Detection of long-range dependence [Elektronische Ressource] : applications in climatology and hydrology / Henning Rust

Henning Rust

Description

PotsdamInstitutfu¨rKlimafolgenforschungDetectionofLong-RangeDependenceApplicationsinClimatologyandHydrologyDissertationzurErlangungdesakademischenGradesDoktorderNaturwissenschaften(Dr.rer.nat.)inderWissenschaftsdisziplinTheoretischePhysikEingereichtanderMathematisch-NaturwissenschaftlichenFakulta¨tderUniversita¨tPotsdamvonHenningRustPotsdam,imJanuar2007Contents1 Introduction 12 TimeSeriesAnalysisandStochasticModelling 52.1 BasicConceptsofTimeSeriesAnalysis . . . . . . . . . . . . . . . . . . . . . 52.1.1 RandomVariables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 StochasticProcesses . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.3 SpectralAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.4 Long-RangeDependence . . . . . . . . . . . . . . . . . . . . . . . . 92.2 SomeStationaryStochasticProcesses . . . . . . . . . . . . . . . . . . . . . . 102.2.1 ProcesseswithShort-RangeDependence . . . . . . . . . . . . . . . 102.2.2 ProcesseswithLong-RangeDependence . . . . . . . . . . . . . . . 152.2.3 MotivationforAutoregressiveMovingAverageModels . . . . . . 172.3 ParameterEstimationforStochasticProcesses . . . . . . . . . . . . . . . . 172.3.1 MaximumLikelihoodforFARIMA[p,d,q]Processes . . . . . . . . . 172.3.2 WhittleEstimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3.3 DetrendedFluctuationAnalysis . . . . . . . . . . . . . . . . . . . . 212.

Informations

• Publié par: universitat_potsdam
• Publié le: 01 janvier 2007
• Nombre de lectures: 21
• Langue: English
• Poids de l'ouvrage: 25 Mo

Extrait

PotsdamInstitutfu¨rKlimafolgenforschung

DetectionofLong-RangeDependence

ApplicationsinClimatologyandHydrology

DissertationzurErlangungdesakademischenGrades
DoktorderNaturwissenschaften(Dr.rer.nat.)
inderWissenschaftsdisziplinTheoretischePhysik

Eingereichtander
Mathematisch-NaturwissenschaftlichenFakulta¨t
derUniversita¨tPotsdam

novHenningRust

Potsdam,imJanuar2007

Contents

1Introduction

2TimeSeriesAnalysisandStochasticModelling
2.1BasicConceptsofTimeSeriesAnalysis.....................
2.1.1RandomVariables............................
2.1.2StochasticProcesses...........................
2.1.3SpectralAnalysis.............................
2.1.4Long-RangeDependence........................
2.2SomeStationaryStochasticProcesses......................
2.2.1ProcesseswithShort-RangeDependence...............
2.2.2ProcesseswithLong-RangeDependence...............
2.2.3MotivationforAutoregressiveMovingAverageModels......
2.3ParameterEstimationforStochasticProcesses................
2.3.1MaximumLikelihoodforFARIMA[p,d,q]Processes.........
2.3.2WhittleEstimator.............................
2.3.3DetrendedFluctuationAnalysis....................
2.4SimulationsfromLong-RangeDependentProcesses.............

3ModelSelection
3.1Goodness-of-FitTests...............................
3.1.1HypothesisTesting............................
3.1.2PortmanteauTest.............................
3.1.3SpectralVariantofthePortmanteauTest...............
3.2ModelComparison................................
3.2.1Likelihood-RatioTest...........................
3.2.2InformationCriteria...........................
3.2.3InformationCriteriaandFARIMA[p,d,q]Models..........
3.3Simulation-BasedModelSelection.......................
3.3.1Non-NestedModelSelection......................
3.3.2Simulation-BasedApproachforFARIMA[p,d,q]...........
3.3.3AnIllustratingExample.........................
3.3.4TestingtheTest..............................
3.3.5EstimatingtheRequiredSampleSize.................
3.3.6BootstrappingtheResiduals.......................
3.4Summary......................................

