpapersthatcoverdifferentaspectson theuse offuzzysets andtheir extensionsindatamining,classification,associationrules,non-supervisedclassifi- cation,subgroupdiscovery,etc.,givinganillustrativerevisionofthestateoftheart in the subject. Section 5 is devoted to the emergenttopic of web intelligence and containsfourpapersthatshowtheuseoffuzzysettheoryincertainproblemsthat we can tackle underthis heading(informationretrieval,web meta-searchengines, e-servicesandrecommendersystems).Section6isdevotedtotheuseoffuzzysets andtheirextensionsinthefieldofcomputervision,presentinghowthesecanbea usefultoolinthisarea(imagethresholding,segmentation,fuzzymeasuresandcolor processing). Webelievethatthisvolumepresentsanup-to-datestateofcurrentresearchinthe useoffuzzysetsandtheirextensionsinthewholeprocessofintelligentinformation management. It will be useful to non-expert readers, whatever their background, who are keen to learn more about this area of research. It will also support those specialistswhowishtodiscoverthelatestresultsaswellasthelatesttrendsinthe mentionedareas. Finally,we wouldlike to expressourmostsincere gratitudeto Springer-Verlag andinparticulartoProf.J.Kacprzyk(editor-in-chiefoftheseries“StudiesinFuzzy- nessandSoftComputing”),forhavinggivenustheopportunitytopreparethetext andforhavingsupportedandencouragedusthroughoutitspreparation.Wewould also like to acknowledge our gratitude to all those who have contributed to the booksby producingarticlesthat we considerto be of the highestquality.We also like to mention the somewhat obscure and altruistic, though absolutely essential, task carried out by a group of referees (all the contributions have been reviewed by two referees), who, through their comments, suggestions, and criticisms, have contributedtoraisingthequalityofthisvolume. March1,2007 H.Bustince F.Herrera J.Montero Contents PARTI FOUNDATIONS:REPRESENTATIONANDAGGREGATION 1.ExtendingFuzzySetsRepresentation Type-2FuzzyLogicandtheModellingofUncertainty SimonCouplandandRobertJohn ................................... 3 MyPersonalViewonIntuitionisticFuzzySetsTheory KrassimirT.Atanassov............................................ 23 HybridizationofFuzzyandRoughSets:PresentandFuture EricC.C.Tsang,QingCaiChen,SuyunZhao,DanielS.Yeung andXizhaoWang ................................................ 45 AnOverviewofComputingwithWordsusingLabelSemantics JonathanLawry.................................................. 65 2.Aggregation OntheConstructionofModelsBasedonFuzzyMeasuresandIntegrals Vicenc¸Torra .................................................... 89 InterpolatoryTypeConstructionofGeneralAggregationOperators GlebBeliakovandTomasaCalvo ................................... 99 AReviewofAggregationFunctions RadkoMesiar,AnnaKolesa´rova´,TomasaCalvo andMagdaKomorn´ıkova´ .......................................... 121 IdentificationofWeightsinAggregationOperators TomasaCalvoandGlebBeliakov ................................... 145 LinguisticAggregationOperators:AnOverview ZeshuiXu ...................................................... 163 xi
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