P1:OTA/XYZ P2:ABC JWST076-FM JWST076-Glahn June29,2011 4:3 PrinterName:YettoCome Compositional Data Analysis P1:OTA/XYZ P2:ABC JWST076-FM JWST076-Glahn June29,2011 4:3 PrinterName:YettoCome Compositional Data Analysis Theory and Applications Editedby Vera Pawlowsky-Glahn DepartmentofComputerScienceandAppliedMathematics UniversityofGirona,Spain Antonella Buccianti DepartmentofEarthSciences UniversityofFlorence,Italy A John Wiley & Sons, Ltd., Publication P1:OTA/XYZ P2:ABC JWST076-FM JWST076-Glahn June29,2011 4:3 PrinterName:YettoCome Thiseditionfirstpublished2011 ©2011JohnWiley&Sons,Ltd Registeredoffice JohnWiley&SonsLtd,TheAtrium,SouthernGate,Chichester,WestSussex,PO198SQ,UnitedKingdom Fordetailsofourglobaleditorialoffices,forcustomerservicesandforinformationabouthowtoapplyforpermissionto reusethecopyrightmaterialinthisbookpleaseseeourwebsiteatwww.wiley.com. TherightoftheauthortobeidentifiedastheauthorofthisworkhasbeenassertedinaccordancewiththeCopyright, DesignsandPatentsAct1988. Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmitted,inanyform orbyanymeans,electronic,mechanical,photocopying,recordingorotherwise,exceptaspermittedbytheUKCopyright, DesignsandPatentsAct1988,withoutthepriorpermissionofthepublisher. Wileyalsopublishesitsbooksinavarietyofelectronicformats.Somecontentthatappearsinprintmaynotbeavailablein electronicbooks. Designationsusedbycompaniestodistinguishtheirproductsareoftenclaimedastrademarks.Allbrandnamesandproduct namesusedinthisbookaretradenames,servicemarks,trademarksorregisteredtrademarksoftheirrespectiveowners. Thepublisherisnotassociatedwithanyproductorvendormentionedinthisbook.Thispublicationisdesignedtoprovide accurateandauthoritativeinformationinregardtothesubjectmattercovered.Itissoldontheunderstandingthatthe publisherisnotengagedinrenderingprofessionalservices.Ifprofessionaladviceorotherexpertassistanceisrequired,the servicesofacompetentprofessionalshouldbesought. LibraryofCongressCataloging-in-PublicationData Compositionaldataanalysis:theoryandapplications/editedbyVeraPawlowsky-Glahn,AntonellaBuccianti. p.cm. Includesbibliographicalreferencesandindex. ISBN978-0-470-71135-4(cloth) 1.Multivariateanalysis. 2.Correlation(Statistics) I.Pawlowsky-Glahn,Vera. II.Buccianti,Antonella. QA278.C622011 519.5(cid:2)35–dc23 2011012322 AcataloguerecordforthisbookisavailablefromtheBritishLibrary. PrintISBN:978-0-470-71135-4 ePDFISBN:978-1-119-97647-9 oBookISBN:978-1-119-97646-2 ePubISBN:978-1-119-97761-2 MobiISBN:978-1-119-97762-9 Typesetin10/12ptTimesbyAptaraInc.,NewDelhi,India P1:OTA/XYZ P2:ABC JWST076-FM JWST076-Glahn June29,2011 4:3 PrinterName:YettoCome To John P1:OTA/XYZ P2:ABC JWST076-FM JWST076-Glahn June29,2011 4:3 PrinterName:YettoCome Contents Preface xvii ListofContributors xix PartI Introduction 1 1 AShortHistoryofCompositionalDataAnalysis 3 JohnBacon-Shone 1.1 Introduction 3 1.2 SpuriousCorrelation 3 1.3 LogandLog-RatioTransforms 4 1.4 SubcompositionalDependence 5 1.5 alr,clr,ilr:WhichTransformationtoChoose? 