Statistical Modelling and Regression Structures

Thecollectedcontributionscontainedwithinthisbookhavebeenwrittenbyfriends andcolleaguestoacknowledgeLudwigFahrmeir'swidespreadandimportantimpact onStatisticsasascience,whilecelebratinghis65thBirthday. Asayoungstudent,LudwigstartedhiscareerasaMathematician,buthequickly turnedintoarisingandshiningstarwithintheGermanandinternationalStatistics community. HesoonobtainedbothhisPhDandhisHabilitationattheTechnical UniversityofMunich. AfterashortperiodasavisitingprofessorattheUniversity ofDortmund,hereturnedtohishomelandBavariaandwasappointedFullProfessor ofStatisticsattheUniversityofRegensburg,attheageof32. Someyearslater,hemovedtothecapitalofBavariaandbecameProfessoratthe DepartmentofStatisticsattheUniversityofMunich. Hisappointmenthadsigni?cant impactontheDepartmentsince,soonafterhisarrival,Ludwigstartedaninitiative toestablishacollaborativeresearchcenteronthe"StatisticalAnalysisofDiscrete Structures. "Afterasuccessfulapplicationforinitialfunding,furtherfundingwas extendedseveraltimes,untiltheresearchcenterreachedthemaximumperiodfor fundingin2006. Duringthecompleteduration,Ludwigservedasaspeakerofthe researchcenterand-tociteoneofthe?nalreferees-"manageditinaneasyand ef?cientwayandcontributedseveralimportantresults." Duringthelastfortyyears,Ludwig'sworkhashadtremendousimpactontheS- tisticscommunity. Hewasamongthe?rstresearcherstorecognizetheimportance ofgeneralizedlinearmodelsandcontributedinaseriesofpaperstothetheoretical backgroundofthatmodelclass. Hisinterestinstatisticalmodellingledtothe- ganizationofaworkshopon"StatisticalModellingandGeneralizedLinearModels (GLIM)"inMunichin1992andculminatedinthehighlycitedmonographon"M- tivariateStatisticalModellingBasedonGeneralizedLinearModels"thatsawtwo printingsandremainstobeakeyreferenceonappliedstatisticalmodellingutilizing generalizedlinearmodels. Ludwigalsohadgreatin?uenceonthecreationofthe StatisticalModellingSociety,andiscurrentlyontheadvisoryboardofthecor- spondingjournalon"StatisticalModelling. "Boththesocietyandjournalemerged outoftheearlyGLIMworkshopsandproceedings. v vi Foreword Ofcourse,Ludwig'sworkisde?nitelynotrestrictedtogeneralizedlinearmodels but-onthecontrary-spansawiderangeofmodernStatistics. Heco-authoredor co-editedseveralmonographs,e. g. onMultivariateStatistics,StochasticProcesses, MeasurementofCreditRisks,aswellaspopulartextbooksonRegressionandan IntroductiontoStatistics.Hisrecentresearchcontributionsaremostlyconcentrated insemiparametricregressionandspatialstatisticswithinaBayesianframework. When?rstcirculatingtheideaofaFestschriftforthecelebrationofLudwig's 65thbirthday,allpotentialcontributorswereextremelypositive,manyimmediately agreeingtocontribute. ThesereactionsatesttoLudwig'shighpersonalandp- fessionalappreciationinthestatisticalcommunity. Thefarreachingandvarietyof subjectscoveredwithinthesecontributionsalsorepresentsLudwig'sbroadinterest andimpactinmanybranchesofmodernStatistics. BotheditorsofthisFestschriftwereluckyenoughtoworkwithLudwigatseveral occasionsandinparticularearlyintheircareersasPhDstudentsandPostDocs. His personalandprofessionalmentorshipandhisstrongcommitmentmadehimaperfect supervisorandhispatient,con?dentandencouragingworkingstylewillalwaysbe rememberedbyallofhisstudentsandcolleagues. Ludwigalwaysprovidedafriendly workingenvironmentthatmadeitapleasureandanhonortobeapartofhisworking group. WeareproudtobeabletosaythatLudwigismuchmorethanacolleague butturnedintoafriendforbothofus.OldenburgandMunich,January2010 ThomasKneib,GerhardTutz Acknowledgements Theeditorswouldliketoexpresstheirgratitudeto * allauthorsofthisvolumefortheiragreementtocontributeandtheireasyco- erationatseveralstagesofputtingtogetherthe?nalversionoftheFestschrift. * JohannaBrandt,JanGertheiss,AndreasGroll,FelixHeinzl,SebastianPetry,Jan UlbrichtandStephanieRubenbauerfortheirinvaluablecontributionsinproof- A readingandcorrectionofthepapers,aswellasinsolvingseveralLTX-related E problems. *swidespreadandimportantimpact onStatisticsasascience,whilecelebratinghis65thBirthday. Asayoungstudent,LudwigstartedhiscareerasaMathematician,buthequickly turnedintoarisingandshiningstarwithintheGermanandinternationalStatistics community. HesoonobtainedbothhisPhDandhisHabilitationattheTechnical UniversityofMunich. AfterashortperiodasavisitingprofessorattheUniversity ofDortmund,hereturnedtohishomelandBavariaandwasappointedFullProfessor ofStatisticsattheUniversityofRegensburg,attheageof32. Someyearslater,hemovedtothecapitalofBavariaandbecameProfessoratthe DepartmentofStatisticsattheUniversityofMunich. Hisappointmenthadsigni?cant impactontheDepartmentsince,soonafterhisarrival,Ludwigstartedaninitiative toestablishacollaborativeresearchcenteronthe"StatisticalAnalysisofDiscrete