基于ABC-SMC回归算法具有观测噪声的AR(p)模型的参数估计

基于ABC-SMC回归算法具有观测噪声的AR(p)模型的参数估计Introduction:Autoregressivemodelsarewidelyusedinvariousfieldssuchastimeseriesanalysis,finance,andeconometrics.Theyprovideavaluabletoolformodelingandpredictingtime-dependentdata.Inthispaper,weproposeanextensionoftheApproximateBayesianComputationSequentialMonteCarlo(ABC-SMC)algorithmforparameterestimationinAutoregressivemodelswithobservationnoise,specificallyAR(p)models.Background:Autoregressivemodelsareaclassoftimeseriesmodelsthatusepastobservationstopredictfuturevalues.TheAR(p)modelassumesthatthecurrentobservationisalinearcombinationoftheppreviousobservationsplusanerrorterm.TheequationsforanAR(p)modelcanbewrittenas:Y_t=c+phi_1*Y_{t-1}+phi_2*Y_{t-2}+...+phi_p*Y_{t-p}+epsilon_tWhereY_tisthecurrentobservation,cisaconstantterm,phi_1tophi_paretheautoregressivecoefficients,andepsilon_trepresentstheobservationnoise.Thegoalistoestimatetheunknownparameters(c,phi_1,...,phi_p)basedontheobserveddata.ABC-SMCAlgorithm:ApproximateBayesianComputationSequentialMonteCarlo(ABC-SMC)isasimulation-basedmethodusedforBayesianinferenceincomplexmodels.Itisparticularlyusefulwhenthelikelihoodfunctionisintractableordifficulttocompute.TheABC-SMCalgorithminvolvessimulatingdatafromthemodelwithdifferentparametervaluesandcomparingthesimulateddatatotheobserveddatausingasummarystatistic.AR(p)ModelwithObservationNoise:Inthispaper,weconsiderthecasewheretheAR(p)modelisaffectedbyobservationnoise.WeassumethattheobservationnoisefollowsaGaussiandistributionwithzeromeanandunknownvariancesigma^2.Hence,theequationfortheAR(p)modelwithobservationnoisebecomes:Y_t=c+phi_1*Y_{t-1}+phi_2*Y_{t-2}+...+phi_p*Y_{t-p}+epsilon_tWhereepsilon_t~N(0,sigma^2).ParameterEstimationusingABC-SMC:ThegoalofparameterestimationintheAR(p)modelwithobservationnoiseistofindtheposteriordistributionoftheparametersgiventheobserveddata.However,thelikelihoodfunctionisintractableduetothepresenceofobservationnoise.Therefore,weresorttousingtheABC-SMCalgorithmforapproximateBayesianinference.TheABC-SMCalgorithmproceedsasfollows:1.Initializethepopulationofparticleswithrandomlyselectedparametervaluesfromthepriordistributions.2.Foreachparticle,simulatedatafromtheAR(p)modelwiththegivenparametervalues.3.Computeasummarystatisticofthesimulateddataandtheobserveddata.4.Ifthedifferencebetweenthesummarystatisticsisbelowapredefinedthreshold,accepttheparticle.5.Otherwise,perturbtheparametervaluesoftheparticleandgotostep2.6.Repeatsteps2-5untilasufficientnumberofparticleshavebeenaccepted.7.Updatethethresholdsandrepeatsteps2-6untilconvergenceisreached.Results:WeappliedtheproposedABC-SMCalgorithmtoestimatetheparametersofanAR(p)modelwithobservationnoiseusingsyntheticdata.Theresultsdemonstratetheeffectivenessofthealgorithminaccuratelyestimatingtheparameters.Conclusion:Inthispaper,wehavepresentedanextensionoftheABC-SMCalgorithmforparameterestimationinAR(p)modelswithobservationnoise.Thealgorithmprovidesanefficientandrobustapproachforestimatingtheunknownparameters.Theresultsshowtheaccuracyandeffectivenessofthepropos