Commit 922ac26c authored by MichelJuillard's avatar MichelJuillard
Browse files

adding notes

parent 3361be0c
(TeX-add-style-hook
"notes"
(lambda ()
(TeX-run-style-hooks
"latex2e"
"article"
"art10"
"amsmath"))
:latex)
\relax
\@writefile{toc}{\contentsline {section}{\numberline {1}Basic filter}{1}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {2}Diffuse filter}{1}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {3}Basic smoother}{4}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {4}Diffuse smoother}{5}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {5}Univariate smoother step}{6}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\numberline {6}Univariate diffuse smoother step}{7}\protected@file@percent }
This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020/Debian) (preloaded format=pdflatex 2020.7.9) 25 JUL 2020 16:11
entering extended mode
restricted \write18 enabled.
file:line:error style messages enabled.
%&-line parsing enabled.
**\input notes.tex
(./notes.tex
(/usr/share/texlive/texmf-dist/tex/latex/base/article.cls
Document Class: article 2019/12/20 v1.4l Standard LaTeX document class
(/usr/share/texlive/texmf-dist/tex/latex/base/size10.clo
File: size10.clo 2019/12/20 v1.4l Standard LaTeX file (size option)
)
\c@part=\count167
\c@section=\count168
\c@subsection=\count169
\c@subsubsection=\count170
\c@paragraph=\count171
\c@subparagraph=\count172
\c@figure=\count173
\c@table=\count174
\abovecaptionskip=\skip47
\belowcaptionskip=\skip48
\bibindent=\dimen134
)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty
Package: amsmath 2020/01/20 v2.17e AMS math features
\@mathmargin=\skip49
For additional information on amsmath, use the `?' option.
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty
Package: amstext 2000/06/29 v2.01 AMS text
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty
File: amsgen.sty 1999/11/30 v2.0 generic functions
\@emptytoks=\toks15
\ex@=\dimen135
))
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty
Package: amsbsy 1999/11/29 v1.2d Bold Symbols
\pmbraise@=\dimen136
)
(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty
Package: amsopn 2016/03/08 v2.02 operator names
)
\inf@bad=\count175
LaTeX Info: Redefining \frac on input line 227.
\uproot@=\count176
\leftroot@=\count177
LaTeX Info: Redefining \overline on input line 389.
\classnum@=\count178
\DOTSCASE@=\count179
LaTeX Info: Redefining \ldots on input line 486.
LaTeX Info: Redefining \dots on input line 489.
LaTeX Info: Redefining \cdots on input line 610.
\Mathstrutbox@=\box45
\strutbox@=\box46
\big@size=\dimen137
LaTeX Font Info: Redeclaring font encoding OML on input line 733.
LaTeX Font Info: Redeclaring font encoding OMS on input line 734.
\macc@depth=\count180
\c@MaxMatrixCols=\count181
\dotsspace@=\muskip16
\c@parentequation=\count182
\dspbrk@lvl=\count183
\tag@help=\toks16
\row@=\count184
\column@=\count185
\maxfields@=\count186
\andhelp@=\toks17
\eqnshift@=\dimen138
\alignsep@=\dimen139
\tagshift@=\dimen140
\tagwidth@=\dimen141
\totwidth@=\dimen142
\lineht@=\dimen143
\@envbody=\toks18
\multlinegap=\skip50
\multlinetaggap=\skip51
\mathdisplay@stack=\toks19
LaTeX Info: Redefining \[ on input line 2859.
LaTeX Info: Redefining \] on input line 2860.
)
(/usr/share/texlive/texmf-dist/tex/latex/l3backend/l3backend-pdfmode.def
File: l3backend-pdfmode.def 2020-06-23 L3 backend support: PDF mode
\l__kernel_color_stack_int=\count187
\l__pdf_internal_box=\box47
)
(./notes.aux)
\openout1 = `notes.aux'.
LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 4.
LaTeX Font Info: ... okay on input line 4.
LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 4.
LaTeX Font Info: ... okay on input line 4.
LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 4.
LaTeX Font Info: ... okay on input line 4.
LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 4.
LaTeX Font Info: ... okay on input line 4.
LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 4.
LaTeX Font Info: ... okay on input line 4.
LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 4.
LaTeX Font Info: ... okay on input line 4.
LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 4.
LaTeX Font Info: ... okay on input line 4.
Overfull \hbox (29.60796pt too wide) detected at line 56
[]
[]
[1
{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}] [2] [3]
Overfull \hbox (42.93344pt too wide) detected at line 170
[]
[]
[4]
Overfull \hbox (16.36311pt too wide) detected at line 199
[]
[]
[5] [6]
Overfull \hbox (46.89458pt too wide) detected at line 249
[]
[]
[7] (./notes.aux) )
Here is how much of TeX's memory you used:
999 strings out of 482343
13383 string characters out of 5946347
260649 words of memory out of 5000000
16848 multiletter control sequences out of 15000+600000
534324 words of font info for 31 fonts, out of 8000000 for 9000
59 hyphenation exceptions out of 8191
30i,11n,25p,208b,130s stack positions out of 5000i,500n,10000p,200000b,80000s
</usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/c
m/cmbx10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx
12.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb
></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb></usr/
share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></usr/share/t
exlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/share/texlive/
texmf-dist/fonts/type1/public/amsfonts/cm/cmr5.pfb></usr/share/texlive/texmf-di
st/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/share/texlive/texmf-dist/fonts
/type1/public/amsfonts/cm/cmsy10.pfb></usr/share/texlive/texmf-dist/fonts/type1
/public/amsfonts/cm/cmsy5.pfb></usr/share/texlive/texmf-dist/fonts/type1/public
/amsfonts/cm/cmsy7.pfb>
Output written on notes.pdf (7 pages, 117790 bytes).
PDF statistics:
72 PDF objects out of 1000 (max. 8388607)
51 compressed objects within 1 object stream
0 named destinations out of 1000 (max. 500000)
1 words of extra memory for PDF output out of 10000 (max. 10000000)
File added
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\section{Basic filter}
In Durbin and Koopmans (2012)
\begin{align*}
\nu_t &= y_t - Z_t a_t & \mbox{(DK2012 4.13)}\\
F_t &= Z_tP_tZ'_t + H_t & \mbox{(DK2012 4.16)}\\
a_{t|t} &= a_t + P_tZ'_t F^{-1}_t\nu_t & \mbox{(DK2012 4.17)}\\
P_{t|t} &= P_t - P_tZ'_t F^{-1}_tZ_tP_t & \mbox{(DK2012 4.18)}\\
K_t &= T_tP_tZ'_t F^{-1}_t & \mbox{(DK2012 4.22)}\\
a_{t+1} &= T_ta_t + K_t\nu_t & \mbox{(DK2012 4.21)}\\
