Fuzzy Sets and Their Extensions: Representation, Aggregation and Models

672 Pages·2008·10.681 MB·English

by  Simon Coupland, Robert John (auth.), Humberto Bustince, Francisco Herrera, Javier Montero (eds.)

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
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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
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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