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TIMESERIESINTRODUCTIONSIMPLETIMESERIESMODELSARIMAVALIDATINGAMODELSPECTRALANALYSISWAVELETSDIGITALSIGNALPROCESSINGDSPMODELINGVOLATILITYGARCHMODELSGENERALIZEDAUTOREGRESSIVECONDITIONNALHETEROSCEDASTICITYMULTIVARIATETIMESERIESSTATESPACEMODELSANDKALMANFILTERINGNONLINEARTIMESERIESANDCHAOSOTHERTIMESDISCRETEVALUEDTIMESERIESMARKOVCHAINSANDBEYONDVARIANTSOFMARKOVCHAINSUNTACKLEDSUBJECTSTOSORTTHISCHAPTERCONTRASTSWITHTHETOPICSWEHAVESEENUPTONOWWEWEREINTERESTEDINTHESTUDYOFSEVERALINDEPENDANTREALIZATIONSOFASIMPLESTATISTICALPROCESSEG,AGAUSSIANRANDOMVARIABLE,ORAMIXTUREOFGAUSSIANS,ORALINEARMODELWESHALLNOWFOCUSONASINGLEREALIZATIONOFAMORECOMPLEXPROCESSHEREISTHESTRUCTUREOFTHISCHAPTERAFTERANINTRODUCTION,MOTIVATINGTHENOTIONOFATIMESERIESANDGIVINGSEVERALEXAMPLES,SIMULATEDORREAL,WESHALLPRESENTTHECLASSICALMODELSOFTIMESERIESAR,MA,ARMA,ARIMA,SARIMA,THATPROVIDERECIPESTOBUILDTIMESERIESWITHDESIREDPROPERTIESWESHALLTHENPRESENTSPECTRALMETHODS,THATFOCUSONTHEDISCOVERYOFPERIODICELEMENTSINTIMESERIESTHESIMPLICITYOFTHOSEMODELSMAKESTHEMAMENABLE,BUTTHEYCANNOTDESCRIBETHEPROPERTIESOFSOMEREALWORLDTIMESERIESNONLINEARMETHODS,BUILTUPONTHECLASSICALMODELSGARCHARECALLEDFORSTATESPACEMODELSANDTHEKALMANFILTERFOLLOWTHESAMEVEINTHEYASSUMETHATTHEDATAISBUILDFROMLINEARALGEBRA,BUTTHATWEDONOTOBSERVEEVERYTHINGTHEREARE“HIDDEN“UNOBSERVED,LATENTVARIABLESSOMEOFTHOSEMETHODSREADILYGENERALIZETOHIGHERDIMENSIONS,IE,TOTHESTUDYOFVECTORVALUEDTIMESERIES,IE,TOTHESTUDYOFSEVERALRELATEDTIMESERIESATTHESAMETIMEBUTSOMENEWPHENOMENAAPPEAREG,COINTEGRATIONFURTHERMORE,IFTHENUMBEROFTIMESERIESTOSTUDYBECOMESTOOLARGE,THEVECTORMODELSHAVETOOMANYPARAMETERSTOBEUSEFULWEENTERTHEREALMOFPANELDATAWESHALLTHENPRESENTSOMELESSMAINSTREAMIDEASINSTEADOFLINEARALGEBRA,TIMESERIESCANBEPRODUCEDBYANALYTICALREADDIFFERENTIALEQUATIONSORPROCEDURALREADCHAOS,FRACTALSMEANSWEFINALLYPRESENTGENERALIZATIONSOFTIMESERIESSTOCHASTICPROCESSES,INWHICHTIMEISCONTINUOUSIRREGULARTIMESERIES,INWHICHTIMEISDISCRETEBUTIRREGULARANDDISCRETEVALUEDTIMESERIESWITHMARKOVCHAINSANDHIDDENMARKOVMODELSINSTEADOFARANDSTATESPACEMODELSINTRODUCTIONEXAMPLESINPROBABILITYTHEORY,ATIMESERIESYOUWILLALSOHEARMENTIONOF“STOCHASTICPROCESS“INATIMESERIES,TIMEISDISCRETE,INASTOCHASTICPROCESS,ITISCONTINUOUSISASEQUENCEOFRANDOMVARIABLESINSTATISTICS,ITISAVARIABLETHATDEPENDSONTIMEFORINSTANCE,THEPRICEOFASTOCK,THATCHANGESEVERYDAYTHEAIRTEMPERATURE,MEASUREDEVERYMONTHTHEHEARTRATEOFAPATIENT,MINUTEAFTERMINUTE,ETCPLOTLAKEHURON,YLAB“,MAIN“LEVELOFLAKEHURON“SOMETIMES,ITISSONOISYTHATYOUDONOTSEEMUCH,XBOXTESTZBOXPIERCEBOXPIERCETESTXSQUARED0014,DF1,