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1、J.R.Statist.Soc.B(2006)68,Part5,pp.837857Afunctionalwavelet–kernelapproachfortimeseriespredictionAnestisAntoniadis,JosephFourierUniversity,Grenoble,FranceandEfstathiosPaparoditisandTheofanisSapatinasUniversityofCyprus,Nicosia,Cyprus[ReceivedApril2005.Fin
2、alrevisionJune2006]Summary.Weconsiderthepredictionproblemofatimeseriesonawholetimeintervalintermsofitspast.Theapproachthatweadoptisbasedonfunctionalkernelnonparametricregressionestimationtechniqueswhereobservationsarediscreterecordingsofsegmentsofanunder
3、-lyingstochasticprocessconsideredascurves.Thesecurvesareassumedtoliewithinthespaceofcontinuousfunctions,andthediscretizedtimeseriesdatasetconsistsofarelativelysmall,comparedwiththenumberofsegments,numberofmeasurementsmadeatregulartimes.Weestimateconditio
4、nalexpectationsbyusingappropriatewaveletdecompositionsofthesegmentedsamplepaths.Anotionofsimilarity,basedonwaveletdecompositions,isusedtocalibratetheprediction.Asymptoticpropertieswhenthenumberofsegmentsgrowsto1areinvestigatedundermildconditions,andanonp
5、arametricresamplingprocedureisusedtogenerate,inaflexibleway,validasymptoticpointwisepredictionintervalsforthetrajectoriespredicted.Weillustratetheusefulnessoftheproposedfunctionalwaveletkernelmethodologyinfinitesamplesituationsbymeansofasimulatedexampleand
6、tworeallifedatasets,andwecomparetheresultingpredictionswiththoseobtainedbythreeothermethodsintheliterature,inparticularwithasmoothingsplinemethod,withanexponentialsmoothingprocedureandwithaseasonalautoregressiveintegratedmovingaveragemodel.Keywords:α-mix
7、ing;Besovspaces;Exponentialsmoothing;Functionalkernelregression;Pointwisepredictionintervals;Resampling;Seasonalautoregressiveintegratedmovingaveragemodels;Smoothingsplines;Timeseriesprediction;Wavelets1.IntroductionInmanyreallifesituationsweseekinformat
8、ionontheevolutionofa(real-valued)continuoustimestochasticprocessX=.X.t/;t∈R/inthefuture.GivenatrajectoryofXobservedontheinterval[0,T],wewouldliketopredictthebehaviourofXontheentireinterval[T,T+δ],whereδ>0,ratherthanatspeci
