Table Of Contentmore information – www.cambridge.org/9780521519267
AggregationFunctions
Aggregationistheprocessofcombiningseveralnumericalvaluesintoasingle
representativevalue,andanaggregationfunctionperformsthisoperation.These
functionsarisewhereveraggregatinginformationisimportant:appliedandpure
mathematics(probability,statistics,decisiontheory,functionalequations),
operationsresearch,computerscience,andmanyappliedfields(economicsand
finance,patternrecognitionandimageprocessing,datafusion,etc.).
Thisreadablebookprovidesacomprehensive,rigorousandself-contained
expositionofaggregationfunctions.Classesofaggregationfunctionscovered
includetriangularnormsandconorms,copulas,meansandaverages,andthose
basedonnonadditiveintegrals.Thepropertiesofeachmethod,aswellastheir
interpretationandanalysis,arestudiedindepth,togetherwithconstructionmethods
andpracticalidentificationmethods.Specialattentionisgiventothenatureofscales
onwhichvaluestobeaggregatedaredefined(ordinal,interval,ratio,bipolar).Itis
anidealintroductionforgraduatestudentsandauniqueresourceforresearchers.
encyclopedia of mathematics and its applications
AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridge
UniversityPress.Foracompleteserieslistingvisit
http://www.cambridge.org/uk/series/sSeries.asp?code=EOM
66 D.Cvetkovic,P.RowlinsonandS.SimicEigenspacesofGraphs
67 F.Bergeron,G.LabelleandP.LerouxCombinatorialSpeciesandTree-LikeStructures
68 R.GoodmanandN.WallachRepresentationsandInvariantsoftheClassicalGroups
69 T.Beth,D.Jungnickel,andH.LenzDesignTheory1,2ndedn
70 A.PietschandJ.WenzelOrthonormalSystemsforBanachSpaceGeometry
71 G.E.Andrews,R.AskeyandR.RoySpecialFunctions
72 R.TicciatiQuantumFieldTheoryforMathematicians
73 M.SternSemimodularLattices
74 I.LasieckaandR.TriggianiControlTheoryforPartialDifferentialEquationsI
75 I.LasieckaandR.TriggianiControlTheoryforPartialDifferentialEquationsII
76 A.A.IvanovGeometryofSporadicGroupsI
77 A.SchinzelPolynomialswithSpecialRegardtoReducibility
78 H.Lenz,T.Beth,andD.JungnickelDesignTheoryII,2ndedn
79 T.PalmerBanachAlgebrasandtheGeneralTheoryof∗-AlbegrasII
80 O.StormarkLie’sStructuralApproachtoPDESystems
81 C.F.DunklandY.XuOrthogonalPolynomialsofSeveralVariables
82 J.P.MayberryTheFoundationsofMathematicsintheTheoryofSets
83 C.Foias,O.Manley,R.RosaandR.TemamNavier–StokesEquationsandTurbulence
84 B.PolsterandG.SteinkeGeometriesonSurfaces
85 R.B.ParisandD.KaminskiAsymptoticsandMellin–BarnesIntegrals
86 R.McElieceTheTheoryofInformationandCoding,2ndedn
87 B.MagurnAlgebraicIntroductiontoK-Theory
88 T.MoraSolvingPolynomialEquationSystemsI
89 K.BichtelerStochasticIntegrationwithJumps
90 M.LothaireAlgebraicCombinatoricsonWords
91 A.A.IvanovandS.V.ShpectorovGeometryofSporadicGroupsII
92 P.McMullenandE.SchulteAbstractRegularPolytopes
93 G.Gierzetal.ContinuousLatticesandDomains
94 S.FinchMathematicalConstants
95 Y.JabriTheMountainPassTheorem
96 G.GasperandM.RahmanBasicHypergeometricSeries,2ndedn
97 M.C.PedicchioandW.Tholen(eds.)CategoricalFoundations
98 M.E.H.IsmailClassicalandQuantumOrthogonalPolynomialsinOneVariable
99 T.MoraSolvingPolynomialEquationSystemsII
100 E.OlivieriandM.EuláliaVaresLargeDeviationsandMetastability
101 A.Kushner,V.LychaginandV.RubtsovContactGeometryandNonlinear
DifferentialEquations
102 L.W.BeinekeandR.J.Wilson(eds.)withP.J.CameronTopicsinAlgebraicGraphTheory
103 O.StaffansWell-PosedLinearSystems
104 J.M.Lewis,S.LakshmivarahanandS.DhallDynamicDataAssimilation
105 M.LothaireAppliedCombinatoricsonWords
106 A.MarkoeAnalyticTomography
107 P.A.MartinMultipleScattering
108 R.A.BrualdiCombinatorialMatrixClasses
110 M.-J.LaiandL.L.SchumakerSplineFunctionsonTriangulations
111 R.T.CurtisSymmetricGenerationofGroups
112 H.Salzmann,T.Grundhöfer,H.HählandR.LöwenTheClassicalFields
113 S.PeszatandJ.ZabczykStochasticPartialDifferentialEquationswithLévyNoise
114 J.BeckCombinatorialGames
116 D.Z.ArovandH.DymJ-ContractiveMatrixValuedFunctionsandRelatedTopics
117 R.Glowinski,J.-L.LionsandJ.HeExactandApproximateControllabilityfor
