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1、IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.46,NO.12,DECEMBER20011887AsymptoticVarianceExpressionsforEstimatedFrequencyFunctionsLiang-LiangXieandLennartLjung,Fellow,IEEEAbstract—Expressionsforthevarianceofanestimatedfre-wherethespectraofandareand,respectively.que
2、ncyfunctionarenecessaryformanyissuesinmodelvalidationThenthebasicresultisthatthevarianceoftheestimatedfre-andexperimentdesign.Ageneralresultisthatasimpleexpressionquencyfunctionisgivenbyforthisvariancecanbeobtainedasymptoticallyasthemodelordertendstoinfinity
3、.Thisexpressionshowsthatthevarianceisinverselyproportionaltothesignal-to-noiseratiofrequencyby(3)frequency.Still,forlowordermodelstheactualvariancemaybequitedifferent.Thishasalsobeenpointedoutinseveralrecentpublications.Inthiscontributionwederiveanexactexpre
4、ssionwheretheexpressionisasymptoticinboththemodelorder,forthevariance,whichisnotasymptoticinthemodelorder.Thisandthenumberofdata.Forthisresulttohold,itisessentiallyexpressionappliestoarestrictedclassofmodels:AR-models,onlyrequiredthatthemodelparameterization
5、in(1)hasablockaswellasfixedpolemodelswithapolynomialnoisemodel.Itbringsoutthecharacterofthesimpleapproximationandtheshiftstructure,sothatthegradientw.r.t.theparameterblock,convergenceratetothelimitasthemodelorderincreases.Italso,isashiftedversionofthegradien
6、tw.r.t.(see[7,p.providesnonasymptoticlowerboundsforthegeneralcase.The293]).Thisstructureisathandformostofthetypicalblackboxcalculationsareillustratedbynumericalexamples.modelparameterizations,likeARMAX,Box–Jenkins,etc.IndexTerms—Accuracy,asymptoticvariance,F
7、IRmodels,Asimilarresultholdsifthefrequencyfunctionisestimatedsystemidentification.byspectralanalysis,[2],[7,Ch.6]:I.INTRODUCTION(4)XPRESSIONSfortheaccuracyofanestimatedfrequencyEfunctionareimportantforseveralpurposes.Theyarewhereisanumberthatdependsonthetype
8、ofwindowneededtounderstandthereliabilityofthemodel,e.g.,forcon-used,aswellasonitssize.troldesign.TheyarealsoimportantforvariousexperimentalTheseresultsareintuitivelyappealing,andmeanthatthevari-
