Analyzing Compositional Data with R

269 Pages·2013·4.27 MB·English

by  K. Gerald van den Boogaart

UseR!
K. Gerald van den Boogaart
Raimon Tolosana-Delgado
Analyzing
Compositional
Data with R
Use R!
SeriesEditors
RobertGentleman KurtHornik GiovanniG.Parmigiani
Forfurthervolumes:
http://www.springer.com/series/6991

K. Gerald van den Boogaart
Raimon Tolosana-Delgado
Analyzing Compositional
Data with R
123
K.GeraldvandenBoogaart
RaimonTolosana-Delgado
FreibergforResourcesTechnology
HelmholtzInstitute
Freiberg
Germany
SeriesEditors:
RobertGentleman KurtHornik
PrograminComputationalBiology DepartmentofStatistikandMathematik
DivisionofPublicHealthSciences Wirtschaftsuniversita¨tWien
FredHutchinsonCancerResearchCenter Augasse2-6
1100FairviewAvenue,N.M2-B876 A-1090Wien
Seattle,Washington98109 Austria
USA
GiovanniParmigiani
TheSidneyKimmelComprehensive
CancerCenteratJohnsHopkinsUniversity
550NorthBroadway
Baltimore,MD21205-2011
USA
ISBN978-3-642-36808-0 ISBN978-3-642-36809-7(eBook)
DOI10.1007/978-3-642-36809-7
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013940100
MathematicsSubjectClassification:62H99(generalmultivariatemethods),62J05(linearregression),
62P12(environmentalapplications)
©Springer-VerlagBerlinHeidelberg2013
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Acknowledgments
We would like to thank Regina van den Boogaart for her infinite patience and
supportto theauthorsduringmanyperiodsofhardworkandMatevzBren forhis
help with the package and the fruitful collaboration on the special compositions.
WeareespeciallyindebtedtoVeraPawlowsky-GlahnandJuan-JoseEgozcuefora
longrunninggeneralsupport,withoutwhichnoneofuswouldbewritingabookon
compositionaldataanalysis.
v

