STATISTICAL METHODS FOR CLIMATE SCIENTISTS This book provides a comprehensive introduction to the most commonly used statistical methodsrelevantinatmospheric,oceanic,andclimatesciences.Eachmethodisdescribed step-by-step using plain language, and illustrated with concrete examples, with relevant statisticalandscientificconceptsexplainedasneeded.Particularattentionispaidtonuances andpitfalls,withsufficientdetailtoenablethereadertowriterelevantcode.Topicscovered includehypothesistesting,timeseriesanalysis,linearregression,dataassimilation,extreme valueanalysis,PrincipalComponentAnalysis,CanonicalCorrelationAnalysis,Predictable ComponentAnalysis,andCovarianceDiscriminantAnalysis.Thespecificstatisticalchal- lengesthatariseinclimateapplicationsarealsodiscussed,includingmodelselectionprob- lemsassociatedwithCanonicalCorrelationAnalysis,PredictableComponentAnalysis,and CovarianceDiscriminantAnalysis.Requiringnopreviousbackgroundinstatistics,thisis ahighlyaccessibletextbookandreferenceforstudentsandearlycareerresearchersinthe climatesciences. timothy m. delsole isProfessorintheDepartmentofAtmospheric,Oceanic andEarthSciences,andSeniorScientistattheCenterforOceanicAtmospheric,and Land Studies, at George Mason University, Virginia. He has published more than 100peer-reviewedpapersinclimatescienceandservedasco-editor-in-chiefofthe JournalofClimate. michael k. tippett is an associate professor at Columbia University. His researchincludesforecastingElNiñoandrelatingextremeweather(tornadoesand hurricanes) with climate, now and in the future. He analyzes data from computer modelsandweatherobservationstofindpatternsthatimproveunderstanding,facil- itateprediction,andhelpmanagerisk. Includesboththemathematicsandtheintuitionneededforclimatedataanalysis. –ProfessorDennisLHartmann,UniversityofWashington STATISTICAL METHODS FOR CLIMATE SCIENTISTS TIMOTHY M. DELSOLE GeorgeMasonUniversity MICHAEL K. TIPPETT ColumbiaUniversity UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 314-321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre,NewDelhi–110025,India 103PenangRoad,#05–06/07,VisioncrestCommercial,Singapore238467 CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781108472418 DOI:10.1017/9781108659055 ©CambridgeUniversityPress2022 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2022 PrintedintheUnitedKingdombyTJBooksLimited,PadstowCornwall AcataloguerecordforthispublicationisavailablefromtheBritishLibrary. LibraryofCongressCataloging-in-PublicationData Names:DelSole,TimothyM.,author. Title:Statisticalmethodsforclimatescientists/TimothyM.DelSoleandMichaelK.Tippett. Description:NewYork:CambridgeUniversityPress,2021.|Includes bibliographicalreferencesandindex. Identifiers:LCCN2021024712(print)|LCCN2021024713(ebook)| ISBN9781108472418(hardback)|ISBN9781108659055(epub) Subjects:LCSH:Climatology–Statisticalmethods.|Atmospheric science–Statisticalmethods.|Marinesciences–Statisticalmethods.