From c31ac80cee87d22a9fa6201138aa970db85591aa Mon Sep 17 00:00:00 2001 From: MichelJuillard Date: Sat, 3 Dec 2022 21:45:31 +0100 Subject: [PATCH] fix notes on diffuse filter --- doc/notes.tex | 9 +++++---- 1 file changed, 5 insertions(+), 4 deletions(-) diff --git a/doc/notes.tex b/doc/notes.tex index c6e6cae..8332dfc 100644 --- a/doc/notes.tex +++ b/doc/notes.tex @@ -57,12 +57,12 @@ \section{Diffuse filter} When $F^{-1}_{\infty,t} = \mathbf{0}$ \begin{align*} K^{(0)}_t &= T_tM_{\star,t}F^{-1}_{\star,t}& \mbox{(DK2012 5.15)}\\ - L^{(0}_t &= T_t - K^{(0)}_tZ_t & \mbox{(DK2012 5.12)}\\ - L^{(1}_t &= - K^{(1)}_tZ_t & \mbox{(DK2012 5.12)}\\ + L^{(0)}_t &= T_t - K^{(0)}_tZ_t & \mbox{(DK2012 5.12)}\\ + L^{(1)}_t &= - K^{(1)}_tZ_t & \mbox{(DK2012 5.12)}\\ a^{(0)}_{t|t} &= a^{(0)}_t + M_{\star,t}F^{-1}_{\star,t}\nu^{(0)}_t\\ P_{\infty,t|t} &= P_{\infty,t} \\ P_{\star,t|t} &= P_{\star,t} - P_{\star,t}Z'_t - F^{-1}_{\star,t}Z_t \\ + F^{-1}_{\star,t}Z_t P_{\star,t}\\ a^{(0)}_{t+1} &= T_ta^{(0)}_t + K^{(0)}_t\nu^{(0)}_t & \mbox{(DK2012 p. 129)}\\ P_{\infty,t+ 1} &= T_tP_{\infty,t}T'_t & \mbox{(DK2012 5.14)}\\ @@ -85,7 +85,8 @@ \section{Diffuse filter} When $F^{-1}_{\infty,t}$ is regular \begin{align*} \tilde K_{\infty,t} &= F^{-1}_{\infty,t}(ZP_{\infty,t}) \\ - \tilde K_{\star,t} &= F^{-1}_{\infty,t}((ZP_{\star,t}) - F_{\star,t}K_{\infty,t}) \\ + \tilde K_{\star,t} &= F^{-1}_{\infty,t}((ZP_{\star,t}) - + F_{\star,t}\tilde K_{\infty,t}) \\ a_{t|t} &= a_t + \tilde K_{\infty,t}'\nu_t\\ P_{\infty,t|t} &= P_{\infty,t} - \tilde K'_{\infty,t}(ZP_{\infty,t}) \\ P_{\star,t|t} &= P_{\star,t} - \tilde K_{\infty,t}'ZP_{\star,t}