第11章Gauss系与时间序列

284–11Gauss,,1.Hilbert1,10,L2L2)(),(xhhxEΔ=,(∈∀hx,L2).L221),(||||xxxΔ=|||||||||),(|hxhx⋅≤.:||||||||||||hxhx+≤+.||||),(hxhx-=ΔdL20||||→-xxnxx→n0||2→-xxnEL2}{nxL2exxe-∃∀||||,,,,0mnNmnNihxhxVi+=hxViEEE+=22}(varhxVVVEEE+==ΔaVaaVEE=)(21,VV)(),(2121VVVVEΔ=.,.}:{I∈axaL2Φ)(xΦ)(xΦ)(xL2Φ)(xL2}:{I∈axa.285}:{I∈axa,0h0,,.}:{I∈axah^h.∨hhL2?HilbertΦ)(x}:{I∈axaΦ)(x∨hh||||min||||)(VhhhxV-=-Φ∈∨∨h∈Φ)(x⊥-∨hhΦ)(x,nnBorel)(ng0)],,()[(1)(=-∨nngEaaxxhhL.∨hhx)(PrΦojh}:{I∈axa,(11.1)hx)(PrΦoj):|(IE∈=axha,(11.2)h}:{I∈axa.}:{I∈axax),(hxijjipyxP==)),(),((hx∑∑=∨iiiiippxhhh.),(hx),(yxpdxduupxpx]),(),([∫∫=∨hhh⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=nhhhM1n),(SmhN,llmlhlΣ-=TTTieEe21286mnΣnn×nx0≥ΣxxT⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=nhhhM1nnaa,,1L∑=nkkka1hGauss,n),(SmhN,,mm)1,0(NmZZZ,,,21Lmn×()),mjniaij≤≤==hm+Z=ΣT⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=mZZZM1Σ),(SmhN(1)Σ=--=]))([(,TEEmhmhmh.(2)),(~TAAbANbAΣ++mh.(3))(nxGauss,xx─→─pn)((),xGauss.xx─→─pn)(),(,0)|(|)()(∞→→-mnPmnexx)(,)()(2)()(jiijjiijVarEEmxxsxx-=-=∫∞--=Φxuduex2221)(p∞→ji,∫--→essp||2)(02122xmxijdxeijij∫+-esp||2221ijijmyydye)()](1[ijijijijmmsese+-Φ++-Φ-=NNji,))1()(1(1-Φ=Φ-)1(1)](1[Φ-+-Φ-ijijmse)1()(-Φ+-ΦijijmseNji,2871+-ijijmse1+ijijmsees±ijijm2esijNji,2e±ijm2||eijmNji,=-2)()()(jiExx),(,022∞→→+jimijijs}{)(nxL2Cauchy.0||2)(→-xxnExxxxVarVarEEnn→→)(,)()(xlxlxlxllxlxVarEiVarEininieeEeEennn2)92)()(21)(21limlim--∞→∞→===x}:{Itt∈xGauss,Ittnn∈∀∀L,,1,),,(1nttxxLGauss.Gauss),0[∞=I,.tEtmxΔ=)()(),(tsEtsBxxΔ=⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛Σ⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛≤njijintttttmtmNn,1),(,)()(~1MMxx,)()(),(),(jijijitmtmttBtt-=Σ,.(1)),(hx,),,(hxhxx+-Gauss(2)}1:{≥nnxGauss,∑==nkknkna1xh,}1:{≥nnhGauss.0GaussL2L2}:{)(ItLt∈=Δxx;}:{Itt∈x.)(xL)(xL,:0||||),()()(→-∈hhxhnnL,)(xhL∈.)(xL}:{Itt∈x}:{Itt∈x}:{Itt∈x)(xΦ}:{Itt∈x)(xL288(1)},:,{JI∈∈bahxbaGauss,}:{I∈axa}:{I∈bhb:bahx,0)(=bahxCov.(2)}:{I∈axa}:{J∈bhb,Φ)(xΦ)(x.3.2GaussU}{):{hxItt∈0GausshHilbert)(xL,^h,h}:{I∈axa^hhx)(PrLoj,),(^xhL∈)(^xhhL⊥-.(11,1)’(11.1)’U}{):{hxItt∈0Gauss^hh=∨,:⊂∈)(^xhLΦ)(x)(^xhhL⊥-,)(,0])[(^ItEt∈∀=-xhh..^hh-}:{Itt∈x,Φ)(x.∈VΦ)(x0])[(^=-VhhE.⊥-^hhΦ)(x^hh=∨.)2,1(,)()()(=+=kiktktkthxV)(),()2()1(tsEtsBVVΔ=}{)1(tV}{)2(tVmmtt,,1Lmaa,,1L0),(1,≥∑=lklkmlkttBaatttihxV+=}{tx}{th0),(tsRGausstx)(tFGausstx)(0)(tdFitTEeFxx∫=ΦΔGauss)(0tdFtTx∫10)]([0=∫tdFEtTx)()(),()]([000tdFsdFtsRtdFVarTTtT∫∫∫=x289)()(),(2100)(tdFsdFtsRTTeF∫∫=Φ-x)(∞-∞nnxkm,m},,(1mkk++xxL},,(1mxxL)(∞-∞nnxkn,)()(,kRnEmEknnn==+xxx,)(∞-∞nnx,)(kR.tEtmxΔ=)()()(),(kREkttBktt==++Δxxt.,.L2,,0,)(kR)(lfFourier:∫=plll20)()(defkRik,(11.4))(lf.,)(lf.,nx,∑=-+=nkikkenf02^||11)(lxl(11.5))(lf,.,.,.0,)11(,|)(|11)(002^^∑∑==-+=-+=niinkikknenfxxxxll(11.4)’..m:0||1||0→-+++mnnxxL,L2?1mnn≈+++10xxL.2900||)(1||0──→─-+++∞→+nknnkkRnxxxxL,1)(10kRnknnk≈++++xxxxL.10^+++=nmnxxL,1)(0^+++=+nkRknnkxxxxL.(11.6)m)(kB.Gauss.Gauss,1)1(0=→+++mnPnxxL,1))(1(0=→++++kRnPknnkxxxxL.(,∫--∞pplldf)(ln)5.3ARMA,ARMA.4.2)(∞-∞nnx,km,,∞→t),,(1mtt++xxL},,(1mktkt++++xxL..)(∞-∞nnx,kn,m)(kR)()(lim,limkREmEkntnttntt==+++∞→+∞→xxxm,)(kR..Markov,,;.,,.,.4.3,,:1-Δ-=nnnxxh,,,,,,.0),()(==+nknnEkRExxx),,1(NnnL=x.)(kRllijjilnnRnn≤-))((,,,1L.}{nw)0(,02REwEwnn==,11.-+++=nknkkwcwcLx,),,(1nccL))[0()])([()1(1121111nknkkjjNjikiNiccccccRwcwcEkR+-+=-+=+++==-∑∑L,2915.ARMA(Auto-RegressionMovingAverage)5ARMA(p,q)nx,)(∞-∞nne(,nx),2,0see==nnVarE,qnqnnpnpnnbbbaa----+++=---eeexxxLL11011,(11.7)ppppzazazazA----=--1111)(L1||≤z(nx,,,),)(∞-∞nnx.).,,),,,(),,,(101qpbbbaaLLnx.ARMA(p,q)22|)(||)(|)(llliieAeBCf=,(11.8)=)(zAppppzazaza------1111L1||≤z,qqqqzbzbzbbzB++++=--1110)(L.0),,(1=qbbLARMA,,;0),,(1=paaLARMA,.,,ARMAARMA.,,ARMAp,q.p,q,,.p,q,,AR(p),ne,pnn--xx,,1L,nxpnpnpnnnpnpnpnnnaabaaEE--------++=+++=xxxxexxxxxLLLL1110111)),,(|)](([)),,(|(.(11.8),,n,nne,AR(p),pnn--xx,,1L,nx:),(22011sxxbaaNpnpn--++L,(20)(sb,10=b,10=b),52ARnx(ARMA))()(kjkj≤a∑∑=-=--=-kjjnjnkjccjnkjncEEk121,2)(||inf||,1xxxaxL,(11.9))(kkak292AR,AR;p();AR(p)p+1.nxAR(p),,p,211||),,(∑=--=kjjnjnkkcEccFxxL.),,()()(1kkkaaL,),,()()(1kkkaaL:),,1(0kjcFjkL==∂∂:⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛=⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛----)()2()1()0()2()1()2()0()1()1()1()0()()(2)(1kRRRRkRkRkRRRkRRRkkkkMMLMMMMLLaaa,(11.10).k)(kka.,Nxx,,1L,))((mjjR≤))((^mjjR≤.k)(kkaYule-Walker))((kjjR≤,,k)(kka^)(kka,.ppppzazaza------1111L1||≤z,.1||1∑=piiappppzazaza------1111L1||≤z.(11.10),k.1+k1+kYule-Walker⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛+=⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛---++++)1()()1()0()1()()1()2()1()()1()0()1(1)1()1(1kRkRRRkRkRRkRkRkRRRkkkkkMMLLMLMMLaaa)}1(,{)1(+≤+kjkja}(,{)(kjkj≤aToepliz,Toeolitz,kR,ARMA().kTk⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=11NkT,=kr⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛)()1(kRRM,=kx⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛-)(1)(1kkkaaM.293⎥⎦⎤⎢⎣⎡+=⎥⎦⎤⎢⎣⎡⎥⎦⎤⎢⎣⎡+++)1()0()1(11kRrxRTrrTRkkkkTkTkkkka.kkkkkkkrrTxR=++++)1(11a,)1()0()1(11+=++++kRRxTrkkkTkTka.kRkT,,⎥⎦⎤⎢⎣⎡=-)(1kkkkkxrRa,kkkkkkkkrRTrRx1)1(111-++-+-=a⎥⎦⎤⎢⎣⎡-⎥⎦⎤⎢⎣⎡=-++kkkkkkkkkxTx1)()1(1)(aaa,(11.11-1))0()()1()1(11)1(11RrRTrRTrkRkkkkkkkkkTkTk++-++-+-=+aakkTkTkkkTkkkrRTrrRrR11)1(1))0((--+++-=a.⎥⎦⎤⎢⎣⎡-⎥⎦⎤⎢⎣⎡-+=++)()()1(1)0()1(kkkTkkkkTkTkkkxrRxTrkRaaa∑∑==--+-+=kjkjkjkjjRRjkRkR1)(1)()()0()1()1(aa.(11.11-2)(11.11-1)(11.11-2),⎪⎪⎪⎪⎪⎪⎪⎪⎩⎪⎪⎪⎪⎪⎪⎪⎪⎨⎧≤-=--+-+=--==--+++==++∑∑)()()0()1()1()1()0()1()2()0()1()()1()1(1)()1(1)(1)()1(1)1(1)1(1)2(2)1(1kjjRRjkRkRRRRRRRkjkkkkjkjkj