| BISAC:SCIENCE/EarthSciences/Meteorology&Climatology Classification:LCCQC866.D382021(print)|LCCQC866(ebook)| DDC551.601/5118–dc23 LCrecordavailableathttps://lccn.loc.gov/2021024712 LCebookrecordavailableathttps://lccn.loc.gov/2021024713 ISBN978-1-108-47241-8Hardback Additionalresourcesforthispublicationatwww.cambridge.org/9781108472418. 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Contents Preface pagexiii 1 BasicConceptsinProbabilityandStatistics 1 1.1 GraphicalDescriptionofData 2 1.2 MeasuresofCentralValue:Mean,Median,andMode 4 1.3 MeasuresofVariation:PercentileRangesandVariance 6 1.4 PopulationversusaSample 8 1.5 ElementsofProbabilityTheory 8 1.6 Expectation 11 1.7 MoreThanOneRandomVariable 13 1.8 Independence 16 1.9 EstimatingPopulationQuantitiesfromSamples 18 1.10 NormalDistributionandAssociatedTheorems 20 1.11 IndependenceversusZeroCorrelation 27 1.12 FurtherTopics 28 1.13 ConceptualQuestions 29 2 HypothesisTests 30 2.1 TheProblem 31 2.2 IntroductiontoHypothesisTesting 33 2.3 FurtherCommentsonthet-test 40 2.4 ExamplesofHypothesisTests 43 2.5 SummaryofCommonSignificanceTests 49 2.6 FurtherTopics 50 2.7 ConceptualQuestions 51 3 ConfidenceIntervals 52 3.1 TheProblem 53 v vi Contents 3.2 ConfidenceIntervalforaDifferenceinMeans 53 3.3 InterpretationoftheConfidenceInterval 55 3.4 APitfallaboutConfidenceIntervals 57 3.5 CommonProceduresforConfidenceIntervals 57 3.6 BootstrapConfidenceIntervals 64 3.7 FurtherTopics 67 3.8 ConceptualQuestions 68 4 StatisticalTestsBasedonRanks 69 4.1 TheProblem 70 4.2 ExchangeabilityandRanks 71 4.3 TheWilcoxonRank-SumTest 73 4.4 StochasticDominance 78 4.5 Comparisonwiththet-test 79 4.6 Kruskal–WallisTest 81 4.7 TestforEqualityofDispersions 83 4.8 RankCorrelation 85 4.9 DerivationoftheMeanandVarianceoftheRankSum 88 4.10 FurtherTopics 92 4.11 ConceptualQuestions 93 5 IntroductiontoStochasticProcesses 94 5.1 TheProblem 95 5.2 StochasticProcesses 100 5.3 WhyShouldICareifMyDataAreSeriallyCorrelated? 105 5.4 TheFirst-OrderAutoregressiveModel 109 5.5 TheAR(2)Model 117 5.6 PitfallsinInterpretingACFs 119 5.7 SolutionsoftheAR(2)Model 121 5.8 FurtherTopics 122 5.9 ConceptualQuestions 124 6 ThePowerSpectrum 126 6.1 TheProblem 127 6.2 TheDiscreteFourierTransform 129 6.3 Parseval’sIdentity 133 6.4 ThePeriodogram 134 6.5 ThePowerSpectrum 135 6.6 PeriodogramofGaussianWhiteNoise 138 6.7 ImpactofaDeterministicPeriodicComponent 139 Contents vii 6.8 EstimationofthePowerSpectrum 140 6.9 PresenceofTrendsandJumpDiscontinuities 144 6.10 LinearFilters 146 6.11 TyingUpLooseEnds 150 6.12 FurtherTopics 152 6.13 ConceptualQuestions 155 7 IntroductiontoMultivariateMethods 156 7.1 TheProblem 157 7.2 Vectors 159 7.3 TheLinearTransformation 160 7.4 LinearIndependence 163 7.5 MatrixOperations 166 7.6 InvertibleTransformations 168 7.7 OrthogonalTransformations 170 7.8 RandomVectors 172 7.9 DiagonalizingaCovarianceMatrix 175 7.10 MultivariateNormalDistribution 178 7.11 Hotelling’sT-squaredTest 179 7.12 MultivariateAcceptanceandRejectionRegions 181 7.13 FurtherTopics 182 7.14 ConceptualQuestions 183 8 LinearRegression:LeastSquaresEstimation 185 8.1 TheProblem 186 8.2 MethodofLeastSquares 188 8.3 PropertiesoftheLeastSquaresSolution 