state-space models

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1、ChapterState-SpaceModels3Instate-spacemodelswehave,ingeneral,anonobservabletargetprocess(Xt)andanobservableprocess(Yt).Theyarelinkedbytheassumptionthat(Yt)isalinearfunctionof(Xt)withanaddednoise,wherethelinearfunctionmayvaryintime.Theaimisthederivationofbestlinearestimatesof

2、Xt,basedon(Ys)st.3.1TheState-SpaceRepresentationManymodelsoftimeseriessuchasARMA(p;q)-processescanbeembeddedinstate-spacemodelsifweallowinthefollowingsequencesofrandomvectorsX2RkandY2Rm.ttAmultivariatestate-spacemodelisnowdenedbythestateequationkXt+1=AtXt+Bt"t+12R;(3.1)des

3、cribingthetime-dependentbehaviorofthestateX2Rk,andthetobservationequationmYt=CtXt+t2R:(3.2)Weassumethat(At),(Bt)and(Ct)aresequencesofknownmatrices,("t)and(t)areuncorrelatedsequencesofwhitenoiseswithmeanvectors0andknowncovariancematricesCov(")=E(""T)=:Q;ttttCov()=E(T)=:R

4、.ttttWesupposefurtherthatX0and"t,t,t1,areuncorrelated,wheretworandomvectorsW2RpandV2Rqaresaidtobeuncorrelatediftheircomponentsarei.e.,ifthematrixoftheircovariancesvanishesTE((WE(W)(VE(V)))=0:122State-SpaceModelsByE(W)wedenotethevectorofthecomponentwiseexpectationsofW.Wes

5、aythatatimeseries(Yt)hasastate-spacerepresentationifitsatisestherepresentations(3.1)and(3.2).Example3.1.1.Let(t)beawhitenoiseinRandputYt:=t+twithlineartrendt=a+bt.Thissimplemodelcanberepresentedasastate-spacemodelasfollows.DenethestatevectorXtastXt:=;1andput1bA:=0

6、1Fromtherecursiont+1=t+bwethenobtainthestateequationt+11btXt+1===AXt;1011andwithC:=(1;0)theobservationequationtYt=(1;0)+t=CXt+t:1NotethatthestateXtisnonstochastici.e.,Bt=0.Thismodelismoreovertime-invariant,sincethematricesA;B:=BtandCdonotdependont.Example3.1.2

7、.AnAR(p)-processYt=a1Yt1+


+apYtp+"twithawhitenoise("t)hasastate-spacerepresentationwithstatevectorTXt=(Yt;Yt1;:::;Ytp+1):3.1TheState-SpaceRepresentation123Ifwedenethepp-matrixAby01a1a2:::ap1apB10:::00CBCA:=B0100CBC@............A00:::10andthep1-matricesB;CTbyTTB:=(1

8、;0;:::;0)=:C;thenwehavethestateequationXt+1=AXt+B"t+1andtheobservationequat