P_{t+1} &= T_tP_t(T_t - K_tZ_t)' + R_tQ_tR_t' & \mbox{(DK2012 4.23)}
\end{align*}
Our in place algorithm
\begin{align*}
\nu_t &= y_t - c_t - Z_t a_t \\
ZP &= Z_tP_t \\
F_t &= ZPZ'_t + H_t \\
iF\nu_t &= F^{-1}_t\nu_t\\
\tilde K_t &= F^{-1}_tZP & \mbox{alternative } K\\
a_{t|t} &= a_t + \tilde K'_t\nu_t) \\
P_{t|t} &= P_t - \tilde K'_t (ZP) \\
a_{t+1} &= d_t + T_ta_{t|t}\\
P_{t+1} &= T_tP_{t|t}T_t + R_tQ_tR_t'
\end{align*}
\section{Diffuse filter}
In Durbin and Koopmans (2012)
\begin{align*}
\nu^{(0)}_t &= y_t - Z_t a^{(0)}_t & \mbox{(DK2012 p. 128)}\\
F_{\infty,t} &= Z_tP_{\infty,t}Z'_t & \mbox{(DK2012 5.7)}\\
F_{\star,t} &= Z_tP_{\star,t}Z'_t + H_t & \mbox{(DK2012 5.7)}\\
M_{\infty,t } &= P_{\infty,t}Z'_t & \mbox{(DK2012 5.7)}\\
M_{\star,t } &= P_{\star,t}Z'_t & \mbox{(DK2012 5.7)}\\
\end{align*}
When $F^{-1}_{\infty,t}$ is regular
\begin{align*}
F^{(1)}_t &= F^{-1}_{\infty, t} & \mbox{(DK2012 5.10)}\\
F^{(2)}_t &= -F^{-1}_{\infty, t}F_{\star, t}F^{-1}_{\infty, t} & \mbox{(DK2012 5.10)}\\
K^{(0)}_t &= T_tM_{\infty,t}F^{(1)}_t & \mbox{(DK2012 5.12)}\\
K^{(1)}_t &= T_tM_{\star,t}F^{(1)}_t + T_tM_{\infty,t}F^{(2)}_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_{\infty,t}F^{(1)}_t\nu^{(0)}_t\\
P_{\infty,t|t} &= P_{\infty,t} - P_{\infty,t}Z'_tF^{(1)}_tM_{\infty,t}' \\
P_{\star,t|t} &= P_{\star,t} - P_{\star,t}Z'_t
F^{(1)}_tZ_tP_{\infty,t} -
P_{\infty,t}Z'_t(F^{(1)}Z_tP_{\star,t} + F^{(2)}Z_tP_{\infty})\\
a^{(0)}_{t+1} &= T_ta^{(0)}_t + K^{(0)}_t\nu^{(0)}_t & \mbox{(DK2012 5.13)}\\
P_{\infty,t+ 1} &= T_tP_{\infty,t}L^{(0)'}_t & \mbox{(DK2012 5.14)}\\
P_{\star, t+1} &= T_tP_{\infty,t}L^{(1)'}_t + T_tP_{\star,t}L^{(0)'}_t + R_tQ_tR_t' & \mbox{(DK2012 5.14)}
\end{align*}
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)}\\
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_tP_{\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)}\\
P_{\star, t+1} &= T_tP_{\star,t}L^{(0)'}_t + R_tQ_tR_t' & \mbox{(DK2012 5.17).}
\end{align*}
When $F^{-1}_{\infty,t}$ is singular but different from zero, one uses
a univariate step.
The diffuse filter is used only for few iterations at the beginning of
the computation of the filter. For some of the arrays we use the same
one that will be used for the rest of the computation. Our in place
algorithm is
\begin{align*}
\nu_t &= y_t - c_t - Z_t a_t \\
F_{\infty,t} &= Z_tP_{\infty,t}Z'_t \\
F_{\star,t} &= Z_tP_{\star,t}Z'_t + H_t \\
