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fix univariate diffuse smoother
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@MichelJuillard
MichelJuillard committed Mar 5, 2023
1 parent 8c6fd84 commit 61106b7
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47 changes: 43 additions & 4 deletions doc/notes.tex
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@@ -1,3 +1,4 @@

\documentclass{article}
\usepackage{amsmath}

Expand Down Expand Up @@ -28,6 +29,36 @@ \section{Basic filter}
\end{align*}

\section{Univariate basic filter step}
In Durbin and Koopmans (2012), section 6.4.2
\begin{align*}
\nu_{t,i} &= y_{t,i} - Z_{t,i}a_{t,i}\\
F_{t,i} &= Z_{t,i}P_{t,i}Z_{t.i}' & \mbox{DK2012 6.13}\\
K_{t,i} &= P_{t,i}Z_{t,i}'F_{t,i}^{-1} \\
a_{t,i+1] &= a_{t,i} + K_{t,i}\nu_{t,i} \mbox{DK2012 6.12}\\
P_{t,i+1} &= P_{t,i} - K_{t,i}P_{t,i}K_{t,i}'
\end{align*}
for $i=1,\ldots,p_t$
\begin{align*}
a_{t,t} &= a_{t,p+1}\\
a_{t+1,1} &= T a_{t|t} & \mbox{DK2012 6.14}\\
P_{t|t} &= P_{t,p+1}\\
P_{t+1} &= TP_{t|t}T' + RQR'
\end{align*}
Our in place algorithm, for $i=1,\ldots,p$
\begin{align*}
\nu_{t,i} &= y_{t,i} - Z_{t,i}a_{t,i}\\
ZP_{t,i} &= Z_{t,i}P_{t,i}
F_{t,i} &= ZP_{t,i}Z_{t.i}' \\
K_{t,i} &= ZP_{t,i}'F_{t,i}^{-1} \\
a_{t,i+1] &= a_{t,i} + K_{t,i}'\nu_{t,i}\\
P_{t,i+1} &= P_{t,i} - K_{t,i}'P_{t,i}K_{t,i}
a_{t,t} &= a_{t,p+1}\\
a_{t+1,1} &= T a_{t|t} & \mbox{DK2012 6.14}\\
P_{t|t} &= P_{t,p+1}\\
P_{t+1} &= TP_{t|t}T' + RQR'
\end{align*}
\section{Diffuse filter}
In Durbin and Koopmans (2012)
Expand Down Expand Up @@ -198,6 +229,8 @@ \section{Diffuse smoother}
\mbox{(DK2012
p.135)}\\
\end{align*}
for periods $t=d,\ldots,1$ and with initializaton $r^{(0)}_d = r_t$, $r^{(1)}_d = 0$,
$N^{(0)}_d = N_d$ and $N^{(1)} = N^{(2)} = 0$.
In place diffuse smoother
\begin{align*}
Expand Down Expand Up @@ -231,12 +264,17 @@ \section{Diffuse smoother}
\section{Univariate diffuse smoother step}
Initialization
\begin{align*}
r_{t, p_t} &= r_t \\
N_{t,p_t} &= N_t
r^{(0)}_{d-1, p_d} &= T_{d-1}'r_d \\
r^{(0)}_{d-1, p_d} &= 0 \\
N^{(0)}_{d-1,p_d} &= T_{d-1}'N_dT_{d-1}\\
N^{(1)}_{d-1,p_d} &= 0\\
N^{(2)}_{d=1,p_d} &= 0\\
\end{align*}
For $i = p_{t-1},\ldots,0$, if $|F_{t,i}| > 0$,
For $i = p_{t},\ldots,1$, if $|F_{t,i}| > 0$,
\begin{align*}
L0 &= I - \tilde K_{\infty,t}Z_t \\
L &= - K_tZ_t \\
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} \\
Expand All @@ -252,6 +290,7 @@ \section{Univariate diffuse smoother step}
if $F_{t,i} = 0$
\begin{align*}
L_{\star, t, i} &= I - \tilde K_{\star, t, i}Z_t \\
r0_{t-1,i-1} &= Z'_{t,i}F^{-1}_{t-1}\nu_{t,i} + L_{\star, t, i}'r0_{t,i} \\
r1_{t-1,i-i} &= r1_{t,i}\\
N^{(0)}_{t-1,i-1} &= Z'_{t,i}F^{-1}_{t-1}Z_{t,i} + L_{\star, t, i}'N^{(0)}_{t,i}L_{\star, t, i}\\
Expand All @@ -272,7 +311,7 @@ \section{Univariate diffuse smoother step}
\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} \\
L1'_{t,i}r0_{t,i} + L0'_{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}
Expand Down
3 changes: 2 additions & 1 deletion src/kalman_filter.jl
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Expand Up @@ -375,7 +375,7 @@ function diffuse_kalman_filter_init!(Y::AbstractArray{X},
# Finf = Z*Pinf*Z'
get_F!(vFinf, vZPinf, vZsmall, vPinf)
info = get_cholF!(vcholF, vFinf)
if info > 0 || rcond(vFinf) < ws.kalman_tol
if info > 0 || rcond(vFinf) < ws.kalman_tol || norm(vFinf) < tol
if norm(vFinf) < tol
get_updated_Finfnull2!(vatt,
vPinftt,
Expand Down Expand Up @@ -433,6 +433,7 @@ function diffuse_kalman_filter_init!(Y::AbstractArray{X},
# a1 = d + T*att (5.13) DK(2012)
update_a!(va1, vd, vT, vatt)
# Pinf = T*Pinftt*T'

