Aggregation Functions PDF
Preview Aggregation Functions
more 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
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