ZP_\infty &= Z_tP_{\infty,t}\\
ZP_\star &= Z_tP_{\star,t} \\
\end{align*}
When $F^{-1}_{\infty,t}$ is regular
\begin{align*}
\tilde K_{\infty,t} &= F^{(1)}_t(ZP_{\infty}) \\
\tilde K_{\star,t} &= F^{(1)}_t((ZP_{\star}) + F_{\star,t}K_{\infty,t}) \\
a_{t|t} &= a_t + K_{\infty,t}'\nu_t\\
P_{\infty,t|t} &= P_{\infty,t} - \tilde K'_{\infty,t}(ZP_{\infty}) \\
P_{\star,t|t} &= P_{\star,t} - (ZP_{\star})'\tilde K_{\infty,t}
- (ZP_{\infty})'\tilde K_{\star,t}\\
a_{t+1} &= d_t + T_ta_{t|t} \\
P_{\infty,t+ 1} &= T_tP_{\infty,t}T'_t\\
P_{\star, t+1} &= T_tP_{\star,t|t}T'_t + R_tQ_tR_t'
\end{align*}
When $F^{-1}_{\infty,t} = \mathbf{0}$
\begin{align*}
K_{\infty,t}^{(0)} &= T_tM_{\star,t}F^{-1}_{\star,t}& \mbox{(DK2012 5.15)}\\
a_{t|t} &= a_t + K^{-1}_{\infty,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_tP_{\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)}\\
P_{\star, t+1} &= T_tP_{\star,t}L^{(0)'}_t + R_tQ_tR_t' & \mbox{(DK2012 5.17).}
\end{align*}
\section{Basic smoother}
\begin{align*}
L_t &= T_t - K_tZ_t & \mbox{(DK2012 p. 87)}\\
r_{t-1} &= Z'_tF^{-1}_t\nu_t + L'_tr_t & \mbox{(DK2012 4.38)}\\
\hat a_t &= a_t + P_tr_{t-1} & \mbox{(DK2012 4.35)}\\
N_{t-1} &= Z'_tF^{-1}_tZ_t + L'_tN_tL_t & \mbox{(DK2012 4.42)}\\
V_t &= P_t - P_tN_{t-1}P_t & \mbox{(DK2012 4.44)}\\
u_t &= F^{-1}_t\nu_t - K'_tr_t & \mbox{(DK2012 4.59)}\\
\hat \epsilon_t &= H_tu_t & \mbox{(DK2012 4.58)}\\
D_t &= F^{-1}_t + K'_tN_tK_t & \mbox{(DK2012 4.66)}\\
\mbox{Var}(\epsilon_t|Y_n) &= H_t - H_tD_tH_t & \mbox{(DK2012 4.65)}\\
\hat \eta_t &= Q_t'R'_tr_t & \mbox{(DK2012 4.63)}\\
\mbox{Var}(\eta_t|Y_n) &= Q_t - Q_tR'_tN_tR_tQ_t & \mbox{(DK2012 4.68)}
\end{align*}
In place basic smoother
\begin{align*}
K_t &= T\tilde K'_t \\
L_t &= T_t - K_tZ_t \\
r_{t-1} &= Z'_t(iF\nu)_t + L'_tr_t \\
\hat a_t &= a_t + P_tr_{t-1} \\
N_{t-1} &= Z'_t(iFZ)_t + L'_tN_tL_t \\
V_t &= P_t - P_tN_{t-1}P_t \\
\hat \epsilon_t &= H_t((iF\nu)_t - K'r_t) \\
D_t &= F^{-1}_t + K'_tN_tK_t \\
(V\epsilon)_t &= H_t - H_tD_tH_t \\
\hat \eta_t &= Q_tR'_tr_t \\
(V\eta)_t &= Q_t - Q_tR'_tN_tR_tQ_t \\
\end{align*}
\section{Diffuse smoother}
\begin{align*}
L^{(0)}_t &= T_t - K^{(0)}_tZ_t & \mbox{(DK2012 5.12)}\\
L^{(1)}_t &= - K^{(1)}_tZ_t & \mbox{(DK2012 5.12)}\\
r^{(0)}_{t-1} &= L^{(0)'}_tr^{(0)}_t & \mbox{(DK2012 5.21)}\\
r^{(1)}_{t-1} &= Z'_tF^{(1)}\nu^{(0)}_t + L^{(0)'}_tr^{(1)}_t +
L^{(1)'}_tr^{(0)}_t & \mbox{(DK2012 5.21)}\\