Structures."Afterasuccessfulapplicationforinitialfunding,furtherfundingwas extendedseveraltimes,untiltheresearchcenterreachedthemaximumperiodfor fundingin2006. Duringthecompleteduration,Ludwigservedasaspeakerofthe researchcenterand-tociteoneofthe?nalreferees-"manageditinaneasyand ef?cientwayandcontributedseveralimportantresults. " Duringthelastfortyyears,Ludwig'sworkhashadtremendousimpactontheS- tisticscommunity. Hewasamongthe?rstresearcherstorecognizetheimportance ofgeneralizedlinearmodelsandcontributedinaseriesofpaperstothetheoretical backgroundofthatmodelclass. Hisinterestinstatisticalmodellingledtothe- ganizationofaworkshopon"StatisticalModellingandGeneralizedLinearModels (GLIM)"inMunichin1992andculminatedinthehighlycitedmonographon"M- tivariateStatisticalModellingBasedonGeneralizedLinearModels"thatsawtwo printingsandremainstobeakeyreferenceonappliedstatisticalmodellingutilizing generalizedlinearmodels. Ludwigalsohadgreatin?uenceonthecreationofthe StatisticalModellingSociety,andiscurrentlyontheadvisoryboardofthecor- spondingjournalon"StatisticalModelling. "Boththesocietyandjournalemerged outoftheearlyGLIMworkshopsandproceedings. v vi Foreword Ofcourse,Ludwig'sworkisde?nitelynotrestrictedtogeneralizedlinearmodels but-onthecontrary-spansawiderangeofmodernStatistics.Heco-authoredor co-editedseveralmonographs,e. g. onMultivariateStatistics,StochasticProcesses, MeasurementofCreditRisks,aswellaspopulartextbooksonRegressionandan IntroductiontoStatistics. Hisrecentresearchcontributionsaremostlyconcentrated insemiparametricregressionandspatialstatisticswithinaBayesianframework. When?rstcirculatingtheideaofaFestschriftforthecelebrationofLudwig's 65thbirthday,allpotentialcontributorswereextremelypositive,manyimmediately agreeingtocontribute. ThesereactionsatesttoLudwig'shighpersonalandp- fessionalappreciationinthestatisticalcommunity. Thefarreachingandvarietyof subjectscoveredwithinthesecontributionsalsorepresentsLudwig'sbroadinterest andimpactinmanybranchesofmodernStatistics. BotheditorsofthisFestschriftwereluckyenoughtoworkwithLudwigatseveral occasionsandinparticularearlyintheircareersasPhDstudentsandPostDocs. His personalandprofessionalmentorshipandhisstrongcommitmentmadehimaperfect supervisorandhispatient,con?dentandencouragingworkingstylewillalwaysbe rememberedbyallofhisstudentsandcolleagues. Ludwigalwaysprovidedafriendly workingenvironmentthatmadeitapleasureandanhonortobeapartofhisworking group.WeareproudtobeabletosaythatLudwigismuchmorethanacolleague butturnedintoafriendforbothofus. OldenburgandMunich,January2010 ThomasKneib,GerhardTutz Acknowledgements Theeditorswouldliketoexpresstheirgratitudeto * allauthorsofthisvolumefortheiragreementtocontributeandtheireasyco- erationatseveralstagesofputtingtogetherthe?nalversionoftheFestschrift. * JohannaBrandt,JanGertheiss,AndreasGroll,FelixHeinzl,SebastianPetry,Jan UlbrichtandStephanieRubenbauerfortheirinvaluablecontributionsinproof- A readingandcorrectionofthepapers,aswellasinsolvingseveralLTX-related E problems. * theSpringerVerlagforagreeingtopublishthisFestschriftandinparticularNils- PeterThomas,AliceBlanck and FrankHolzwarthfor the smooth collabo- tion in preparing th emanuscript. vii Contents ListofContributors...xix TheSmoothComplexLogarithmandQuasi-PeriodicModels ...1 PaulH. C. Eilers 1 Foreword...1 2 Introduction...1 3 DataandModels...2 3. 1 TheBasicModel...3 3. 2 SplinesandPenalties...3 3. 3 StartingValues...7 3. 4 SimpleTrendCorrectionandPriorTransformation...8 3. 5 AComplexSignal...8 3. 6 Non-normalDataandCascadedLinks...10 3. 7 AddingHarmonics...11 4 MoretoExplore...12 5 Discussion...15 References...17 P-splineVaryingCoef?cientModelsforComplexData...19 BrianD. Marx 1 Introduction ...19 2 "LargeScale"VCM,withoutBack?tting...22 3 NotationandSnapshotofaSmoothingTool:B-splines...24 3. 1 GeneralKnotPlacement...25 3. 2 SmoothingtheKTBData...25 4 UsingB-splinesforVaryingCoef?cientModels...26 5 P-splineSnapshot:Equally-SpacedKnots&Penalization...28 5. 1 P-splinesforAdditiveVCMs...30 5. 2 StandardErrorBands...30 6 OptimallyTuningP-splines ...31 7 MoreKTBResults...33 8 ExtendingP-VCMintotheGeneralizedLinearModel...33 9 Two-dimensionalVaryingCoef?cientModels...36 ix x Contents 9. 1 Mechanicsof2D-VCMthroughExample ...37 9. 2 VCMsandPenaltiesasArrays...39 9. 3 Ef?cientComputationUsingArrayRegression...40 10 DiscussionTowardMoreComplexVCMs...41 References...42 PenalizedSplines,MixedModelsandBayesianIdeas...45 .. GoranKauermann 1 Introduction...45 2 NotationandPenalizedSplinesasLinearMixedModels...46 3 Classi?cationwithMixedModels...48 4 VariableSelectionwithSimplePriors...50 4. 1 MarginalAkaikeInformationCriterion...50 4. 2 ComparisoninLinearModels...