A STRUCTURE THEORY FOR LINEAR DYNAMIC ERRORS-IN-VA

ASTRUCTURETHEORYFORLINEARDYNAMICERRORS-IN-VARIABLESMODELS∗W.SCHERRER†ANDM.DEISTLER†SIAMJ.CONTROLOPTIM.c1998SocietyforIndustrialandAppliedMathematicsVol.36,No.6,pp.2148–2175,November1998016Abstract.Wedealwithproblemsconnectedwiththeidentificationoflineardynamicsystemsinsituationswheninputsandoutputsmaybecontaminatedbynoise.Thecaseofuncorrelatednoisecomponentsandtheboundednoisecaseisconsidered.Ifalsotheinputsmaybecontaminatedbynoise,anumberofadditionalcomplicationsinidentificationarise,inparticulartheunderlyingsystemisnotuniquelydeterminedfromthepopulationsecondmomentsoftheobservations.Adescriptionofclassesofobservationallyequivalentsystemsisgiven,continuitypropertiesofmappingsrelatingclassesofobservationallyequivalentsystemstothespectraldensitiesoftheobservationsarederivedandtheclassesofspectraldensitiescorrespondingtoagivenmaximumnumberofoutputsarestudied.Keywords.errors-in-variables,factoranalysis,systemidentificationAMSsubjectclassifications.93B30,93B15,62H25PII.S03630129942624641.Introduction.Inthe“mainstreamapproach”tolinearsystemsidentifica-tion(seeDeistler[7])oneofthebasicassumptionsisthatallnoiseisaddedtotheoutputs(ortotheequations,whichisthesameforourpurpose).Thenoisetherebyisassumedtobeorthogonaltotheinputs.Ineconometricsthisiscalledtheerrors-in-equationsapproach.Hereweareconcernedwithadifferentand,inprinciple,moregeneralapproachtonoisemodeling,whereallvariablesmaybecontaminatedbynoise.Modelsofthiskindarecallederrors-in-variables(EV)orlatentvariablesmodels,orinadifferentbutequivalentformulation,factormodels.Forthecaseofstaticsystems,suchmodelshavebeenanalyzedandusedforalongtimeinstatis-tics,science(inparticular,chemistry),psychometrics,andeconometrics(see,e.g.,Adcock[1],Spearman[26],Gini[14],Frisch[13]).Inthelasttwodecadestherehasbeenaresurginginterestinsuchmodels(see,e.g.,Aigneretal.[2],Anderson[5]).Recently—mainlytriggeredbyKalman’swork[18,19,20]—EVmodelshavealsobeenanalyzedinsystemsengineering.Thedynamiccasehasbeentreated,e.g.,inAndersonandDeistler[3]andDeistlerandAnderson[9].Thetraditionalerrors-in-equationsapproachisjustifiedinagreatnumberofapplicationsdealing,forinstance,withprediction.Ontheotherhand,inanumberofcasestheasymmetryinerrors-in-equationsmodelingcannotbejustifiedandmayleadto“prejudiced”results(Kalman[20]).Forexample,insonararrayprocessing,whenanarrayofnsensorsisassumedtoreceivenoisysignalsfromn−msources(Haykin[16]),EVmodelsariseinanaturalway.Moregenerally,wecandistinguishthefollowingthreemainareasforEVmodeling:1.Ifweareinterestedinthe“truesystem”underlyingthedata(ratherthan,forinstance,inprediction)andifwecannotbesureapriorithattheinputshavebeenobservedfreeofnoise.Thisisthe“classical”motivationforEVmodels,forexample,ineconometrics.∗ReceivedbytheeditorsJanuary31,1994;acceptedforpublication(inrevisedform)October10,1997;publishedelectronicallySeptember3,1998.ThisresearchwassupportedbytheAustrian“FondszurF¨orderungderwissenschaftlichenForschung”ProjektP11213-MAT.†Institutf¨ur¨Okonometrie,OperationsResearchundSystemtheorie,TechnischeUniversit¨atWien,Argentinierstr.8,A-1040Vienna,Austria([e-mail],[e-mail]).2148ERRORS-IN-VARIABLESMODELS21492.Ifwewanttoapproximateahighdimensionaldatavectorbyasmallnumberoffactors.Thisisthe“classical”motivationforfactoranalysis(e.g.,inpsychometrics,whereanexamplewouldbedeterminingtheintelligencefactorsunderlyingthetestscores).ArelatedissueisthatEVmodelingmayconsiderablyreducethedimensionofparameterspacesincomparisonwithmultivariateARorARMAmodels.3.Inanumberofcases,nosufficientaprioriinformationaboutthenumberofequationsand/orabouttheclassificationofthevariablesintoinputsandoutputsisavailable.Then,onehastouseasymmetricsystemmodelwhichinturndemandsasymmetricnoisemodel.ThispointhasbeenemphasizedinparticularbyKalman[18].Thesystemsconsideredareoftheformw(z)ˆxt=0,(1.1)whereˆxtisann-dimensionalvectoroflatent(i.e.,notnecessarilyobserved)realvaluedrandomvariables,zisusedforthebackward-shiftontheintegersZ(i.e.,z(ˆxt|t∈Z)=(ˆxt−1|t∈Z))aswellasforacomplexvariable,andwherew(z)=∞Xj=−∞Wjzj;Wj∈Rm×nand∞Xj=−∞kWjk∞.(1.2)Wewillcallw(z)therelationfunction;itrepresentsanexact(i.e.,deterministic)linearsystemofaverygeneralform.Clearly,systemsoftheform(1.1)aresymmetricinthesensethatnoaprioriclassificationofthevariablesˆxtasinputsandoutputsandnoaprioriinformationaboutcausalityareneeded.Herealsothenumberofequations,m,in(1.1)isnotassumedtobeknownapriori.Withoutrestrictionofgenerality,wewillassumethat1≤m≤nholdsandthatw(z)containsnolinearlydependentrows.Theobservedvariablesxtareoftheformxt=ˆxt+ut,(1.3)whereutisthen-dimensionalnoisevector.Throughoutthepaperwewillassumethefollowing:(a.1)Theprocesses(xt),(ˆxt),and(ut)are(widesense)stationarywithabsolutelysummableautocovariancefunctions.Thus,inparticular,thespectraldensitiesΣ,ˆΣ,and˜Σof(xt),(ˆxt),and(ut),respectively,existandareboundedcontinuousfunctions.(Inaddition,limitsofrandomvariablesareunderstoodinthesenseofmeansquareconvergence.)(a.2)Eˆxt=0andEut=0.(a.3)Eˆxtu′s=0.(a.4)U