Jos_von_Asmuth_Thesis.pdf - repository

4^=Åçåíáåìçìë=åçáëÉ=ãçÇÉä=Chapter4 A continuous noise model forautocorrelated data of irregularA continuous noise model forfrequencyautocorrelated data of irregular==frequency======= VN===Adopted from:=Von Asmuth, J.R., and M.F.P. Bierkens (2005)=Modeling irregularly spaced residual series as a continuous=stochastic process.^ÇçéíÉÇ=ÑêçãW=Water Resources Research, 41(12), W12404, doi: 10.1029 /Von Asmuth, J. R., and M. F. P. Bierkens (2005), Modeling irregularly spaced residual series2004WR003726.as a continuous stochastic process, t~íÉê=oÉëçìêÅÉë=oÉëÉ~êÅÜ, 41(12), W12404,doi:10.1029/2004WR003726. Reproduced Reproduced by permission by permission of American of American Geophysical Geophysical Union Union,copyright © 2005 AmericancopyrightGeophysical2002 AmericanUnion.Geophysical Union.==Abstract: Abstract=få=íÜáë=ÅÜ~éíÉêI=íÜÉ=Ä~ÅâÖêçìåÇ=~åÇ=ÑìåÅíáçåáåÖ=çÑ=~=ëáãéäÉ=Äìí=ÉÑÑÉÅíáîÉ=Åçåíáåìçìë=íáãÉ=~ééêç~ÅÜ=Ñçê=ãçÇÉäáåÖ=áêêÉÖìä~êäó=ëé~ÅÉÇ=êÉëáÇì~ä=ëÉêáÉë=~êÉ=In this chapter, the background and functioning of a simple but effective continuous timeéêÉëÉåíÉÇK=qÜÉ=Ä~ëáÅ=Éèì~íáçåë=ïÉêÉ=éìÄäáëÜÉÇ=É~êäáÉê=áå=xsçå=^ëãìíÜ=Éí=~äKI=OMMOz=approach for modeling irregularly spaced residual series are presented. The basicïÜÉêÉ=íÜÉó=ïÉêÉ=ìëÉÇ=~ë=é~êí=çÑ=~=Åçåíáåìçìë=íáãÉ=íê~åëÑÉê=ÑìåÅíáçå=åçáëÉ=EqckF=equations were published earlier in [ Von Asmuth et al., 2002] where they were used asãçÇÉäK=fí=áë=ëÜçïå=íÜ~í=íÜÉ=ãÉíÜçÇë=ÄÉÜáåÇ=íÜÉ=ãçÇÉä=~êÉ=ÄìáäÇ=çå=íïç=éêáåÅáéäÉëW=part of a continuous time transfer function noise (TFN) model. It is shown that the methodsÑáêëíI=íÜÉ=Ñ~Åí=íÜ~í=íÜÉ=Éèì~íáçåë=çÑ=~=h~äã~å=ÑáäíÉê=ÇÉÖÉåÉê~íÉ=íç=~=Ñçêã=íÜ~í=áë=behind the model are build on two principles: first, the fact that the equations of a KalmanÉèìáî~äÉåí=íç=ÚÅçåîÉåíáçå~äÛ=~ìíçêÉÖêÉëëáîÉ=ãçîáåÖ=~îÉê~ÖÉ=E^oj^F=ãçÇÉäë=ïÜÉå=íÜÉ=filter degenerate to a form that is equivalent to 'conventional' autoregressive movingãçÇÉäÉÇ=Ç~í~=~êÉ=ÅçåëáÇÉêÉÇ=íç=ÄÉ=ÑêÉÉ=çÑ=ãÉ~ëìêÉãÉåí=ÉêêçêëK=qÜáë=~ëëìãéíáçåI=áå=average (ARMA) models when the modeled data are considered to be free ofÅçãé~êáëçå=íç=íÜÉ=ÚÑìääÛ=h~äã~å=ÑáäíÉê=~äëç=óáÉäÇë=~=ÄÉííÉê=éêÉÇáÅíáçå=ÉÑÑáÅáÉåÅó=x^Üë~å=measurement errors. This assumption, in comparison to the 'full' Kalman filter also yields a~åÇ=lD`çååçêI=NVVQzX=ëÉÅçåÇI=íÜÉ=ã~íÜÉã~íáÅ~ä=Éèìáî~äÉåÅÉ=ÄÉíïÉÉå=ÇáëÅêÉíÉJíáãÉ=better prediction efficiency [ Ahsan and O'Connor, 1994]; second, the mathematical^o=é~ê~ãÉíÉêë=~åÇ=Åçåíáåìçìë=ÉñéçåÉåíá~äë=~åÇ=íÜÉ=éçáåí=íÜ~í=Åçåíáåìçìë=íáãÉ=equivalence between discrete-time AR parameters and continuous exponentials and theãçÇÉäë=éêçîáÇÉ=~å=ÉäÉÖ~åí=ëçäìíáçå=Ñçê=ãçÇÉäáåÖ=áêêÉÖìä~êäó=ëé~ÅÉÇ=çÄëÉêî~íáçåë=xÉKÖKI=point that continuous time models provide an elegant solution for modeling irregularlye~êîÉóI=NVUVzK=_ÉÅ~ìëÉ=ëáãéäÉ=äÉ~ëíJëèì~êÉë=ãÉíÜçÇë=Çç=åçí=~ééäó=áå=Å~ëÉ=çÑ=spaced observations [e.g., Harvey, 1989]. Because simple least-squares methods do notãçÇÉäáåÖ=áêêÉÖìä~ê=Ç~í~I=~=ëìã=çÑ=ïÉáÖÜíÉÇ=ëèì~êÉÇ=áååçî~íáçåë=EptpfF=ÅêáíÉêáçå=áë=apply in case of modeling irregular data, a sum of weighted squared innovations (SWSI)áåíêçÇìÅÉÇ=~åÇ=ÇÉêáîÉÇ=Ñêçã=íÜÉ=äáâÉäáÜççÇ=ÑìåÅíáçå=çÑ=íÜÉ=áååçî~íáçåëK=få=~å=Éñ~ãéäÉ=criterion is introduced and derived from the likelihood function of the innovations. In an~ééäáÅ~íáçåI=áí=áë=ëÜçïå=íÜ~í=íÜÉ=Éëíáã~íÉë=çÑ=íÜÉ=ptpf=ÅêáíÉêáçå=ÅçåîÉêÖÉ=íç=ã~ñáãìã=example application, it is shown that the estimates of the SWSI criterion converge toäáâÉäáÜççÇ=Éëíáã~íÉë=Ñçê=ä~êÖÉê=ë~ãéäÉ=ëáòÉëK=cáå~ääóI=ïÉ=éêçéçëÉ=íç=ìëÉ=íÜÉ=ëçJÅ~ääÉÇ=maximum likelihood estimates for larger sample sizes. Finally, we propose to use the socalledinnovation variance function as an additional diagnostic check, next to the well-áååçî~íáçå=î~êá~åÅÉ=ÑìåÅíáçå=~ë=~å=~ÇÇáíáçå~ä=Çá~ÖåçëíáÅ=ÅÜÉÅâI=åÉñí=íç=íÜÉ=ïÉääJâåçïå=~ìíç=~åÇ=ÅêçëëÅçêêÉä~íáçå=ÑìåÅíáçåëK==known auto and crosscorrelation functions.KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK