Jos_von_Asmuth_Thesis.pdf - repository

^=Åçåíáåìçìë=åçáëÉ=ãçÇÉä=ñ = W== ÇÉîá~íáçåë=çÑ=íÜÉ=êÉëáÇì~ä=ëÉêáÉë= n Ñêçã=áíë=ãÉ~å= E( n t) =ϕ = W== ~ìíçêÉÖêÉëëáîÉ=é~ê~ãÉíÉê=a = W== ÇáëÅêÉíÉ=ïÜáíÉ=åçáëÉ=éêçÅÉëë=ïáíÜ=éêçéÉêíáÉë= E{ at} = 0 =Eëç=íÜ~í= at = ã tFI==2 2= = E{ at} = σaI= E[ ata t + i] = 0 Ñçê i ≠ 0 ==få=^oj^=ãçÇÉäë=áå=ÖÉåÉê~äI=~=ïÜáíÉ=åçáëÉ=ëÉêáÉë=~í=Éèì~äë=íÜÉ=ëÉêáÉë=çÑ=çåÉJëíÉéJ~ÜÉ~Ç=éêÉÇáÅíáçå=ÉêêçêëI=çê=áååçî~íáçåëK=cçê=íÜÉ=^oENF=ãçÇÉäI=~å=Éëíáã~íÉ=çÑ=~í==áë=çÄí~áåÉÇ=Äó=ëìÄíê~ÅíáåÖ=íÜÉ=çåÉ=ëíÉé=~ÜÉ~Ç=éêÉÇáÅíáçå= ñ ˆ = ϕñ Ñêçã=íÜÉ=̃−t| nt−1t 1çÄëÉêîÉÇ=î~äìÉ=çÑ= ñtK=_ÉÅ~ìëÉ=çÑ=íÜáëI=~=ãáëëáåÖ=çÄëÉêî~íáçå=çÑ= ñ =~í=íáãÉ= t =áãéäáÉë=íÜ~í= at~åÇ== at + 1=Å~å=åçí=ÄÉ=Å~äÅìä~íÉÇK=mêÉÇáÅíáçå=çÑ= ñt=ìëáåÖ= ñt − 2çê=ãçêÉ=íáãÉ=ëíÉéë=Ä~Åâ=ÇçÉë=åçí=çÑÑÉê=~=ëíê~áÖÜíÑçêï~êÇ=ëçäìíáçåI=ÄÉÅ~ìëÉ=íÜÉ=äÉ~Ç=íáãÉ=áåÑäìÉåÅÉë=íÜÉ=éêÉÇáÅíáçå=Éêêçê=çê=î~êá~åÅÉ=çÑ= atI=ïÜáÅÜ=ëÜçìäÇ=ÄÉ=ëí~íáçå~êóK=qÜÉêÉÑçêÉI=~=íáãÉ=ëÉêáÉë=ãìëí=ÄÉ=ÅçãéäÉíÉ=~åÇ=ÖáîÉå=~í=êÉÖìä~ê=áåíÉêî~äë=áå=çêÇÉê=íç=ÄÉ=~ÄäÉ=íç=Ñáí=~å=^oj^=ãçÇÉäë=çå=íÜÉ=Ç~í~K=qÜÉ=ÑçêÉÅ~ëíáåÖ=ãçÇÉ=çÑ=^oj^=ãçÇÉäëI=ÜçïÉîÉêI=ÇçÉë=éêçîáÇÉ=íÜÉ=åÉÅÉëë~êó=Éèì~íáçåë=íç=éêÉÇáÅí=çîÉê=~=î~êá~ÄäÉ=äÉ~Ç=íáãÉ=~åÇ=íç=èì~åíáÑó=íÜÉ=~ÅÅçãé~åóáåÖ=éêÉÇáÅíáçå=Éêêçê=xëÉÉ=_çñ=~åÇ=gÉåâáåëI=NVTMzK=cçê=íÜÉ=^oENF=ãçÇÉäI=íÜÉ=éêÉÇáÅíáçå=~åÇ=áíë=î~êá~åÅÉ=~êÉ=~=ÑìåÅíáçå=çÑ=íÜÉ=íáãÉ=ä~Ö=ä=~åÇ=ÖáîÉå=ÄóW==⎧ ˆ lñt+l= φ ñt⎪2l⎨ 2 2 (1 −ϕ)(4.2)⎪σe= σt+l a 2⎩ (1 −ϕ)=t 2=== VR=n tnt-3*a t 2n t-2v*t?*+time indexKKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK+2 vt1 n 2t-3t-2t-2==figure 4.2: Schematic representation of the functioning of an AR(1) model combined with a‘degenerate’ Kalman filter. The (*) denotes a residual, whereas the (+) denotes a predicted residual.The model predicts missing values of the residual series at every time step, along with the predictionerror varianceK==nnt+n2 vt