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CONTENTS

4DetectionofLong-RangeDependence43
4.1ConstructingtheExampleProcess.......................44
4.2DetrendedFluctuationAnalysis.........................45
4.3Log-PeriodogramRegression..........................46
4.4Full-ParametricModellingApproach......................47
4.4.1ModellingtheARMA[3,2]Realisation.................47
4.4.2ModellingtheFAR[1]Realisation....................50
4.5Summary......................................50
5ModellingTemperatureRecords53
5.1PragueDailyMaximumTemperature.....................54
5.1.1StochasticModelling...........................54
5.1.2DetectingLong-RangeDependence..................58
5.1.3ConservativeTrendTest.........................60
5.2NorthernHemisphereMeanTemperature...................61
5.2.1TemperatureAnomalies.........................62
5.2.2TemperatureResiduals–AccountingforaLinearTrend......66
5.2.3Long-RangeDependenceandtheLinear-TrendAssumption....69
5.3Summary......................................70
6ModellingRun-offRecords71
6.1DanubeDailyMeanRun-offatAchleiten...................73
6.1.1StochasticModelling...........................73
6.1.2DetectingLong-RangeDependence..................78
6.2GroßeVilsDailyMeanRun-offatVilsbiburg.................78
6.2.1StochasticModelling...........................78
6.2.2DetectingLong-RangeDependence..................83
6.2.3WeeklyCyclesinRiverRun-Off....................83
6.3WislaMonthlyMeanRun-offatTczew.....................84
6.3.1StochasticModelling...........................84
6.3.2DetectingLong-RangeDependence..................88
6.4Summary......................................89
7BootstrapBasedCondenceIntervals91
7.1SketchoftheBootstrapApproach........................92
7.2CaseStudy:Vilsbiburg/GroßeVils.......................93
7.2.1ModellingtheMaximaDistribution..................93
7.2.2ModellingtheACFoftheMaximaSeries...............94
7.2.3CondenceIntervalsforReturnLevelEstimates...........95
7.3Summary......................................95
8ConclusionsandOutlook99
8.1SummaryandConclusions...........................99
8.2Outlook.......................................102
Appendices105

CONTENTS

AppendixAReviewofDetrendedFluctuationAnalysis
A.1BiasandVarianceforSelf-SimilarIncrementProcesses............
A.2DFAandtheDetectionofLong-RangeDependence.............
A.2.1InferenceofScaling............................
A.2.2Pitfalls...................................
A.3InvestigatingthePragueTemperatureAnomalies..............
A.3.1DetectingLong-RangeDependenceusingDFA............
A.3.2DetectingLong-RangeDependenceusingParametricModels...
A.4Summary......................................
AppendixBLong-RangeDependence–Effects,Methods,Mechanisms
B.1EffectsofLong-RangeDependence.......................
B.2PhysicalExplanationsofthePhenomenon...................
B.3FurtherHeuristicandSemi-ParametricMethods...............
B.3.1RescaledAdjustedRange........................
B.3.2Log-PeriodogramRegression......................
B.4SpecicTestStatisticsforModelSelection...................
B.4.1Log-PeriodogramRegression......................
B.4.2DetrendedFluctuationAnalysis....................
B.5ConstructingtheExampleProcess–DetailedDescription..........
AppendixCBootstrapMethodsforCondenceIntervalEstimation
C.1ExtremeValueStatistics.............................
C.1.1TheFisher-TippettTheorem.......................
C.1.2TheFisher-TippettTheoremforDependentSeries..........
C.1.3ParameterEstimationfortheGeneralExtremeValueDistribution
C.2BootstrappingtheEstimatorsVariance.....................
C.2.1EmulatingDependence.........................
C.2.2ModellingtheDistribution.......................
C.2.3ModellingtheACFusingFARIMA[p,d,q]Processes.........
C.2.4CombiningDistributionandAutocorrelation.............
C.2.5GeneratingBootstrapEnsembles....................
C.3ComparisonoftheBootstrapApproaches...................
C.3.1MonteCarloReferenceEnsemble....................
C.3.2TheBootstrapEnsembles........................
C.3.3EnsembleVariabilityandDependenceonEnsembleSize......
C.4Summary......................................
AppendixDDataSourcesandPreprocessing
D.1DataSources....................................
D.2Preprocessing...................................
D.2.1EstimationofPeriodicCycles......................
D.2.2Box-CoxTransformation.........................
D.3PreprocessingofRun-OffandTemperatureRecords.............
D.3.1PragueDailyMaximumTemperatures.................
D.3.2DanubeDailyMeanRun-OffatAchleiten...............
D.3.3GroßeVilsDailyMeanRun-OffatVilsbiburg.............
D.3.4WislaMonthlyMeanRun-OffatTczew................

