when-the-data-are-functions

PSYCHOMETRIKA--VOL.47,NO.4.DECEMBER,1982WHENTHEDATAAREFUNCTIONSJ.O.RAMSAYMCGILLUNIVERSITYAdatumisoftenacontinuousfunctionx(t)ofavariablesuchastimeobservedoversomeinterval.Oneormoresuchfunctionsareobservedforeachsubjectorunitofobservation.Theextensionofclassicaldataanalytictechniquesdesignedforp-variateobservationstosuchdataisdiscussed.Theessentialstepistheexpressionoftheclassicalprobleminthelanguageoffunction-alanalysis,afterwhichtheextensiontofunctionsisastraightforwardmatter.Aschematicdevicecalledthedualitydiagramisaveryusefultoolfordescribingananalysisandforsuggestingnewpossibilities.Leastsquaresapproximation,descriptivestatistics,principalcomponentsanalysis,andcanonicalcorrelationanalysisarediscussedwithinthisbroaderframework.Keywords:continuousdata,functionalanalysis,dualitydiagram.IntroductionSophisticateddatacollectionhardwareoftenproducedatawhichareasetofcon-tinuousfunctions.Iamsurethatallofushaveseensuchdata:EEGandEMGrecords,learningcurves,pathsinspace,subjectresponsescontinuousintime,speechproductionmeasurementsduringvocalization,bioassaydata,andsoon.Considerasafurtherexam-plethecurvesdisplayedinFigure1.Theseindicatetheheightofthetonguedorsumduringtendifferentutterancesofthesoundah-kahbyasinglesubject[Keller&Ostry,Note1].Itisnaturaltoconsidereachcurveasasingleobservation,tosummarizethetencurvesintermsofanaveragecurve,andtomeasureinsomewaythevariationofthecurvesaboutthisaverage.Thispaperconsiderstheextensionofclassicalstatisticaltechniquestoincludefunc-tionaldata.Itwillbeanelementaryandsimplifiedtreatment,whichmayannoythosewantingmoresubtletyandrigor.Imustwarnyou,however,thatafundamentalchangeofpointofviewaboutwhatdataarewillberequired,andifyouleavemyaddressawarethatanalteredstateofstatisticalconsciousnessispossible,Ishallbecontent.IndealingwithfunctionaldataIwillreferfrequentlytotwolinesofdevelopment.Thefirstistheexpressionoftraditionaldataanalytictechnologyinthelanguageoffunc-tionalanalysis.MuchofthisworkhastakenplaceinFranceandisnotavailableinEnglish.IamparticularlyindebtedtothemongraphsofCailliezandPages1-1976]andDauxoisandPousse[1976].Weareveryfortunatetohavewithusforthesemeetingsanumberofthoseassociatedwiththiswork,andinpartmytalkisonlyanintroductiontotomorrow'ssymposium.*Thesecondlineofdevelopmentthathasfascinatedmycol-leagueSuzanneWinsbergandIinrecentyearshasbeenstatisticalapplicationsofsplinefunctions.Ifeelthatsplinesaredestinedtoplayafundamentalroleintheanalysisoffunctionaldata,butIwilltrytoshowhowinonlyavaguewayatthispoint.Finally,this*Newglancesatprincipalcomponentsandcorrespondenceanalysiswasasymposiumatthe1982JointMeetingsoftheClassificationSocietyandPsychometricSociety,Montreal,Canada.PresentedasthePresidentialAddresstothePsychometricSociety'sAnnualMeeting,May,1982.IwishtoexpressmygratitudetomycolleaguesinFrance,especiallyattheUniversityofGrenoble,fortheirwarmhospitalityduringmysabbaticalleave.PreparationofthispaperwassupportedbyGrantAPA0320fromtheNaturalSciencesandEngineeringResearchCouncilofCanada.Requestsforreprintsshouldbesentto:J.O.Ramsay,Dept.ofPsychology,1205Dr.PenfieldAve.,Mon-treal,Qurbec,CanadaH3A1B1.0033-3123/82/1200-5004500.75/0©1982ThePsychometricSociety379380PSYCHOMETRIKATaNGUEMOVEMENTDURINGRH-KRHB.OI--I,iJ',r-E(f)0hJZIDI..-5.55.0q.5tl.00.0II.......!I0.2O.q0.60.8TIMEFIGURE11.0Theheightofthetonguedorsumovera400millisecondintervaloftimeduringwhichthesoundah-kahwasuttered.Eachcurverepresentsasingleutterance.Thesamesubjectwasinvolvedinalltenreplications.Theaveragecurveisrepresentedbyadashedlineandwascomputedbyaveragingthetencurvesateachpointintime.Thetimeunitshavebeenarbitrarilyscaledtotheinterval[0,1].paperwillbecorrectlyperceivedbymanyofthereadersofPsychometrikaasbeingageneralizationofthepioneeringworkofTucker[1958],anditisaprivilegetoagainacknowledgetheworkofsomeonewhohassooftenbeentherefirst.Figure2offersanapproachtotheconceptofafunctionaldatum.Intheupperleftcornerwehavethedomainoftheclassicaldatamatrix:eachofnsubjectsispairedwitheachofpvariablesandtoeachpairanumberxuisassignedastheconsequenceofanexperimentordatacollection.Asonemovesdownfromthiscorner,wecometothesituationwherenisineffectinfinityandwearediscussingpopulationcharacteristics.Letusnowfixthenumberofsubjectsnandallowthenumberofvariablesptoincreasewithoutlimit.Thisprocesscanbeextendedevenbeyondcountabilitytothe(/)Q)t/)t3¢,30E::3Zn(30J.O.RAMSAYDomainofNumberofpa)ooe0OOOQOQO~eSQ$$0D0beeooQ$O$On=a)XVariablesp=Co//381DatumXil,---,Xipxi(t),Ot=TFIGURE2Possibledomainsforstatisticalobservations.Thesedomainsdependonwhetherthenumberofreplicationsorcases(n)isfiniteorinfiniteandwhetherthenumberofvariablesorpointsofobservation(p)isfiniteorcorrespondstothepointsonacontinuum.situationwherethevariablesdefineacontinuum.AstheupperrightcornerofFigure2indicates,thedatanowofferanumberforeachpointonacontinuumforeachsubject,anditbecomesnaturaltousefunctionnotationx,.(t)todesignatethevalueassignedtoindividualiatpointtonthiscontinuum.Thelowerrightcornershowsthatonemayevenhaveaninfinityofsubjectsorcasestoconsider,althoughinthistalkIwi