5 1.6 Principles,PerturbationsandBacktotheSimplex 6 1.7 BiplotsandSingularValueDecompositions 7 1.8 Mixtures 7 1.9 DiscreteCompositions 8 1.10 CompositionalProcesses 8 1.11 Structural,CountingandRoundedZeros 8 1.12 Conclusion 9 Acknowledgement 9 References 9 2 BasicConceptsandProcedures 12 JuanJose´EgozcueandVeraPawlowsky-Glahn 2.1 Introduction 12 2.2 ElectionDataandRawAnalysis 13 2.3 TheCompositionalAlternative 15 2.3.1 ScaleInvariance:VectorswithProportionalPositiveComponents RepresenttheSameComposition 15 2.3.2 SubcompositionalCoherence:AnalysesConcerningaSubsetof PartsMustNotDependonOtherNon-InvolvedParts 16 P1:OTA/XYZ P2:ABC JWST076-FM JWST076-Glahn June29,2011 4:3 PrinterName:YettoCome viii CONTENTS 2.3.3 PermutationInvariance:TheConclusionsofaCompositional AnalysisShouldNotDependontheOrderoftheParts 17 2.4 GeometricSettings 17 2.5 CentreandVariability 22 2.6 Conclusion 27 Acknowledgements 27 References 27 PartII Theory–StatisticalModelling 29 3 ThePrincipleofWorkingonCoordinates 31 Glo`riaMateu-Figueras,VeraPawlowsky-GlahnandJuanJose´Egozcue 3.1 Introduction 31 3.2 TheRoleofCoordinatesinStatistics 32 3.3 TheSimplex 33 3.3.1 BasisoftheSimplex 34 3.3.2 WorkingonOrthonormalCoordinates 35 3.4 MoveorStayintheSimplex 38 3.5 Conclusions 40 Acknowledgements 41 References 41 4 DealingwithZeros 43 JosepAntoniMart´ın-Ferna´ndez,JavierPalarea-AlbaladejoandRicardoAntonioOlea 4.1 Introduction 43 4.2 RoundedZeros 44 4.2.1 Non-ParametricReplacementofRoundedZeros 45 4.2.2 ParametricModifiedEMAlgorithmforRoundedZeros 47 4.3 CountZeros 50 4.4 EssentialZeros 53 4.5 Difficulties,TroublesandChallenges 55 Acknowledgements 57 References 57 5 RobustStatisticalAnalysis 59 PeterFilzmoserandKarelHron 5.1 Introduction 59 5.2 ElementsofRobustStatisticsfromaCompositionalPointofView 60 5.3 RobustMethodsforCompositionalData 63 5.3.1 MultivariateOutlierDetection 64 5.3.2 PrincipalComponentAnalysis 64 5.3.3 DiscriminantAnalysis 65 5.4 CaseStudies 66 5.4.1 MultivariateOutlierDetection 66 5.4.2 PrincipalComponentAnalysis 68 5.4.3 DiscriminantAnalysis 68 5.5 Summary 70 P1:OTA/XYZ P2:ABC JWST076-FM JWST076-Glahn June29,2011 4:3 PrinterName:YettoCome CONTENTS ix Acknowledgement 71 References 71 6 GeostatisticsforCompositions 73 RaimonTolosana-Delgado,KarlGeraldvandenBoogaartandVeraPawlowsky-Glahn 6.1 Introduction 73 6.2 ABriefSummaryofGeostatistics 74 6.3 CokrigingofRegionalisedCompositions 76 6.4 StructuralAnalysisofRegionalisedComposition 76 6.5 DealingwithZeros:ReplacementStrategiesandSimplicial IndicatorCokriging 78 6.6 Application 79 6.6.1 DelimitingtheBody:SimplicialIndicatorKriging 81 6.6.2 InterpolatingtheOil–Brine–SolidContent 82 6.7 Conclusions 84 Acknowledgements 84 References 84 7 CompositionalVARIMATimeSeries 87 CarlesBarcelo´-Vidal,Luc´ıaAguilarandJosepAntoniMart´ın-Ferna´ndez 