MMSE estimator - tips til start
Posted: 25/04-2010 16:27
Oppgaven lyder slik:
Spørsmålet blir da om noen har et hint til hvordan jeg skal begynne på denne. Synes oppgaven er litt annereldes enn alle andre jeg tidligere har gjort om samme emne, så står litt fast.Consider a parameter [tex]\mathbf{\theta}[/tex] which changes with time according to the deterministic relation
[tex]\mathbf{\theta}\left[n\right] = \mathbf{A}\mathbf{\theta}\left[n-1\right]\; n\geq 1[/tex],
where [tex]\mathbf{A}[/tex] is a known [tex]p\times p[/tex] matrix and [tex]\mathbf{\theta}\left[0\right][/tex] is an unknown parameter which is modeled as a random ([tex]p\times 1[/tex]) vector. Note that once [tex]\mathbf{\theta}\left[0\right][/tex] is specified, so is [tex]\mathbf{\theta}\left[n\right][/tex] for [tex]n\geq 1[/tex]. Prove that the MMSE estimator of [tex]\mathbf{\theta}\left[n\right][/tex] is
[tex]\mathbf{\hat{\theta}}\left[n\right] =\mathbf{A}^n\mathbf{\hat{\theta}}\left[0\right][/tex],
where [tex]\mathbf{\hat{\theta}}\left[0\right][/tex] is the MMSE estimator of [tex]\mathbf{\theta}\left[0\right][/tex], or equivalently,
[tex]\mathbf{\hat{\theta}}\left[n\right] = \mathbf{A}\mathbf{\hat{\theta}}\left[n-1\right][/tex]