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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
