Wavelet Analysis on Stochastic Time Series(随机时间序列的

ANovelWaveletBasedApproachforTimeSeriesDataAnalysisZurErlangungdesakademischenGradeseinesDoktorsderWirtschaftswissenschaften(Dr.rer.pol.)vonderFakult¨atf¨urWirtschaftswissenschaftendesKarlsruheInstitutsf¨urTechnologie(KIT)genehmigteDissertationvonDipl.-Math.ThomasMeinlTagderm¨undlichenPr¨ufung:10.05.2011Referent:Prof.Dr.SvetlozarRachevKorreferent:Prof.Dr.Karl-HeinzWaldmann2011KarlsruheToJack.AcknowledgementsThisworkwouldnothavebeenpossiblewithoutthelovingsupportofallmyfamily,mydog,andmyex-wife.Thankyouall,Iammostdeeplyobligedtoyouforbeingalongmysideoveralltheseyears.IalsothankmysupervisorProf.Dr.SvetlozarRachevandDr.EdwardSunfortheirguid-anceandtheirinvaluableinputwhichwithoutthisthesiscouldnothavebeenfinishedinsuchashorttimeandwithsuchexcellentresults.IalsothankChristofWeinhardtwhoprovidedmewiththeopportunitytoundertakesomeseriousresearchathisinstitute,nottomentionthefunIhadwithallthenicecolleagues.Thankyou,wehadsometimestogetherwewillrememberforsure.DuringthetimethisthesiswaswrittentheauthorundertookastayabroadinBrazilaswellasinJapan,whichwasonlymadefeasiblethankstothehighlyappreciatedsupportoftheKarlsruheHouseofYoungScientists.Mythanksalsogoestoallthepeopletherewhowelcomedmesoheartily.Karlsruhe,May2011ThomasMeinlAbstractTimeseriesanalysisisstillaverywidefieldofresearchfrombothatheoreticalpointofviewaswellasamongstpractitioners.Amongtheveryfirsttasksintheanalysisproce-dureistheestimationoflong-termtrends,thatis,theseparationofthisgenerallyslowlyevolvingcomponentfromanyshort-termfluctuations.Usually,thetrendcurve,whichinmostcasesisexpectedtobesmooth,canbeextractedbyavarietyofdifferentmethods.However,inmanyapplicationscenariosthetrendmustalsoaccountforsuddenchanges.Thesesuddenchangescompriseofnotonlyjumps,butalsootherphenomenalikesteepslopesandvalleys.Thischallengeconstitutesanon-goingproblemfortraditionaltrendestimationmethods.Whileestablishedfilteringtechniqueseitherfailtocapturethesesuddenchangesaccuratelyoraresensitivetohigh-amplitudefluctuations,theapplica-tionofparametricmethodsischallengingduetothegenerallyunknowntrendandtheinnumerableshapesthatthesesuddenchangescanassume.Thisthesisproposesatrendextractionapproachbasedonwaveletmethods.Thenewalgorithm,namedlocallinearscalingapproximation(LLSA),isdevelopedbyanalyzingspecificwaveletcoefficientstepresponsestructuresandbytransferringthesestructuresontorealsignals.Thisprocedureenablestheanalysttoextractatrendwhosesmooth-nessiscomparabletotheoutputoflinearfilteringtechniques,whileatthesametimecapturingthedetailsofsuddenchangeswitharbitraryshapes,anareainwhichusu-allymostnonlinearfiltersexcel.Therefore,LLSAcanbeseenasanovelapproachtobridgethegapbetweenlinearandnonlinearfilters.Thealgorithmwasdevelopedtobeapplicableonhomogeneoustimeserieswithoutanyfurtherrequirementsonthese,andtoworkwithonlytwoadditionalinputparameters,whichcanalsobesetinaheuristicmanner,yieldingadirectlyimplementableandusablemethod.vMoreover,thealgorithm’spropertiesareshown,namelyitscomputationalcomplexity,itslocallinearity,anditsimpulseandstepresponse.TherobustnessofLLSAisfirstshownanalytically,andthensubstantiatedbyseveralanalysesperformedonsimulatedsignalsaswellasonempiricaldata.LLSA’sperformanceisfurtherevaluatedintwoseparateapplicationscenarios,thatare,pricevolatilityestimationandvalueatrisk.Thealgorithm’ssuperiorperformanceinrelationtotwobenchmarkfilteringtechniquesisshownforaconsiderablenumberofcases,andseveralaspects(i.e.,possibilitiesandlimitations)ofLLSA’sgeneralapplicationarediscussed.viContentsListofFiguresxiListofTablesxiii1.Introduction11.1.RequirementsandResearchQuestions....................31.2.ContributionsofthisThesis..........................51.3.Structure....................................62.MethodsofTrendExtraction92.1.TimeSeriesAnalysisandTrendExtraction.................92.2.LinearFilters..................................222.2.1.GeneralFormulation..........................222.2.2.TransferFunctions:Timevs.FrequencyDomain..........232.3.NonlinearFilters................................262.3.1.GeneralPerception...........................262.3.2.FilterExamples.............................302.4.FurtherRelatedMethods...........................342.4.1.AlgorithmsforJumpDetectionandModeling............352.4.2.AlternativeMethodsforTrendEstimation..............372.5.Summary....................................423.WaveletsandTheirTransforms453.1.Wavelets.....................................453.2.WaveletTransforms..............................51viiContents3.3.WaveletTrendExtractionandDenoisingMethods.............623.4.Summary....................................664.TheLocalLinearScalingApproximation674.1.MethodologyandImplementation.......................674.1.1.Derivation................................674.1.2.FinalFormulationandRemarks...................754.1.3.ImplementationandUsage......................784.2.Properties....................................814.2.1.ComputationalComplexity......................814.2.2.LocalLinearity.............................814.2.3.ImpulseandStepResponse..........