1 maximum likelihood and bayesian methods for mixtures of normal distributions.

《1 maximum likelihood and bayesian methods for mixtures of normal distributions.》由会员上传分享，免费在线阅读，更多相关内容在教育资源-天天文库。

1、MAXIMUMLIKELIHOODANDBAYESIANMETHODSFORMIXTURESOFNORMALDISTRIBUTIONS.PeterM.Saama.UCLAOfficeofAcademicComputing.November,1997.A.ABSTRACTDatawerewaitingtimesbetweeneruptionsoftheOldFaithfulgeyserinYellowstoneNationalPark,Wyoming,USA.Thesamplehistogramshowedevidenceofbimodalit

2、y.Forthwith,atwo-componentnormalmixturemodelwasfittedtothedata.TheGauss-Newtonalgorithmwasusedtoobtainthemaximumlikelihoodestimateofthenuisanceparametersinthemixturemodel.AnalternativemethodwhichusestheGibbsSamplertoobtainparameterestimatesaswellas100(1-FUHGLEOHEDQGVIR

3、UWKHSDUDPHWHUVZDVLPSOHPHQWHGDQGLVSUHVHQWHGB.INTRODUCTIONBecauseofoverdispersionandheterogeneityinthepopulation,amixtureofdistributionsisoftenusedtomodelthequantitativeresponse.Suchdistributionsareoftenconsideredappropriatemodelsforthoughttoconsistofanumberofrelative

4、lydistinctsub-populations(c).Insituationswherethenumberofcomponentsisunknown,mixturedensitiesoftheformk2∑πjN(θj,σj)j=1havefoundtheirwidestapplicationsasamodelbasedclusteringprocedure;πisthejthprobabilitythatobservationycomesfromcomponentjofthemixture.Hereinθ=λ,τ,πiwilldenot

5、ethesetofallunknownparametersandp(..)isusedtodenoteagenericconditionalprobabilitydensityfunction.AmixtureoftwonormaldensitieswasfirstconsideredbyPearsonin1894withparameterestimatesobtainedfromthemethodofmomentsandinvolvedthesolutionofaninth-degreepolynomial.Theseminalpapero

6、ntheEMalgorithm(Dempster,LairdandRubin,1977)hasgreatlystimulatedworkonfinitemixturesofdistributions.Applicationsofmixturemodels1reportedbyTitterington,SmithandMakov(1985)andMcLachlanandBasford(1988)usetheExpectationMaximization(EM)algorithm.Itsdisadvantagesinclude:•extremes

7、lownessofconvergencewhentheproportionofmissingdataishigh;•absenceofstandarderrorsfromtheinformationmatrixatconvergence.CompetitorsofEMareGauss-Newton(Lois,1982;Aitkinetal,1994),FisherScoring(Rao,1948),andDifferentialEvolution(PriceandStorn,1997).TheGauss-Newton(GN)algorithm

8、,isnotguaranteedtoconvergewhenthelog-likelihoodisnotconcavebutwhenitdoesconverge,t