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1、TimeseriesIntroductionSimpletimeseriesmodelsARIMAValidatingamodelSpectralAnalysisWaveletsDigitalSignalProcessing(DSP)Modelingvolatility:GARCHmodels(GeneralizedAutoRegressiveConditionnalHeteroscedasticity)MultivariatetimeseriesState-SpaceModelsandKalmanFilteringNon-lineart
2、imeseriesandchaosOthertimesDiscrete-valuedtimeseries:MarkovchainsandbeyondVariantsofMarkovchainsUntackledsubjectsTOSORTThischaptercontrastswiththetopicswehaveseenuptonow:wewereinterestedinthestudyofseveralindependantrealizationsofasimplestatisticalprocess(e.g.,agaussianra
3、ndomvariable,oramixtureofgaussians,oralinearmodel);weshallnowfocusonasinglerealizationofamorecomplexprocess.Hereisthestructureofthischapter.Afteranintroduction,motivatingthenotionofatimeseriesandgivingseveralexamples,simulatedorreal,weshallpresenttheclassicalmodelsoftimes
4、eries(AR,MA,ARMA,ARIMA,SARIMA),thatproviderecipestobuildtimeserieswithdesiredproperties.Weshallthenpresentspectralmethods,thatfocusonthediscoveryofperiodicelementsintimeseries.Thesimplicityofthosemodelsmakesthemamenable,buttheycannotdescribethepropertiesofsomereal-worldti
5、meseries:non-linearmethods,builtupontheclassicalmodels(GARCH)arecalledfor.State-SpaceModelsandtheKalmanfilterfollowthesamevein:theyassumethatthedataisbuildfromlinearalgebra,butthatwedonotobserveeverything--thereare"hidden"(unobserved,latent)variables.Someofthosemethodsrea
6、dilygeneralizetohigherdimensions,i.e.,tothestudyofvector-valuedtime-series,i.e.,tothestudyofseveralrelatedtimeseriesatthesametime--butsomenewphenomenaappear(e.g.,cointegration).Furthermore,ifthenumberoftimeseriestostudybecomestoolarge,thevectormodelshavetoomanyparameterst
7、obeuseful:weentertherealmofpaneldata.Weshallthenpresentsomelessmainstreamideas:insteadoflinearalgebra,timeseriescanbeproducedbyanalytical(read:differentialequations)orprocedural(read:chaos,fractals)means.Wefinallypresentgeneralizationsoftimeseries:stochasticprocesses,inwh
8、ichtimeiscontinuous;irregulartimeseries,inwhichtimeisdiscretebutirregular;anddiscrete-valuedtime