7.1 Introduction 87 7.2 TheSimplexSD asaCompositionalSpace 89 7.2.1 BasicConceptsandNotation 89 7.2.2 TheCovarianceStructureontheSimplex 90 7.3 CompositionalTimeSeriesModels 91 7.3.1 C-StationaryProcesses 92 7.3.2 C-VARIMAProcesses 93 7.4 CTSModelling:AnExample 94 7.4.1 ExpenditureSharesintheUK 94 7.4.2 ModelSelection 95 7.4.3 EstimationofParameters 96 7.4.4 InterpretationandComparison 96 7.5 Discussion 99 Acknowledgements 99 References 100 Appendix 102 8 CompositionalDataandCorrespondenceAnalysis 104 MichaelGreenacre 8.1 Introduction 104 8.2 ComparativeTechnicalDefinitions 105 8.3 PropertiesandInterpretationofLRAandCA 107 8.4 ApplicationtoFattyAcidCompositionalData 107 8.5 DiscussionandConclusions 111 Acknowledgements 112 References 112 P1:OTA/XYZ P2:ABC JWST076-FM JWST076-Glahn June29,2011 4:3 PrinterName:YettoCome x CONTENTS 9 UseofSurveyWeightsfortheAnalysisofCompositionalData 114 MoniqueGraf 9.1 Introduction 114 9.2 ElementsofSurveyDesign 115 9.2.1 Randomization 115 9.2.2 Design-BasedEstimation 118 9.3 ApplicationtoCompositionalData 122 9.3.1 WeightedArithmeticandGeometricMeans 123 9.3.2 ClosedArithmeticMeanofAmounts 123 9.3.3 CentredLog-RatiooftheGeometricMeanComposition 124 9.3.4 ClosedGeometricMeanComposition 124 9.3.5 Example:SwissEarningsStructureSurvey(SESS) 125 9.4 Discussion 126 References 126 10 NotesontheScaledDirichletDistribution 128 GiannaSerafinaMonti,Glo`riaMateu-FiguerasandVeraPawlowsky-Glahn 10.1 Introduction 128 10.2 GenesisoftheScaledDirichletDistribution 129 10.3 PropertiesoftheScaledDirichletDistribution 131 10.3.1 GraphicalComparison 131 10.3.2 MembershipintheExponentialFamily 133 10.3.3 MeasuresofLocationandVariability 134 10.4 Conclusions 136 Acknowledgements 137 References 137 PartIII Theory–AlgebraandCalculus 139 11 ElementsofSimplicialLinearAlgebraandGeometry 141 JuanJose´Egozcue,CarlesBarcelo´-Vidal,JosepAntoniMart´ın-Ferna´ndez, EusebiJarauta-Bragulat,Jose´LuisD´ıaz-BarreroandGlo`riaMateu-Figueras 11.1 Introduction 141 11.2 ElementsofSimplicialGeometry 142 11.2.1 n-PartSimplex 142 11.2.2 VectorSpace 143 11.2.3 CentredLog-RatioRepresentation 146 11.2.4 Metrics 147 11.2.5 OrthonormalBasisandCoordinates 149 11.3 LinearFunctions 151 11.3.1 LinearFunctionsDefinedontheSimplex 152 11.3.2 SimplicialLinearFunctionDefinedonaRealSpace 153 11.3.3 SimplicialLinearFunctionDefinedontheSimplex 154 11.4 Conclusions 156 Acknowledgements 156 References 156 P1:OTA/XYZ P2:ABC JWST076-FM JWST076-Glahn June29,2011 4:3 PrinterName:YettoCome CONTENTS xi 12 CalculusofSimplex-ValuedFunctions 158 JuanJose´Egozcue,EusebiJarauta-BragulatandJose´LuisD´ıaz-Barrero 12.1 Introduction 158 12.2 Limits,ContinuityandDifferentiability 161 12.2.1 LimitsandContinuity 161 12.2.2 Differentiability 163 12.2.3 HigherOrderDerivatives 169 12.3 Integration 171 12.3.1 Antiderivatives.IndefiniteIntegral 171 12.3.2 IntegrationofContinuousSVFunctions 172 12.4 