Contents
1 Introduction .................................................................. 1
1.1 WhatAreCompositionalData?........................................ 1
1.1.1 CompositionsArePortionsofaTotal........................ 1
1.1.2 CompositionsAreMultivariatebyNature ................... 3
1.1.3 TheTotalSumofaCompositionIsIrrelevant............... 3
1.1.4 PracticallyAllCompositionsAreSubcompositions......... 4
1.1.5 TheVeryBriefHistoryofCompositionalDataAnalysis.... 5
1.1.6 SoftwareforCompositionalDataAnalysis .................. 6
1.2 GettingStartedwithR.................................................. 6
1.2.1 SoftwareNeededfortheExamplesinThisBook............ 6
1.2.2 InstallingRandtheExtensionPackages..................... 7
1.2.3 BasicRUsage................................................. 7
1.2.4 GettingHelpforCompositionSpecificCommands......... 10
1.2.5 Troubleshooting............................................... 11
References..................................................................... 11
2 FundamentalConceptsofCompositionalDataAnalysis................ 13
2.1 APracticalViewtoCompositionalConcepts ......................... 13
2.1.1 DefinitionofCompositionalData ............................ 13
2.1.2 SubcompositionsandtheClosureOperation................. 14
2.1.3 CompletionofaComposition ................................ 15
2.1.4 CompositionsasEquivalenceClasses........................ 17
2.1.5 PerturbationasaChangeofUnits............................ 18
2.1.6 Amalgamation................................................. 19
2.1.7 MissingValuesandOutliers.................................. 20
2.2 PrinciplesofCompositionalAnalysis.................................. 20
2.2.1 ScalingInvariance............................................. 20
2.2.2 PerturbationInvariance........................................ 21
2.2.3 SubcompositionalCoherence................................. 22
2.2.4 PermutationInvariance........................................ 23
vii
viii Contents
2.3 ElementaryCompositionalGraphics................................... 23
2.3.1 SenseandNonsenseofScatterplotsofComponents ........ 24
2.3.2 TernaryDiagrams ............................................. 24
2.3.3 Log-RatioScatterplots ........................................ 27
2.3.4 BarPlotsandPieCharts ...................................... 28
2.4 MultivariateScales...................................................... 29
2.4.1 ClassicalMultivariateVectorialData(rmult)................ 31
2.4.2 PositiveDatawithAbsoluteGeometry(rplus)............... 31
2.4.3 PositiveDatawithRelativeGeometry(aplus)............... 32
2.4.4 CompositionalDatawithAbsoluteGeometry(rcomp)...... 32
2.4.5 CompositionalDatawithAitchisonGeometry(acomp) .... 33
2.4.6 CountCompositions(ccomp)................................. 34
2.4.7 PracticalConsiderationsonScaleSelection ................. 34
2.5 TheAitchisonSimplex ................................................. 37
2.5.1 TheSimplexandtheClosureOperation ..................... 37
2.5.2 PerturbationasCompositionalSum.......................... 37
2.5.3 PoweringasCompositionalScalarMultiplication........... 39
2.5.4 CompositionalScalarProduct,Norm,andDistance......... 39
2.5.5 TheCenteredLog-RatioTransformation(clr)............... 41
2.5.6 TheIsometricLog-RatioTransformation(ilr)............... 42
2.5.7 TheAdditiveLog-RatioTransformation(alr) ............... 44
2.5.8 GeometricRepresentationofStatisticalResults............. 45
2.5.9 ExpectationandVarianceintheSimplex .................... 47
References..................................................................... 49
3 DistributionsforRandomCompositions.................................. 51
3.1 ContinuousDistributionModels ....................................... 51
3.1.1 TheNormalDistributionontheSimplex..................... 51
3.1.2 TestingforCompositionalNormality ........................ 53
3.1.3 TheDirichletDistribution..................................... 58
3.1.4 TheAitchisonDistribution.................................... 61
3.2 ModelsforCountCompositions ....................................... 62
3.2.1 TheMultinomialDistribution................................. 62
3.2.2 TheMulti-PoissonDistribution............................... 64
3.2.3 DoubleStochasticCountDistributions....................... 66
3.3 RelationsBetweenDistributions....................................... 67
3.3.1 MarginalizationProperties.................................... 67
3.3.2 ConjugatedPriors ............................................. 70
References..................................................................... 70
4 DescriptiveAnalysisofCompositionalData.............................. 73
4.1 DescriptiveStatistics.................................................... 73
4.1.1 CompositionalMean.......................................... 74
4.1.2 MetricVarianceandStandardDeviation..................... 75
4.1.3 VariationMatrixandItsRelatives............................ 76
4.1.4 VarianceMatrices ............................................. 80
4.1.5 Normal-BasedPredictiveandConfidenceRegions.......... 82
Contents ix
4.2 ExploringMarginals.................................................... 85
4.2.1 TheThreeTypesofCompositionalMarginals............... 85
4.2.2 TernaryDiagramMatricesforMultidimensional
Compositions.................................................. 87
4.3 ExploringProjections................................................... 89
4.3.1 ProjectionsandBalances...................................... 89
4.3.2 BalanceBases ................................................. 91
4.3.3 TheCoda-Dendrogram........................................ 92
References..................................................................... 93
5 LinearModelsforCompositions........................................... 95
5.1 Introduction ............................................................. 95
5.1.1 ClassicalLinearRegression(ContinuousCovariables) ..... 95
5.1.2 ClassicalAnalysisoftheVariance
(DiscreteCovariables)......................................... 100
5.1.3 The Different Types of Compositional
RegressionandANOVA ...................................... 102
5.2 CompositionsasIndependentVariables ............................... 103
5.2.1 Example........................................................ 103
5.2.2 DirectVisualizationoftheDependence...................... 103
5.2.3 TheModel..................................................... 106
5.2.4 EstimationofRegressionParameters......................... 107
5.2.5 DisplayingtheModel ......................................... 108
5.2.6 PredictionandPredictiveRegions............................ 110
5.2.7 ModelChecks ................................................. 112
5.2.8 TheStrengthoftheRelationship ............................. 112
5.2.9 GlobalandIndividualTests................................... 114
5.2.10 ModelDiagnosticPlots ....................................... 115
5.2.11 CheckingtheNormalityAssumptions ....................... 117
5.2.12 CheckingConstantVariance.................................. 118
5.2.13 RobustRegression............................................. 118
5.2.14 QuadraticRegression.......................................... 120
5.3 CompositionsasDependentVariables................................. 122
5.3.1 Example........................................................ 122
5.3.2 DirectVisualizationoftheDependence...................... 125
5.3.3 TheModelandItsRInterface................................ 129
5.3.4 EstimationandRepresentationoftheModelParameters ... 134
5.3.5 TestingtheInfluenceofEachVariable....................... 138
5.3.6 ComparingPredictedandObservedValues.................. 140
5.3.7 PredictionofNewValues..................................... 142
5.3.8 Residuals....................................................... 145
5.3.9 MeasuresofCompositionalDetermination.................. 152
5.4 CompositionsasBothDependentandIndependentVariables........ 154
5.4.1 Example........................................................ 154
5.4.2 Visualization................................................... 155