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《8.Stochastic Modeling》由會(huì)員上傳分享，免費(fèi)在線閱讀，更多相關(guān)內(nèi)容在學(xué)術(shù)論文-天天文庫。
1、Stochastic?modeling?A.W.Jayawardena?MastersCourseinDisasterManagementInternationalCentreforWaterHazardandRiskManagement(ICHARM)STOCHASTICCOMPONENTAfterthetrendandtheperiodiccomponenthavebeenremovedtheremainingstochasticcomponentwhichisassumedtobecovariancestationarymayconsistofadependent(cor
2、related)partandanindependent(uncorrelated)randompart.Fourdifferenttypesofstochasticmodelscanbeusedtodescribethedependentpart;namely,?Autoregressive(AR)?MovingAverage(MA)?AutoregressiveandMovingAverage(ARMA)?AutoregressiveIntegratedMovingAverage(ARIMA)Inalltheabovefourtypesofmodels,thepresent
3、valueofthestochasticvariableislinearlyrelatedtothepastvaluesinsomeform.Theyarethereforeidentifiedaslinearstochasticmodels.StationarityisalsoimpliedintheAR,MAandARMAtypesmodels.1.AUTOREGRESSIVE(AR)MODELSInautoregressivemodels,thecurrentvalueofthevariableislinearlyrelatedtotheweightedsumofanum
4、berofpastvaluesandanindependentrandomvalue.Thegeneralp'thorderARmodelhastheform:zt=φp,1zt-1+φp,2zt-2+......+φp,pzt-p+ηtp=∑φp,izt?i+ηt(1)i=1whereφp,i'sarecalledtheautoregressivecoefficients,ηtisanindependent(uncorrelated)randomnumberandztisthestochasticcomponentwhichisobtainedbysubtractingany
5、trendsandperiodicitiesfromtheoriginaltimeseries.Forconvenience,ztisreducedtozeromeanandunitvariance(normalised).(i)PropertiesofAutoregressivemodelsThefollowingpropertiesformthebasisofmodeldevelopment:2E(zt)=E(ηt)=0?()()22?Varz=Ez=σttz?22Var()η=E()η=σ?ttη2?ρk=E()ztzt?k/σz?(2)E()()ηη=Eηz=0fork
6、=1,2,3...?tt?ktt?k?p22???ση=σz??1?∑φp,iρi????i=1??Inthelastoftheaboveequations,thesummationiscalledthecoefficientof2determination,R.Itisthesquareofthemultiplecorrelationcoefficient.An2unbiasedestimateofσmaybeobtainedbymultiplyingbyN/(N-p).AnARprocessη2iscompletelyknownifφp,i'sandσareknown.η(
7、ii)EstimationofparametersMultiplyingthegeneralautoregressiveequationbyzt-1,zt-2,....,zt-p,inturn,andtaking1expectations,thefollowingpequationscanbeobtained.TheyarecalledtheYule-Walkerequations(AfterYule(1927),andWalker(1931)).?ρ1??1ρ1ρ2....ρp?1??φp