Conclusions 174 Acknowledgements 175 References 175 13 CompositionalDifferentialCalculusontheSimplex 176 CarlesBarcelo´-Vidal,JosepAntoniMart´ın-Ferna´ndezandGlo`riaMateu-Figueras 13.1 Introduction 176 13.2 Vector-ValuedFunctionsontheSimplex 177 13.2.1 Scale-InvariantVector-ValuedFunctionsonRn 177 + 13.2.2 Vector-ValuedFunctionsonSn 178 13.3 C-DerivativesontheSimplex 178 13.3.1 DerivativeofaScale-InvariantVector-ValuedFunctiononRn 178 + 13.3.2 DirectionalC-Derivatives 180 13.3.3 C-Derivative 182 13.3.4 C-Gradient 184 13.3.5 CriticalPointsofaC-DifferentiableReal-ValuedFunction onSn 184 13.4 Example:ExperimentswithMixtures 185 13.4.1 PolynomialofDegreeOne 185 13.4.2 PolynomialofDegreeTwo 186 13.4.3 PolynomialofDegreeOneinLogarithms 187 13.4.4 AnumericalExample 188 13.5 Discussion 189 Acknowledgements 190 References 190 PartIV Applications 191 14 Proportions,Percentages,PPM:DotheMolecularBiosciences TreatCompositionalDataRight? 193 DavidLovell,WarrenMu¨ller,JenTaylor,AlecZwartandChrisHelliwell 14.1 Introduction 193 14.2 TheOmicsImpandTwoBioscienceExperimentParadigms 194 14.3 TheImpactofCompositionalConstraintsintheOmics 197 14.3.1 UnivariateImpactofCompositionalConstraints 197 14.3.2 ImpactofCompositionalConstraintsonMultivariate DistanceMetrics 199 P1:OTA/XYZ P2:ABC JWST076-FM JWST076-Glahn June29,2011 4:3 PrinterName:YettoCome xii CONTENTS 14.4 ImpactofCompositionalConstraintsonCorrelationandCovariance 201 14.4.1 CompositionalConstraints,Covariance,Correlation andLog-TransformedData 202 14.4.2 ASimulationApproachtoUnderstandingtheImpactof Closure 202 14.5 Implications 204 14.5.1 GatheringInformationtoInferAbsoluteAbundance 204 14.5.2 AnalysingCompositionalOmicsDataAppropriately 205 Acknowledgements 206 References 206 15 Hardy–WeinbergEquilibrium:ANonparametricCompositionalApproach 208 JanGraffelmanandJuanJose´Egozcue 15.1 Introduction 208 15.2 GeneticDataSets 209 15.3 ClassicalTestsforHWE 210 15.4 ACompositionalApproach 210 15.5 Example 214 15.6 ConclusionandDiscussion 215 Acknowledgements 215 References 215 16 CompositionalAnalysisinBehaviouralandEvolutionaryEcology 218 MicheleEdoardoRaffaelePierottiandJosepAntoniMart´ın-Ferna´ndez 16.1 Introduction 218 16.2 CODAinPopulationGenetics 219 16.3 CODAinHabitatChoice 222 16.4 MultipleChoiceandIndividualVariationinPreferences 224 16.5 EcologicalSpecialization 228 16.6 TimeBudgets:MoreonSpecialization 229 16.7 Conclusions 231 Acknowledgements 231 References 231 17 FlyinginCompositionalMorphospaces:EvolutionofLimbProportions inFlyingVertebrates 235 LuisAzevedoRodrigues,JosepDaunis-i-Estadella,Glo`riaMateu-Figueras andSantiagoThio´-Henestrosa 17.1 Introduction 235 17.2 FlyingVertebrates–GeneralAnatomicalandFunctionalCharacteristics 236 17.3 Materials 236 17.4 Methods 238 17.5 AitchisonDistanceDisparityMetrics 239 17.5.1 IntragroupAitchisonDistance 239 17.5.2 IntergroupAitchisonDistance 240 17.6 StatisticalTests 243