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depmixS4AflexiblepackagetoestimatemixtureandhiddenMarkovmodelsMaartenSpeekenbrink(m.speekenbrink@ucl.ac.uk)andIngmarVisserLondonR,10September2013DepmixS4TheobjectiveofthepackageistoprovideaflexibleimplementationofmixtureandhiddenMarkovmodels.Mixturecomponentsare(mostly)implementedasgeneralizedlinearmodels.UsingS4,userscaneasilydefinetheirownobservationmodels.AnRpackagetoestimatedependentmixturemodelsWrittenbyIngmarVisser(UniversityofAmsterdam)andMaartenSpeekenbrink(UniversityCollegeLondon)Firstversion(0.1.0)wasreleasedonCRANon26-Mar-2008.Thecurrentdevelopmentversionis1.3.0(availableonRForgeandsoononCRAN).···2/49MixturemodelsInamixturemodel,eachobservationisassumedtobedrawnfromoneofanumberofdistinctsubpopulations(componentdistributions).Whichsubpopulationanobservationisdrawnfromisnotdirectlyobservableandrepresentedbyalatentstate.Amixturedistributionoverobservations,,isdefinedaswhereYtt=1,…,Tp(=y)=p(=y|=i)P(=i)Yt∑i=1NYtStStdenotesthelatentstate(a.k.a.class,component)ofobservationdenotestheprobabilitythatthelatentstateatequalsdenotesthedensityofobservation(evaluatedat),conditionaluponthelatentstatebeing;i.e.,itisthevalueofthe-thcomponentdensity(evaluatedat).·∈{1,…,N}Stt·P(=i)Stti·p(=y|=i)YtStYty=iStiy3/49Mixturedistribution:p()=p(|=i)P(=i)Yt∑i=1NYtStSt4/49Mixturedistribution:Modelcomponents:p()=p(|=i)P(=i)Yt∑i=1NYtStSt·p(|=1)=N(5.48,0.13)YtSt5/49Mixturedistribution:Modelcomponents:p()=p(|=i)P(=i)Yt∑i=1NYtStSt·p(|=1)=N(5.48,0.13)YtSt·P(=1)=0.33St6/49Mixturedistribution:Modelcomponents:p()=p(|=i)P(=i)Yt∑i=1NYtStSt·p(|=1)=N(5.48,0.13)YtSt·P(=1)=0.33St·p(|=2)=N(6.31,0.32)YtSt7/49Mixturedistribution:Modelcomponents:p()=p(|=i)P(=i)Yt∑i=1NYtStSt·p(|=1)=N(5.48,0.13)YtSt·P(=1)=0.33St·p(|=2)=N(6.31,0.32)YtSt·P(=2)=0.67St8/49Mixturedistribution:Modelcomponents:p()=p(|=i)P(=i)Yt∑i=1NYtStSt·p(|=1)=N(5.48,0.13)YtSt·P(=1)=0.33St·p(|=2)=N(6.31,0.32)YtSt·P(=2)=0.67St·p()Yt9/49DependentmixturemodelsInadependentmixturemodel,statesareassumedtobestatisticallydependent.Theprocessunderlyingstatetransitionsisahomogenousfirst-orderMarkovprocess.ThisprocessiscompletelydefinedbytheinitialstateprobabilitiesandthestatetransitionmatrixP(=1),…,P(=N)S1S1⎛⎝⎜⎜⎜⎜P(=1|=1)StSt−1P(=1|=2)StSt−1⋮P(=1|=N)StSt−1P(=2|=1)StSt−1P(=2|=2)StSt−1⋮P(=2|=N)StSt−1⋯⋯⋱⋯P(=N|=1)StSt−1P(=N|=2)StSt−1⋮P(=N|=N)StSt−1⎞⎠⎟⎟⎟⎟10/49ImplementationindepmixS4ModelsareestimatedusingExpectation-Maximization(EM)ornumericaloptimization(whenparametersareconstrained).Thestructureofadependentmixturemodelallowsittobedividedintothreesubmodels:Thepriorandtransitionmodelsareimplementedasmultinomialregressionmodels,andtheresponsemodelsasgeneralizedlinearmodels.WithintheEMalgorithm,themaximization-stepcanbeperformedbyweightedmaximumlikelihood(asine.g.,glm.fit()).Thepriormodel:Thetransitionmodel:Theresponsemodel:·P(|x,)S1θprior·P(|x,,)StSt−1θtrans·P(|,x,)YtStθresp11/49UsingdepmixS4Mixturemodelscanbeconstructedusingthemix()function,andhiddenMarkovmodelswiththedepmix()function.Thedepmix()functiontakesthefollowingarguments:MoregeneralmodelscanbeconstructedwiththemakeDepmix()function.Modelsareestimatedbycallingthefit()functiononaconstructedmodel.response:aformulaspecifyingtheresponsemodels(univariate)oralistwithformulae(multivariate)1.nstates:thenumberofstates/components2.data:thedataframecontainingthevariablesintheresponsemodels3.family:thefamilyoftheresponsemodels(asintheglmfunction)oralistwithfamilies(multivariate)4.ntimes:avectorwiththelengthofeachtime-seriesinthedata.5.12/49Example1:ClimatechangeClimatechangeDatafromtheWaterCorporationofWesternAustraliacontainingtheyearlyinflowinwatercatchmentdamsaroundPerth,South-westAustraliafrom1911throughto2012.14/49ClimatechangeIsthereatrendinthesedata?1.Aretheresuddenshiftsinthecatchmentamounts?2.15/49Typicalanalyses:linearmodels(bothlinearandquadratictrendsaresignificant)lm1-lm(water~1,data=perth)lmyr-lm(water~yr,data=perth)lmyr2-lm(water~yr+I(yr^2),data=perth)anova(lm1,lmyr,lmyr2)AnalysisofVarianceTableModel1:water~1Model2:water~yrModel3:water~yr+I(yr^2)Res.DfRSSDfSumofSqFPr(F)1101381169521003257024155467118.73.7e-05***3992939677131734710.70.0015**---Signif.codes:0'***'0.001'**'0.01'*'0.05'.'0.1''116/49Typicalanalyses:linearmodels17/49Typicalanalyses:ARmodelsarOrder-ar(perth$water)$orderar1-arima(perth$water,c(arOrder,0,0))aryr-arima(perth$water,c(arOrder,0,0),xreg=perth$yr)aryr2-arima(perth$water,c(arOrder,0,0),xreg=cbind(yr=scale(perth$yr),yr2=scale(perth$yr)^2))print(c(ar1=AIC(ar1),aryr=AIC(aryr),aryr2=AIC(aryr2)))ar1aryraryr2135513491344#Comparetolinearmodels:innerCodeprint(c(lm1=AIC(lm1),lmyr=AIC(lmyr),lmyr2=AIC(lmyr2)))lm1lmyrlmyr213671353134518/49Typicalanalyses:ARmodels19/49Transitionmatrixfor1changepointTransitionmatrixfor2changepointsTransitionmatrixfor3changepointsEtceteraInHMMliteraturethesemodelsarealsocalledleft-rightmodelsorBakismodelsChangepointmodelsCanbeestimatedwithhiddenMarkovmodelsbyrestrictingthetransitionmatrixAssumeoneormorediscretechangepointsinthedata1.Mean,trend,and/orotherparameters(egarparameters)oftheprocessma
本文标题:depmixS4-的使用例程
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