DistributedParameterSystems
118 A.A.BorovkovandK.A.BorovkovAsymptoticAnalysisofRandomWalks
119 M.DezaandM.DutourSikiric´GeometryofChemicalGraphs
120 T.NishiuraAbsoluteMeasurableSpaces
121 M.PrestPurity,SpectraandLocalisation
122 S.KhrushchevOrthogonalPolynomialsandContinuedFractions:FromEuler’sPointofView
123 H.NagamochiandT.IbarakiAlgorithmicAspectsofGraphConnectivity
124 F.W.KingHilbertTransformsI
125 F.W.KingHilbertTransformsII
126 O.CalinandD.-C.ChangSub-RiemannianGeometry
127 M.Grabisch,J.-L.Marichal,R.MesiarandE.PapAggregationFunctions
encyclopedia of mathematics and its applications
Aggregation Functions
MICHEL GRABISCH
UniversityofPanthéon-Sorbonne
Paris,France
JEAN-LUC MARICHAL
UniversityofLuxembourg
Luxembourg
RADKO MESIAR
SlovakUniversityofTechnology
Bratislava,Slovakia
ENDRE PAP
UniversityofNoviSad
NoviSad,Serbia
cambridgeuniversitypress
Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,SãoPaulo,Delhi
CambridgeUniversityPress
TheEdinburghBuilding,CambridgeCB28RU,UK
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
www.cambridge.org
Informationonthistitle:www.cambridge.org/9780521519267
©M.Grabisch,J.-L.Marichal,R.MesiarandE.Pap2009
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithout
thewrittenpermissionofCambridgeUniversityPress.
Firstpublished2009
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
ISBN978-0-521-51926-7hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceor
accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto
inthispublication,anddoesnotguaranteethatanycontentonsuch
websitesis,orwillremain,accurateorappropriate.
ToAgnieszka,Francis,Raphaëlle,andRémi
M.G.
ToPascale,Olivia,Jean-Philippe,andClaudia
J.-L.M.
ToAnka,Janka,andAndrejka
R.M.
ToDarinkaandDanijela
E.P.
Contents
Listoffigures pagex
Listoftables xii
Preface xiii
1 Introduction 1
1.1 Mainmotivationsandscope 1
1.2 Basicdefinitionsandexamples 2
1.3 Conventionalnotation 9
2 Propertiesforaggregation 11
2.1 Introduction 11
2.2 Elementarymathematicalproperties 12
2.3 Grouping-basedproperties 31
2.4 Invarianceproperties 41
2.5 Furtherproperties 49
3 Conjunctiveanddisjunctiveaggregationfunctions 56
3.1 Preliminariesandgeneralnotes 56
3.2 Generatedconjunctiveaggregationfunctions 59
3.3 Triangularnormsandrelatedconjunctiveaggregationfunctions 64
3.4 Copulasandquasi-copulas 88
3.5 Disjunctiveaggregationfunctions 100
3.6 Uninorms 106
3.7 Nullnorms 115
3.8 Moreaggregationfunctionsrelatedtot-norms 119
3.9 Restricteddistributivity 123
4 Meansandaverages 130
4.1 Introductionanddefinitions 130
4.2 Quasi-arithmeticmeans 132
vii
viii Contents
4.3 Generalizationsofquasi-arithmeticmeans 139
4.4 Associativemeans 161
4.5 Meansconstructedfromameanvalueproperty 163
4.6 Constructingmeans 166
4.7 Furtherextendedmeans 168
5 Aggregationfunctionsbasedonnonadditiveintegrals 171
5.1 Introduction 171
5.2 Setfunctions,capacities,andgames 172
5.3 Somelineartransformationsofsetfunctions 177
5.4 TheChoquetintegral 181
5.5 TheSugenointegral 207
5.6 Otherintegrals 227
6 Constructionmethods 234
6.1 Introduction 234
6.2 Transformedaggregationfunctions 234
6.3 Composedaggregation 242
6.4 Weightedaggregationfunctions 247
6.5 Someotheraggregation-basedconstructionmethods 252
6.6 Aggregationfunctionsbasedonminimaldissimilarity 257
6.7 Ordinalsumsofaggregationfunctions 261
6.8 Extensionstoaggregationfunctions 266
7 Aggregationonspecificscaletypes 272
7.1 Introduction 272
7.2 Ratioscales 273
7.3 Differencescales 280
7.4 Intervalscales 284
7.5 Log-ratioscales 289
8 Aggregationonordinalscales 292
8.1 Introduction 292
8.2 Orderinvariantsubsets 293
8.3 Latticepolynomialfunctionsandsomeoftheirproperties 296
8.4 Ordinalscaleinvariantfunctions 300
8.5 Comparisonmeaningfulfunctionsonasingleordinalscale 304
8.6 Comparisonmeaningfulfunctionsonindependentordinalscales 308
8.7 Aggregationonfinitechainsbychainindependentfunctions 310
9 Aggregationonbipolarscales 317
9.1 Introduction 317
9.2 Associativebipolaroperators 319
