Bayesian semiparametric regression analysis of mul

BayesianSemiparametricRegressionAnalysisofMulticategoricalTime-SpaceDataLudwigFahrmeirandStefanLangUniversityofMunichLudwigstrasse33D-80539Muniche-mail:[e-mail]@stat.uni-muenchen.deSUMMARYWepresentauniedsemiparametricBayesianapproachbasedonMarkovrandomeldpriorsforanalyzingthedependenceofmulticategoricalresponsevariablesontime,spaceandfurthercovariates.Thegeneralmodelextendsdynamic,orstatespace,modelsforcategoricaltimeseriesandlongitudinaldatabyincludingspatialeectsaswellasnonlineareectsofmetricalcovariatesinexiblesemiparametricform.Trendandseasonalcomponents,dierenttypesofcovariatesandspatialeectsarealltreatedwithinthesamegeneralframeworkbyassigningappropriatepriorswithdierentformsanddegreesofsmoothness.InferenceisfullyBayesianandusesMCMCtechniquesforposterioranalysis.Weprovidetwoapproaches:Therstoneisbasedondirectevaluationofobservationlikelihoods.Thesecondoneisbasedonlatentsemiparametricutilitymodelsandisparticularlyusefulforprobitmodels.Themethodsareillustratedbyapplicationstounemploymentdataandaforestdamagesurvey.KEYWORDS:Categoricaltime-spacedata,forestdamage,Markovrandomelds,MCMC,semiparametricBayesianinference,unemployment.1IntroductionMulticategoricallongitudinaldataconsistsofobservations(Yit;xit),i=1;:::;n,t=1;:::;T,forapopulationofnunitsobservedacrosstime,wheretheresponsevariableYisobservedinorderedorunorderedcategoriesr2f1;:::;kg.Covari-atesmaybetime-constantortime-varying.ForTsmallcomparedton,generalizedestimatingequationapproachesareapopularchoicefordataanalysis.Formod-erateorlargerT,dynamicorstatespacemodelsareausefulalternative,see,e.g.,FahrmeirandTutz(1994,2000,ch.8).Forthespecialcase(n=1)ofcategoricaltimeseries,dynamicgeneralizedlinearmodelsareameanwhilewellestablishedtoolforapproximateorfullBayesianinference.Inthispaper,weconsidermulticategoricaltime-spacedata,wherethespatialloca-tionorsitesonaspatialarrayf1;:::;s;:::;Sgisgivenforeachunitasanadditional1information.Wealsodistinguishbetweenmetricalcovariatesxt=(xt1;:::;xtp)0,whoseeectswillbemodelledandestimatednonparametrically,andafurthervec-torwtofcovariates,whoseeectswillbemodelledparametricallyinusuallinearform.Multicategoricaltime-spacedataonnindividualsorunitsthenconsistsofobservations(Yit;xit;wit;si);i=1;:::;n;t=1;:::;T;(1.1)wheresi2f1;:::;Sgisthelocationofindividuali.AtypicalexamplearemonthlyregisterdatafromtheGermanEmploymentOcefortheyears1980-1995,whereYitistheemploymentstatus(e.g.unemployed,parttimejob,fulltimejob)ofindividualiduringmonthtandsiisthedistrictinGermanywhereihasitsdomicile.Datafromsurveysonforesthealthareafurtherexample:DamagestateYitoftreeiinyeart,indicatedbythedefoliationdegree,ismeasuredinorderedcategories(severetonone)andsiisthesiteofthetreeonalatticemap.Inbothexamples,covariatescanbecategoricalorcontinuous,andpossiblytime-varying.Ingeneral,time-spacedataofthiskindcannotbeanalyzedadequatelywithexistingnonparametricorconventionalparametricmethods.Wepresentauniedsemipara-metricBayesianframeworkforjointlymodellingandanalyzingeectsoftime,spaceanddierenttypesofcovariatesoncategoricalresponses.Trendorseasonalcompo-nents,spatialeects,metricalcovariateswithnonlineareectsandusualcovariateswithxedeectsarealltreatedwithinthesamegeneralframeworkbyassigningappropriatepriorswithdierentformsanddegreesofsmoothness.Thisbroadclassofmodelscontainsstatespacemodelsforcategoricaltimeseriesconsideredinpre-viousworkasaspecialcase.InferenceisfullyBayesianandusesrecentMCMCtechniques.WesuggesttwoapproachesforMCMCinference.Therstoneisuse-fulifthelikelihoodofthedata,givencovariatesandunknownparameters,canbeeasilycomputedasforcumulativeormultinomiallogisticmodels.MarkovchainsamplesarethengeneratedbyanextensionofMetropolis-Hastingsalgorithmsde-velopedinFahrmeirandLang(1999)forunivariateresponses.Thesecondapproachisbasedonlatentvariables,wheretheobservablecategoricalresponsesaregeneratedthroughthresholdorutilitymechanisms.ForlatentGaussianvariablesthisleadstomulticategoricalprobitmodels,seeAlbertandChib(1993)forthesimplercaseoflinearpredictors,andYau,KohnandWong(2000)fornonparametricregressionusingbasisfunctions.ForMCMCinference,Gaussianlatentvariablesareconsid-eredasunknownadditionalparametersandaregeneratedjointlywiththeotherparametersinaGibbssamplingscheme.EcientmethodsforsamplingfromhighdimensionalGaussianMarkovrandomeldsareincorporatedasamajorbuildingblock.Section2describesourBayesiansemiparametricregressionmodelsforcategoricalresponses,observedacrosstimeandspace,anddependingonunknownfunctionsandparameters.MCMCalgorithmsarepresentedinSection3.InSection4,themethodsareappliedtoreemploymentchancesbasedoncategoricaltime-spacedataon(un-)employmentstatusandtodatafromaforesthealthinventory.22SemiparametricBayesianmodelsformulticat-egoricaltime-spacedataCategoricalresponsemodelsmaybemotivatedfromtheconsiderationoflatentvariables.Thisisnotonlyusefulforconstructionofmodels,butalsoforBayesianinference,treatinglatentvariablesasadditionalunknownparameters.ForthecaseofanominalresponseYwithunorderedcategori