\hat a_t &= a^{(0)}_t + P_{\star,t}r^{(0)}_{t-1} +
P_{\infty,t}r^{(1)}_{t-1} & \mbox{(DK2012 5.23)} \\
N^{(0)}_{t-1} &= L^{(0)'}_tN^{(0)}L^{(0)}_t & \mbox{(DK2012 5.26)}\\
N^{(1)}_{t-1} &= Z'_tF^{(1)}_tZ_t + L^{(0)'}_tN^{(1)}L^{(0)}_t
+L^{(1)'}_tN^{(0)}L^{(0)}_t
& \mbox{(DK2012 5.29)}\\
N^{(2)}_{t-1} &= Z'_tF^{(2)}_tZ_t + L^{(0)'}_tN^{(2)}L^{(0)}_t +
L^{(0)'}_tN^{(1)}L^{(1)}_t +
L^{(1)'}_tN^{(1)}L^{(0)}_t \\
&\;\;\;\; L^{(1)'}_tN^{(0)}L^{(1)}_t & \mbox{(DK2012 5.29)}\\
V_t &= P_{\star,t} - P_{\star,t}N^{(0)}_{t-1}P_{\star,t} -
(P_{\star,t}N^{(1)}_{t-1}P_{\infty,t})' -
P_{\infty,t}N^{(1)}_{t-1}P_{\star,t}\\
&\;\;\;\;-P_{\infty,t}N^{(2)}_{t-1}P_{\infty,t} & \mbox{(DK2012
5.28)}\\
\hat\epsilon_t &= -H_tK^{(0)}_tr^{(0)}_t & \mbox{(DK2012 p.135)}\\
\hat\eta_t &= Q_tR'_tr^{(0)}_t& \mbox{(DK2012 p.135)}\\
\mbox{Var}(\epsilon_t|Y_n) &= H_t - H_tK^{(0)}_tN^{(0)}K^{(0)}_t &
\mbox{(DK2012
p.135)}\\
\mbox{Var}(\eta_t|Y_n) &= Q_t - Q_tR'_tN^{(0)}_tR_tQ_t &
\mbox{(DK2012
p.135)}\\
\end{align*}
In place diffuse smoother
\begin{align*}
K_{\infty,t} &= T\tilde K'_{\infty,t} \\
K_t &= T\tilde K'_t \\
L0 &= T_t - K_{\infty,t}Z_t \\
L &= - K_tZ_t \\
r0 &= L0'r0\_1 \\
r1 &= Z'_tF^{(1)}\nu_t + (L0)'r1\_1 +
Lr0\_1 \\
ahat_t &= a^{(0)}_t + P_{\star,t}r0 +
P_{\infty,t}r1 \\
N0 &= (L0)'(N0)(L0) \\
N1 &= Z'_tiFZ_t + (L0)'(N1\_1)(L0)
+L(N0)L0
\\
N2 &= Z_tF^{-1}_{\infty,t}F_{\star,t}F^{-1}_{\infty,t}Z_t + (L0)'(N2\_1)(L0) +
L^{(0)'}_tN^{(1)}L^{(1)}_t +
L^{(1)'}_tN^{(1)}L^{(0)}_t \\
&\;\;\;\; L^{(1)'}_tN^{(0)}L^{(1)}_t \\
V_t &= P_{\star,t} - P_{\star,t}N^{(0)}_{t-1}P_{\star,t} -
(P_{\star,t}N^{(1)}_{t-1}P_{\infty,t})' -
P_{\infty,t}N^{(1)}_{t-1}P_{\star,t}\\
&\;\;\;\;-P_{\infty,t}N^{(2)}_{t-1}P_{\infty,t} \\
\hat\epsilon_t &= -H_tK^{(0)}_tr^{(0)}_t \\
\hat\eta_t &= Q_tR'_tr^{(0)}_t\\
\mbox{Var}(\epsilon_t|Y_n) &= H_t - H_tK^{(0)}_tN^{(0)}K^{(0)}_t\\
\mbox{Var}(\eta_t|Y_n) &= Q_t - Q_tR'_tN^{(0)}_tR_tQ_t \\
\end{align*}
\section{Univariate smoother step}
Initialization
\begin{align*}
r_{t, p_t} &= r_t \\
N_{t,p_t} &= N_t
\end{align*}
For $i = p_{t-1},\ldots,0$, if $|F_{t,i}| > 0$,
\begin{align*}
r_{t-1,i-1} &= Z'_{t,i}F^{-1}_{t,i}\nu_{t,i} + L'_{t,i}r_{t,i} & \mbox{(DK2012 6.15)}\\
N_{t-1,i-1} &= Z'_{t,i}F^{-1}_{t,i}Z_{t,i} + L'_{t,i}N_{t,i}L_{t,i} & \mbox{(DK2012 6.15)}\\
\hat\epsilon_{t,i} & \sigma^2_{t,i}F^{-1}_{t,i}(\nu_{t,i} -
K'_{t,i}r_{t,i} & \mbox{(DK2012 p. 157)}\\