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Chapter1

Introduction

Themysteryofnature’svariabilityhasbeenperceivedasapotentialthreatsinceancient
times(Genesis41:29-30):“Sevenyearsofgreatabundancearecomingthroughoutthe
landofEgypt,butsevenyearsoffaminewillfollowthem.Thenalltheabundancein
Egyptwillbeforgotten,andthefaminewillravagetheland.”Onlythisprophecymade
byJosephintheOldTestamentledtothestorageofexcessfoodproducedintheyearsof
abundanceandsavedtheancientEgyptiansfromstarvingintheyearsoffamine.Like-
wiseinmoderntimestheBritishhydrologistHaroldEdwinHurststudiedtheNileRiver
minimalowsto“prophesy”theinter-annualwaterlevelvariability(Hurst,1951).The
aimofhisworkwastodeterminethesizeofareservoirlargeenoughtoallowaconstant
outowgiventhehighlyvariableinowsandtherebynallyensureaconstantfoodpro-
ductionundervaryingnaturalwaterresources.Duringhisanalysis,Hurstobservedthat
waterlevelsseemtobestronglyinuencedbythoseobservedinthedistantpast.The
slowalgebraicdecayofthisinterdependence(orautocorrelation)wasdifferentfromstan-
dardmodels.ThisislaterreferredtoasHurstphenomenonorlong-rangedependence
contrarytoshort-rangedependencespecifyinganexponentialdecay.
Droughtsandtheresultingscarcityofalimentsarestillchallengestoovercome(e.g.,
Rebetezetal.,2006).Thesechallengesareenhancedbyoodswhichresultinamenacing
excessofwater,andcreatetheneedforhydraulicstructurespotentenoughtowithstand
extreme(i.e.unlikelyhigh)waterdischarges.Itisdesirablethatthestructureitselfis
conserved,incaseofdamsanddykes,aswellasthepropertiesbehindit.Inrecentyears
oodshavebecomemoreabundantandmoredestructive(KundzewiczandSchellnhuber,
2004).ExtremeoodshavebeenrecordedincentralEuropeinthelastdecades,e.g.,at
theDonau1997orattheElbe2002(BeckerandGru¨newald,2003).Thereforeoodrisk
assessmentandtheassociateduncertaintyhavebecomeahighlyrelevanttopiconthesci-
enticaswellasonthepoliticalagenda(e.g.,Apeletal.,2004;Merz,2006;WMO/UNEP,
2002).Floodriskassessmentinvolvesanestimateoftheprobabilityofexceedingacertain
waterlevelordischarge,i.e.anextremevalueanalysis.InasimilarwayasHurst’sstor-
ageproblem,thisanalysisisaffectedbytheautocorrelationoftheobservedwaterlevels
(Koutsoyiannis,2003;ColesandTawn,1999).Itreducestheinformationobtainedfrom
anewobservationbecausetheinterdependencedeterminesto