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1、1164IEEETRANSACTIONSONSIGNALPROCESSING,VOL.48,NO.4,APRIL2000AMetricforARMAProcessesRichardJ.MartinAbstract—Autoregressive-moving-average(ARMA)modelsConcentratingontheARcaseforthemoment,wemightpro-seektoexpressasystemfunctionofadiscretelysampledprocessposeametricoftheformasarationalfun
2、ctioninthe-domain.TreatinganARMAmodelasacomplexrationalfunction,wediscussametricdefinedonthesetofcomplexrationalfunctions.Wegiveanaturalmeasureofthe“distance”betweentwoARMAprocesses.Ourpapercon-(2)centratesonthemathematicsbehindtheproblemandshowsthatthevariousalgebraicstructuresendowo
3、urchoiceofmetricwithsomeinterestingandremarkableproperties,whichwediscuss.Wesuggestthatthemetriccanbeusedinatleasttwocircum-forappropriatepositiveweights.Thisapproachraisesallstances:i)inwhichwehavesignalsarisingfromvariousmodelsthatareunknown(soweconstructthedistancematrixandperforms
4、ortsofproblems.First,thereisnoobviouswaytofinda“good”clusteranalysis)andii)wherethereareseveralpossiblemodelssetofweights.Second,themetric(2)doesnothaveanyuseful,allofwhichareknown,andwewishtofindwhichoftheseissystem-theoreticproperties.Forinstance,wemightarguethatclosesttoanobservedd
5、atasequencemodeledas.themetricshouldbemoresensitivetopolesneartheunitcircleIndexTerms—Autoregression,classification,linearsystems.thenpolesneartheoriginsicnesuchpolesindicatestrongres-onancesinthesystem.Inaddition,themetricdoesnotcarewhetherthemodelsarestableorunstable.Third,itdoesnot
6、I.INTRODUCTIONhaveanyusefulmathematicalproperties.ARmodelsformaA.Generalitiessemigroupundermultiplicationinthe-domain,andARMAUTOREGRESSIVE(AR),moving-average(MA),andmodels,asrationalfunctions,formagroup.Themetric(2)doesAauto-regressive-moving-average(ARMA)modelsarenotrespectthisgroups
7、tructure,norisitaparticularlytrans-usefultime-domainmodelsfortherepresentationofdis-parentfunctionofthepoles.crete-timesignalssuchasspeech,audio,andradar[1]–[7].TheReturningtothegeneralARMAcase,thereisanotherdif-timeseriesissaidtoobeyanARMAmodelifficulty.AnMAmodelmaybeapproximatedasah
8、igh-o