\mbox{Var}(\hat\epsilon_{t,i}) &= \sigma^4_{t,i}F^{-2}_{t,i}(F_{t,i}
+ K'_{t,i}N_{t,i}K_{t,i}) & \mbox{(DK2012 p. 157)}
\end{align*}
if $F_{t,i} = 0$
\begin{align*}
r_{t-1,i-1} &= r_{t,i} \\
N_{t-1,i-1} &= N_{t,i}
\end{align*}
and
\begin{align*}
r_{t-1,p_{t-1}} &= T'_{t-1}r_{t,0} & \mbox{(DK2012 6.15)}\\
N_{t-1,p_{t-1}} &= T'_{t-1}N_{t,0}T_{t-1} & \mbox{(DK2012 6.15)}\\
r_{t-1} &= r_{t-1, p_{t-1}}\\
N_{t-1} &= N_{t-1, p_{t-1}}
\end{align*}
\section{Univariate diffuse smoother step}
Initialization
\begin{align*}
r_{t, p_t} &= r_t \\
N_{t,p_t} &= N_t
\end{align*}
For $i = p_{t-1},\ldots,0$, if $|F_{t,i}| > 0$,
\begin{align*}
r0_{t-1,i-1} &= L'_{\infty,i}r0_{t,i} \\
r1_{t-1,i-1} &= Z'_{t,i}F^{-1}_{t,i}\nu_{t,i} +
L'_{\infty,t,i}r0_{t,i} + L'_{0,t,i}r1_{t,i} \\
N^{(0)}_{t-1,i-1} &= L'_{\infty,t,i}N^{(0)}_{t,i}L_{\infty,t,i} & \mbox{(DK2012 5.26)}\\
N^{(1)}_{t-1,i-1} &= Z'_{t,i}F^{(1)}_{t,i}Z_{t,i} + L'_{\infty,t,i}N^{(0)}L_t
+L_{\infty,t,i}N^{(1)}L_{0,t,i}
& \mbox{(DK2012 5.29)}\\
N^{(2)}_{t-1,-1} &= Z'_{t,i}F^{(2)}_{t,i}Z_{t,i} + L^{(0)'}_{0,t,i}N^{(2)}_{t,i}L^{(0)}_{0,t,i} +
L^{(0)'}_{t,i}N^{(1)}_{t,i}L^{(1)}_{t,i} +
L^{(1)'}_{t,i}N^{(1)}_{t,i}L^{(0)}_{t,i} \\
&\;\;\;\; L^{(1)'}_{t,i}N^{(0)}_{t,i}L^{(1)}_{t,i} & \mbox{(DK2012 5.29)}\\
\end{align*}
if $F_{t,i} = 0$
\begin{align*}
r_{t-1,i-1} &= r_{t,i} \\
N_{t-1,i-1} &= N_{t,i}
\end{align*}
and
\begin{align*}
r_{t-1,p_{t-1}} &= T'_{t-1}r_{t,0} & \mbox{(DK2012 6.15)}\\
N_{t-1,p_{t-1}} &= T'_{t-1}N_{t,0}T_{t-1} & \mbox{(DK2012 6.15)}\\
r_{t-1} &= r_{t-1, p_{t-1}}\\
N_{t-1} &= N_{t-1, p_{t-1}}\\
\end{align*}
In place univariate diffuse smoother step
For $i = p_{t-1},\ldots,0$, if $|F_{t,i}| > 0$,
\begin{align*}
r0_{t-1,i-1} &= L0'_{t,i}r0_{t,i} \\
r1_{t-1,i-1} &= Z'_{t,i}F^{-1}_{t,i}\nu_{t,i} +
L0'_{t,i}r0_{t,i} + L1'_{t,i}r1_{t,i} \\
N^{(0)}_{t-1,i-1} &= L0'_{t,i}N^{(0)}_{t,i}L0_{t,i}\\
N^{(1)}_{t-1,i-1} &= Z'_{t,i}F^{(1)}_{t,i}Z_{t,i} + L'_{\infty,t,i}N^{(0)}L_t
+L_{\infty,t,i}N^{(1)}L_{0,t,i}
& \mbox{(DK2012 5.29)}\\
N^{(2)}_{t-1,-1} &= Z'_{t,i}F^{(2)}_{t,i}Z_{t,i} + L^{(0)'}_{0,t,i}N^{(2)}_{t,i}L^{(0)}_{0,t,i} +
L^{(0)'}_{t,i}N^{(1)}_{t,i}L^{(1)}_{t,i} +
L^{(1)'}_{t,i}N^{(1)}_{t,i}L^{(0)}_{t,i} \\
&\;\;\;\; L^{(1)'}_{t,i}N^{(0)}_{t,i}L^{(1)}_{t,i} & \mbox{(DK2012 5.29)}\\
\end{align*}
\end{document}
%%% Local Variables:
%%% mode: latex
%%% TeX-master: t
%%% End:
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment