Jos_von_Asmuth_Thesis.pdf - repository

_~ÅâÖêçìåÇ=~åÇ=íÜÉçêó=2.2 Statistical, ‘black box‘ viewpoint2.2.1 ARMA models as linear regression equationcêçã=~=ëí~íáëíáÅ~ä=îáÉïéçáåíI=~=íáãÉ=ëÉêáÉë=ãçÇÉä=Å~å=ÄÉ=êÉÖ~êÇÉÇ=~ë=~=î~êá~åí=çÑ=äáåÉ~ê=êÉÖêÉëëáçåK=qÜÉ=Éèì~íáçå=ÇÉÑáåáåÖ=~=ëáãéäÉ=êÉÖêÉëëáçå=äáåÉ=Å~å=ÄÉ=ïêáííÉå=~ëW====y = ω x + µ + n(2.1)=ïÜÉêÉ=y = W=çÄëÉêî~íáçåë=çÑ=íÜÉ=Éñéä~áåÉÇ=î~êá~ÄäÉ=EÉKÖKI=ÖêçìåÇï~íÉê=ÜÉ~ÇF=x = W=çÄëÉêî~íáçåë=çÑ=íÜÉ=Éñéä~å~íçêó=î~êá~ÄäÉ=EÉKÖKI=êÉÅÜ~êÖÉF=ω = W=é~ê~ãÉíÉê=ÇÉÑáåáåÖ=íÜÉ=ëäçéÉ=çÑ=íÜÉ=êÉÖêÉëëáçå=äáåÉ=µ = W=é~ê~ãÉíÉê=ÇÉÑáåáåÖ=íÜÉ=áåíÉêÅÉéí=ïáíÜ=íÜÉ=óJ~ñáë=n = W=Éêêçê=çê=êÉëáÇì~ä=íÉêã==cçê=åçïI=ïÉ=ïáää=áÖåçêÉ=íÜÉ=êÉëáÇì~ä=íÉêãI=çå=ïÜáÅÜ=ïÉ=ïáää=Éä~Äçê~íÉ=áå=íÜÉ=åÉñí=ëÉÅíáçåK=lÑ=ÅçìêëÉI=äáåÉ~ê=êÉÖêÉëëáçå=Å~å=ÄÉ=~ééäáÉÇ=íç=~ää=ëçêíë=çÑ=î~êá~ÄäÉëI=áåÅäìÇáåÖ=íáãÉ=ëÉêáÉëK=få=íÜÉ=ä~ííÉê=Å~ëÉI=íÜÉ=~î~áä~ÄäÉ=çÄëÉêî~íáçåë=~êÉ=áåÇÉñÉÇ=ïáíÜ=íáãÉ= t xJzI=ïÜáÅÜ=ÖáîÉë=íÜÉ=ÑçääçïáåÖ=Çóå~ãáÅ=êÉÖêÉëëáçå=Éèì~íáçåW==y = ω x + µ(2.2)tt=få=ã~åó=Çóå~ãáÅ=ëóëíÉãëI=ÜçïÉîÉêI=íÜÉ=ÉÑÑÉÅí=çÑ=çåÉ=î~êá~ÄäÉ=çåíç=íÜÉ=çíÜÉê=áë=åçí=Eçê=åçí=çåäóF=áåëí~åí~åÉçìëK=få=ëìÅÜ=Å~ëÉëI=íÜÉ=ëóëíÉã=áë=ë~áÇ=íç=Ü~îÉ=~=ãÉãçêóK=qÜÉ=êáëÉ=áå=ï~íÉê=äÉîÉä=áå=~å=~êÄáíê~êó=ÜóÇêçäçÖáÅ=ëóëíÉã=Å~ìëÉÇ=Äó=~=ê~áå=ëÜçïÉêI=Ñçê=áåëí~åÅÉI=ïáää=åçí=Çáë~ééÉ~ê=áããÉÇá~íÉäó=ïÜÉå=íÜÉ=ê~áå=ëíçéëK=`çåëÉèìÉåíäó=~åÇ=îáÅÉ=îÉêë~I=íÜÉ=ÅìêêÉåí=ï~íÉê=äÉîÉä=áå=ÜóÇêçäçÖáÅ=ëóëíÉãë=áë=~=ÑìåÅíáçå=çÑ=éêÉëÉåí=•åÇ=éêÉîáçìë=ê~áåÑ~ää=ÉîÉåíëK=j~íÜÉã~íáÅ~ääóI=íÜÉ=ÉÑÑÉÅíë=çÑ=éêÉîáçìë=ëí~íÉë=çÑ=íÜÉ=Éñéä~å~íçêó=î~êá~ÄäÉ=Å~å=ÄÉ=ëáãéäó=~ÇÇÉÇ=íç=íÜÉ=êÉÖêÉëëáçå=Éèì~íáçå=áå=íÜÉ=ÑçääçïáåÖ=ã~ååÉêW==yt ω0xt ω1xt −1 ..... ωnsxt −nsµ=qÜÉ=é~ê~ãÉíÉêë== + + + (2.3)ωns~êÉ=âåçïå=~ë=ãçîáåÖ=~îÉê~ÖÉ=Ej^F=é~ê~ãÉíÉêë=E~ë=EOKPF=êÉëÉãÄäÉë=íÜÉ=Å~äÅìä~íáçå=çÑ=~=ïÉáÖÜíÉÇ=ãçîáåÖ=~îÉê~ÖÉ=çÑ= x FI=~åÇ=íÜÉ=êÉëìäíáåÖ=ãçÇÉä=áë=ë~áÇ=íç=Ü~îÉ=çêÇÉê=j^E ns FK=fÑ=ïÉ=éäçí=ω =îÉêëìë= ns =EÑáÖìêÉ=OKTFI=íÜÉ=êÉëìäí= t0 0 1 2 31 2 3 4 45 6tfigure 2.7: Transfer function of a hypotheticalMA(4) system.=== PP=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK