ARIMA模型
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2023年3月20日发(作者：外敷)ARIMA（预测时间序列的模型）
Autoregressiveintegratedmovingaverage(⼀种预测时间序列的模型)ARIMAInstatisticsandeconometrics,andinparticular
intimeseriesanalysis,anautoregressiveintegratedmovingaverage(ARIMA)modelisageneralisationofanautoregressive
movingaverage(ARMA)odelsarefittedtotimeseriesdataeithertobetterunderstandthedataortopredict
eappliedinsomecaseswheredatashowevidenceofnon-stationarity,whereaninitial
differencingstep(correspondingtothe"integrated"partofthemodel)canbeappliedtoremovethenon-stationarity.
ThemodelisgenerallyreferredtoasanARIMA(p,d,q)modelwherep,d,andqareintegersgreaterthanorequaltozeroand
refertotheorderoftheautoregressive,integrated,odelsform
eofthetermsiszero,it'susualtodropAR,I
mple,anI(1)modelisARIMA(0,1,0),andaMA(1)modelisARIMA(0,0,1).
Contents
[hide]
1Definition
2ForecastsusingARIMAmodels
3Examples
4Implementationsinstatisticspackages
5Seealso
6References
7Externallinks
Definition
GivenatimeseriesofdataXtwheretisanintegerindexandtheXtarerealnumbers,thenanARMA(p,q)modelisgivenby:
whereListhelagoperator,theαiaretheparametersoftheautoregressivepartofthemodel,theθiaretheparametersofthe
ortermsaregenerallyassumedtobeindependent,
identicallydistributedvariablessampledfromanormaldistributionwithzeromean.
canberewrittenas:
AnARIMA(p,d,q)processexpressesthispolynomialfactorisationproperty,andisgivenby:
andthuscanbethoughtasaparticularcaseofanARMA(p+d,q)processhavingtheauto-regressivepolynomialwithsome
sreasoneveryARIMAmodelwithd>0isnotwidesensestationary.
ForecastsusingARIMAmodels
ARIMAmodelsareusedforobservablenon-stationaryprocessesXtthathavesomeclearlyidentifiabletrends:
constanttrend(-zeroaverage)leadstod=1
lineartrend(rgrowthbehavior)leadstod=2
quadratictrend(aticgrowthbehavior)leadstod=3
InthesecasestheARIMAmodelcanbeviewedasa"cascade"stisnon-stationary:
whilethesecondiswide-sensestationary:
NowstandardforecaststechniquescanbeformulatedfortheprocessYt,andthen(havingthesufficientnumber
ofinitialconditions)Xtcanbeforecastedviaopportuneintegrationsteps.
[edit]Examples
mple,anARIMA(0,1,0)modelisgivenby:
whichissimplyarandomwalk.
mple,ifmultipletimeseries
areusedmes
mple,consideramodelofdailyroadtrafficvolumes.
caseitisoftenconsideredbetterto
useaSARIMA(seasonalARIMA)modelthantoincreasetheorderoftheARorMApartsofthemodel.
Ifthetime-seriesissuspectedtoexhibitlong-rangedependencethenthedparametermaybereplacedby
certainnon-integervaluesinanautoregressivefractionallyintegratedmovingaveragemodel,whichis
alsocalledaFractionalARIMA(FARIMAorARFIMA)model.
[edit]Implementationsinstatisticspackages
InR,ctionisdocumentedin"ARIMA
ModellingofTimeSeries".BesidestheARIMA(p,d,q)part,thefunctionalsoincludesseasonal
factors,aninterceptterm,andexogenousvariables(xreg,called"externalregressors").
[edit]Seealso
Autocorrelation
ARMA
X-12-ARIMA
Partialautocorrelation
[edit]References
Mills,dgeUniversityPress,1990.
Percival,alAnalysisforPhysicalApplications.
CambridgeUniversityPress,1993.
Autoregressiveintegratedmovingaverage(ARIMA)isoneofthepopularlinearmodelsintimeseriesforecasting