PVALUE09059BOXTESTZ,TYPE“LJUNGBOX“BOXLJUNGTESTXSQUARED00142,DF1,PVALUE09051BOXTESTYBOXPIERCETESTXSQUARED415007,DF1,PVALUE1178E10BOXTESTY,TYPELJUNGBOXLJUNGTESTXSQUARED417749,DF1,PVALUE1024E10OPDWTESTLAKEHURON1DURBINWATSONTESTDATALAKEHURON1DW03195,PVALUEDURBINWATSONLMLAKEHURON1LAGAUTOCORRELATIONDWSTATISTICPVALUE108319112031952690ALTERNATIVEHYPOTHESISRHO0OPX50,1,51,COLIFELSEX20X50,“BLACK“,“BLUE“1EXERCISEDOTHESAMEWITHANEXPONENTIALMOVINGAVERAGEEXPONENTIALMOVINGAVERAGETHEMOVINGAVERAGEHASASLIGHTPROBLEMITUSESAWINDOWWITHSHARPEDGESANOBSERVATIONISEITHERINTHEWINDOWORNOTASARESULT,WHENLARGEOBSERVATIONSENTERORLEAVETHEWINDOW,THEREISLARGEJUMPINTHEMOVINGAVERAGETHISISANOTHERMOVINGAVERAGEINSTEADOFTAKINGTHENLATESTVALUES,EQUALLYWEIGHTED,WETAKEALLTHEPRECEDINGVALUESANDGIVEAHIGHERWEIGHTTOTHELATESTVALUESEXPONENTIALMOVINGAVERAGEX50OP3GIRCALLSTRUCTTSXXVARIANCESLEVELSLOPESEASEPSILON00007718000000000001396900000000SUMMARYRFITTED,“SLOPE“MIN1STQUMEDIANMEAN3RDQUMAX000000000091010010210000942000105600011040MATPLOTSTRUCTTSXMINXFITTED,TYPEL,YLAB“,MAIN“STRUCTURALMODELDECOMPOSITIONOFATIMESERIES“YOUCANALSOTRYTOFORECASTFUTUREVALUESBUTITMIGHTNOTBETHATRELIABLELPMINPARMFROWC5,1ELSEPARMFROWC4,1IFISTSXXPMINMAINR1CALLARIMAXCO2,ORDERC1,1,1,SEASONALLISTORDERC2,1,1,PERIOD12COEFFICIENTSAR1MA1SAR1SAR2SMA10259505902001130086908369SE0139001186005580053900332SIGMA2ESTIMATEDAS008163LOGLIKELIHOOD836,AIC1792R2CALLARIMAXCO2,ORDERC1,1,2,SEASONALLISTORDERC2,1,1,PERIOD12COEFFICIENTSAR1MA1MA2SAR1SAR2SMA105935092901412001410087008398SE02325023701084005570053800328SIGMA2ESTIMATEDAS008132LOGLIKELIHOOD8285,AIC1797R3CALLARIMAXCO2,ORDERC2,0,0,SEASONALLISTORDERC1,1,0,PERIOD12COEFFICIENTSAR1AR2SAR1068010308704469SE004460044600432SIGMA2ESTIMATEDAS01120LOGLIKELIHOOD15065,AIC3093FORR4,ITWASEVENERRORINARIMACO2,ORDERC2,0,1,LISTORDERC1,1,1,PERIOD12NONSTATIONARYARPARTFROMCSSTHEAICOFA3ISAPPALLINGLYHIGHWEWANTASLOWAVALUEASPOSSIBLEWEREALLYNEEDTODIFFERENTIATETWICELETUSLOOKATTHEPVALUESROUNDPNORMABSR1COEF,SDSQRTDIAGR1VARCOEF,5AR1MA1SAR1SAR2SMA1003094000000042007005341000000ROUNDPNORMABSR1COEF,SDSQRTDIAGR1VARCOEF,5AR1MA1MA2SAR1SAR2SMA1000535000004009635039989005275000000THISSUGGESTSANSARIMA1,1,10,1,1MODELR3R3CALLARIMAXCO2,ORDERC1,1,1,SEASONALLISTORDERC0,1,1,PERIOD12COEFFICIENTSAR1MA1SMA1023990571008516SE014300123700256SIGMA2ESTIMATEDAS00822LOGLIKELIHOOD8503,AIC17807ROUNDPNORMABSR3COEF,SDSQRTDIAGR3VARCOEF,5AR1MA1SMA1004676000000000000WENOWLOOKATTHERESIDUALSR3RRCALLARIMAXCO2,ORDERC0,1,1,SEASONALLISTORDERC0,1,1,PERIODTCOEFFICIENTSMA1SMA101436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