192 8.4 GeometricInterpretationofLeastSquaresSolutions 196 8.5 IllustrationUsingAtmosphericCO Concentration 199 2 8.6 TheLineFit 205 8.7 AlwaysIncludetheInterceptTerm 206 8.8 FurtherTopics 207 8.9 ConceptualQuestions 209 9 LinearRegression:Inference 210 9.1 TheProblem 211 9.2 TheModel 212 9.3 DistributionoftheResiduals 212 9.4 DistributionoftheLeastSquaresEstimates 213 9.5 InferencesaboutIndividualRegressionParameters 215 viii Contents 9.6 ControllingfortheInfluenceofOtherVariables 216 9.7 Equivalenceto“RegressingOut”Predictors 218 9.8 SeasonalityasaConfoundingVariable 222 9.9 EquivalencebetweentheCorrelationTestandSlopeTest 224 9.10 GeneralizedLeastSquares 225 9.11 DetectionandAttributionofClimateChange 226 9.12 TheGeneralLinearHypothesis 233 9.13 TyingUpLooseEnds 234 9.14 ConceptualQuestions 236 10 ModelSelection 237 10.1 TheProblem 238 10.2 Bias–VarianceTradeoff 240 10.3 Out-of-SampleErrors 243 10.4 ModelSelectionCriteria 245 10.5 Pitfalls 249 10.6 FurtherTopics 253 10.7 ConceptualQuestions 254 11 Screening:APitfallinStatistics 255 11.1 TheProblem 256 11.2 Screeningiid TestStatistics 259 11.3 TheBonferroniProcedure 262 11.4 ScreeningBasedonCorrelationMaps 262 11.5 CanYouTrustRelationsInferredfromCorrelationMaps? 265 11.6 ScreeningBasedonChangePoints 265 11.7 ScreeningwithaValidationSample 268 11.8 TheScreeningGame:CanYouFindtheStatisticalFlaw? 268 11.9 ScreeningAlwaysExistsinSomeForm 271 11.10 ConceptualQuestions 272 12 PrincipalComponentAnalysis 273 12.1 TheProblem 274 12.2 Examples 276 12.3 SolutionbySingularValueDecomposition 283 12.4 RelationbetweenPCAandthePopulation 285 12.5 SpecialConsiderationsforClimateData 289 12.6 FurtherTopics 295 12.7 ConceptualQuestions 297 Contents ix 13 FieldSignificance 298 13.1 TheProblem 299 13.2 TheLivezey–ChenFieldSignificanceTest 303 13.3 FieldSignificanceTestBasedonLinearRegression 305 13.4 FalseDiscoveryRate 310 13.5 WhyDifferentTestsforFieldSignificance? 311 13.6 FurtherTopics 312 13.7 ConceptualQuestions 312 14 MultivariateLinearRegression 314 14.1 TheProblem 315 14.2 ReviewofUnivariateRegression 317 14.3 EstimatingMultivariateRegressionModels 320 14.4 HypothesisTestinginMultivariateRegression 323 14.5 SelectingX 324 14.6 SelectingBothXandY 328 14.7 SomeDetailsaboutRegressionwithPrincipalComponents 331 14.8 RegressionMapsandProjectingData 332 14.9 ConceptualQuestions 333 15 CanonicalCorrelationAnalysis 335 15.1 TheProblem 336 15.2 SummaryandIllustrationofCanonicalCorrelationAnalysis 337 15.3 PopulationCanonicalCorrelationAnalysis 343 15.4 RelationbetweenCCAandLinearRegression 347 15.5 InvariancetoAffineTransformation 349 15.6 SolvingCCAUsingtheSingularValueDecomposition 350 15.7 ModelSelection 357 15.8 HypothesisTesting 359 15.9 ProofoftheMaximizationProperties 362 15.10 FurtherTopics 364 15.11 ConceptualQuestions 364 16 CovarianceDiscriminantAnalysis 366 16.1 TheProblem 367 16.2 Illustration:MostDetectableClimateChangeSignals 370 16.3 HypothesisTesting 378 16.4 TheSolution 382 16.5 SolutioninaReduced-DimensionalSubspace 388 16.6 VariableSelection 392