信号平稳性检验新方法

BORGNATetal.:TESTINGSTATIONARITYWITHSURROGATES:ATIME-FREQUENCYAPPROACH3469

cussed,basedeitheronadistance-basedapproachoronama-chinelearningtechnique(oneclass-SVM).Bothwaysarecom-plementarysincethefirstoneisparametricwhereasthesecondoneisnot.Asisusual,aparametricapproachbasedonaspe-cificmodelforthedistributionisnaturallymoreefficientintermsofrequireddatasize,butitisbasedontheassumptionthattheunderlyingmodeliscorrect,somethingwhichhastobeeitherknownaprioriorassessed.Onthecontrary,anonpara-metricapproachsuchasSVMdoesnotrequiresuchassump-tions,butitismoredemandingintermsofdatasizeandcompu-tation.Moreover,forthedistance-basedapproach,someanalyt-icalstudiescanbeconducted(possiblyinasymptoticsituations,see[34]),whereastheSVMapproachallowsformoreversatilecharacterizationoftypesofnonstationarity.Asforcomparingtoearlyworksusingasymptoticanalysisofevolutionaryspectra[11],[12]toformulateaglobaltestofstationarity,theassump-tionofindependencebetweenthetime-frequencybinsrequiresthatmanydatainthetime-frequencyrepresentationshavetobediscardedforthetest.Ourresamplingmethodwithsurrogatesallowforlighteningthislimitationandusingallthebins,henceprovidingamethodthatusesallavailableinformation.

Thebasicprinciplesofthemethodhavebeenoutlined,withanumberofconsiderationsrelatedtoitsimplementation,butitisclearthattheproposedframeworkstillleavesroomformorethoroughinvestigationsaswellasvariationsand/orextensions.Intermsoftime-frequencydistributionsforinstance,onecouldimaginetogobeyondspectrogramsandtakeadvantageofmorerecentadvances[35].Two-dimensionalextensionscanalsobeenvisionedfortestingstationarityinthesenseofhomogeneityofrandomfields,e.g.,fortextureanalysis.Preliminaryresultsinthisdirectionaregivenin[36].

ACKNOWLEDGMENT

TheauthorsthankA.Ferrari(fromLaboratoireFIZEAU(UMRCNRS6525),ObservatoiredelaCôted’Azur,Univer-sitédeNiceSophia-Antipolis)forinterestingdiscussionsaboutstatisticsofsurrogatedata.

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3470PierreBorgnat(M’06)wasborninPoissy,France,in1974.HereceivedtheProfesseur-AgrégédeSciencesPhysiquesdegree,theM.Sc.degreeinphysics,andthePh.D.degreeinphysicsandsignalprocessing,allfromtheÉcoleNormaleSupérieuredeLyon,France,in1997,1999,and2002,respec-tively.

In2003–2004,hespentoneyearintheSignalandImageProcessinggroupoftheIRS,IST,Lisbon,Por-tugal.SinceOctober2004,hehasbeenafull-timeCNRSresearcherwiththeLaboratoiredePhysique,

ÉNSLyon.Hisresearchinterestsareinstatisticalsignalprocessingofnonsta-tionaryprocesses(time-frequencyrepresentations,timedeformations,station-aritytests)andscalingphenomena(time-scale,wavelets)forcomplexsystems(turbulence,networks,etc.).HeisalsoworkingonInternettrafficmeasurementsandmodeling,andinanalysisandmodelingofdynamicalcomplex

networks.

PatrickFlandrin(M’85–SM’01–F’02)receivedtheEngineerdegreefromICPILyon,France,in1978andtheDoct.-Ing.andDocteurd’ÉtatdegreesfromINPGrenoble,France,in1982and1987,respectively.In1982,hejoinedCNRS,whereheiscurrentlyResearchDirector.Since1991,hehasbeenwiththeSignals,SystemsandPhysicsGroup,withinthePhysicsDepartmentatÉcoleNormaleSupérieuredeLyon,France.In1998,hespentonesemesterinCambridge,U.K.,asaninvitedlong-termresidentoftheIsaacNewtonInstituteforMathematicalSciences

and,from2002to2005,hewasDirectoroftheCNRSnationalcooperativestructure“GdRISIS.”Hisresearchinterestsincludemainlynonstationarysignalprocessing(withemphasisontime-frequencyandtime-scalemethods)andthestudyofself-similarstochasticprocesses.Hepublishedmanyresearchpapersinthoseareas,andheistheauthorofthebookTemps-Fréquence(Paris,France:Hermès,1993and1998),translatedintoEnglishasTime-Fre-quency/Time-ScaleAnalysis(SanDiego,CA:Academic,1999).

Dr.FlandrinhasbeenaGuestCo-EditoroftheSpecialIssueonWaveletsandSignalProcessingoftheIEEETRANSACTIONSONSIGNALPROCESSINGin1993,theTechnicalProgramChairmanofthe1994IEEE-SPInternationalSympo-siumonTime-FrequencyandTime-ScaleAnalysis,andsince2001,hehasbeentheProgramChairmanoftheFrenchGRETSISymposiumonSignalandImageProcessing.HeiscurrentlyanAssociateEditorfortheIEEETRANSACTIONSONSIGNALPROCESSING,andhehasbeenamemberoftheSignalProcessingTheoryandMethodsTechnicalCommitteeoftheIEEESignalProcessingSocietyfrom1993to2004.HewasawardedthePhilipMorrisScientificPrizeinMathematicsin1991,theSPIEWaveletPioneerAwardin2001,andthePrixMichelMon-petitfromtheFrenchAcademyofSciencesin2001.HehasbeenaFellowofEURASIPsince

2009.

PaulHoneine(M’07)wasborninBeirut,Lebanon,onOctober2,1977.HereceivedtheDipl.-Ing.de-greeinmechanicalengineeringandtheM.Sc.degreeinindustrialcontrol,bothfromtheFacultyofEngi-neering,theLebaneseUniversity,Lebanon,in2002and2003,respectively,andthePh.D.degreeinsys-temsoptimizationandsecurityfromtheUniversityofTechnologyofTroyes,France,in2007.

HewasaPostdoctoralResearchAssociatewiththeSystemsModelingandDependabilityLabora-tory,theUniversityofTechnologyofTroyes,from

IEEETRANSACTIONSONSIGNALPROCESSING,VOL.58,NO.7,JULY

2010

2007to2008.SinceSeptember2008,hehasbeenanAssistantProfessorattheUniversityofTechnologyofTroyes.Hisresearchinterestsincludenon-stationarysignalanalysis,nonlinearadaptivefiltering,sparserepresentations,machinelearning,andwirelesssensornetworks.

Dr.Honeineisthecoauthor(withC.Richard)ofthe2009BestPaperAwardattheIEEEWorkshoponMachineLearningforSignal

Processing.

CédricRichard(S’98–M’01–SM’07)wasborninSarrebourg,France,onJanuary24,1970.HereceivedtheDipl.-Ing.andtheM.S.degreesin1994andthePh.D.degreein1998fromtheUniversityofTech-nologyofCompiègne,France,allinelectricalandcomputerengineering.

From1999to2003,hewasanAssociateProfessorattheUniversityofTechnologyofTroyes(UTT),France.From2003to2009,hewasaFullProfessorattheInstitutCharlesDelaunay(CNRSFRE2848)attheUTT,andthesupervisorofagroupconsisting

of60researchersandPh.D.Inwinter2009,hewasaVisitingResearcherwiththeDepartmentofElectricalEngineering,FederalUniversityofSantaCatarina(UFSC),Florianòpolis,Brazil.SinceSeptember2009,hehasbeenaFullProfessoratFizeauLaboratory(CNRSUMR6525,ObservatoiredelaCôted’Azur),UniversityofNiceSophia-Antipolis,France.Hiscurrentresearchinterestsincludestatisticalsignalprocessingandmachinelearning.Heistheauthorofmorethan100papers.

Prof.CédricwastheGeneralChairoftheXXIfrancophoneconferenceGRETSIonSignalandImageProcessing,whichwasheldinTroyes,France,in2007.Since2005,hehasbeeninchargeofthePh.D.studentsnetworkofthefederativeCNRSresearchgroupISISonInformation,Signal,ImagesandVision.HeisamemberoftheGRETSIassociationboardandoftheEURASIPsociety.HehasservedasanAssociateEditoroftheIEEETRANSACTIONSONSIGNALPROCESSINGsince2006andoftheEURASIPSignalProcessingMag-azinesince2009.In2009,hewasnominatedliaisonlocalofficerforEURASIPandmemberoftheSignalProcessingTheoryandMethods(SPTM)TechnicalCommitteeoftheIEEESignalProcessingSociety.He(alongwithcoauthorP.Honeine)receivedtheBestPaperAwardfor“SolvingthePre-ImageProbleminKernelMachines:ADirectMethod”atthe2009IEEEWorkshoponMachineLearningforSignalProcessing(IEEE

MLSP).

JunXiaoreceivedtheB.E.degreeinoptoelectronicsandtheM.S.degreeinoptics(withhighesthons.)fromtheEastChinaNormalUniversity,Shanghai,in2002and2005,respectively,andthePh.D.degreeinsignalprocessingfromtheÉcoleNormaleSupérieuredeLyon,France,andtheEastChinaNormalUniver-sity,Shanghai,in2008.