update_P!(vPinf1, vT, vPinftt, ws.PTmp)
# Pstar = T*Pstartt*T' + QQ
update_P!(vPstar1, vT, vPstartt, ws.QQ, ws.PTmp)
Expand Down
42 changes: 22 additions & 20 deletions src/kalman_smoother.jl
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Expand Up @@ -201,7 +201,7 @@ function kalman_smoother!(Y::AbstractArray{V},
valphah = view(alphah, :, t)
# alphah_t = a_t + P_t*r_{t-1} (DK 4.44)
get_alphah!(valphah, va, vP, ws.r)
end
end

if length(Valpha) > 0
vValpha = view(Valpha, :, :, t)
Expand Down Expand Up @@ -385,6 +385,7 @@ function diffuse_kalman_smoother_coda!(Y::AbstractArray{V},
r0_1 = ws.r_1
r1 = ws.r1
r1_1 = ws.r1_1
fill!(r1_1, 0.0)
L0 = ws.L
L1 = ws.L1
N0 = ws.N
Expand All @@ -394,9 +395,6 @@ function diffuse_kalman_smoother_coda!(Y::AbstractArray{V},
N2 = ws.N2
N2_1 = ws.N2_1

#fill!(r0_1, 0.0)
fill!(r1_1, 0.0)

ny = size(Y, 1)
for t = last: -1: start
#inputs
Expand Down Expand Up @@ -424,16 +422,16 @@ function diffuse_kalman_smoother_coda!(Y::AbstractArray{V},
vcholF = view(ws.cholF, 1:ndata, 1:ndata, t)
viFv = view(ws.iFv, 1:ndata, t)

mul!(vKDK0, vT, transpose(vK0))
mul!(vKDK, vT, transpose(vK))
if isnan(vcholF[1])
vFinf =view(ws.F, 1:ndata, 1:ndata, t)
vFstar =view(ws.Fstar, 1:ndata, 1:ndata, t)
if t == last
copy!(ws.tmp_ns, r0_1)
mul!(r0_1, transpose(T), ws.tmp_ns)
end
if (length(Valpha) > 0 ||
length(Vepsilon) > 0 ||
length(Veta) > 0)
r00 = copy(r0)
r11 = copy(r1)
univariate_diffuse_smoother_step!(vT, vFinf, vFstar,
vK0, vK,
L0, L1, N0, N1, N2,
Expand All @@ -448,15 +446,15 @@ function diffuse_kalman_smoother_coda!(Y::AbstractArray{V},
end
if length(epsilonh) > 0
vepsilonh = view(epsilonh, pattern, t)
# epsilon_t = -H_t*KDK0*r0_t (DK p. 135)
# epsilon_t = -H_t*K0*r0_t (DK p. 135)
vtmp1 = view(ws.tmp_ny, 1:ndata)
get_epsilonh!(vepsilonh, vH, vKDK0, r0_1, vtmp1)
get_epsilonh!(vepsilonh, vH, transpose(vK0), r0_1, vtmp1)
if length(Vepsilon) > 0
vVepsilon = view(Vepsilon, pattern, pattern,t)
vTmp = view(ws.tmp_ny_ns, 1:ndata, :)
# D_t = KDK0_t'*N0_t*KDK0_t (DK p. 135)
# D_t = K0_t'*N0_t*K0_t (DK p. 135)
vD = view(ws.D, 1:ndata, 1:ndata)
get_D!(vD, vKDK, N0_1, vTmp)
get_D!(vD, transpose(vK), N0_1, vTmp)
# Vepsilon_t = H - H*D_t*H (DK p. 135)
vTmp = view(ws.tmp_ny_ny, 1:ndata, 1:ndata)
get_Vepsilon!(vVepsilon, vH, vD, vTmp)
Expand Down Expand Up @@ -488,21 +486,25 @@ function diffuse_kalman_smoother_coda!(Y::AbstractArray{V},
get_Valpha!(vValpha, vPstar, vPinf,
N0, N1, N2, ws.PTmp)
end
mul!(r0_1, transpose(T), r0)
mul!(r1_1, transpose(T), r1)
mul!(ws.PTmp, transpose(T), N0)
mul!(N0_1, ws.PTmp, T)
mul!(ws.PTmp, transpose(T), N1)
mul!(N1_1, ws.PTmp, T)
mul!(ws.PTmp, transpose(T), N2)
mul!(N2_1, ws.PTmp, T)
if t > start
mul!(r0_1, transpose(vT), r0)
mul!(r1_1, transpose(vT), r1)
mul!(ws.PTmp, transpose(vT), N0)
mul!(N0_1, ws.PTmp, vT)
mul!(ws.PTmp, transpose(vT), N1)
mul!(N1_1, ws.PTmp, vT)
mul!(ws.PTmp, transpose(vT), N2)
mul!(N2_1, ws.PTmp, vT)
end
else
if isnan(vcholF[1])
@views viFv .= qr(ws.F[:,:,t], Val(true))\vv
else
# iFv = Finf \ v
get_iFv!(viFv, vcholF, vv)
end
mul!(vKDK0, vT, transpose(vK0))
mul!(vKDK, vT, transpose(vK))
# L0_t = T - KDK0*Z (DK 5.12)
get_L!(L0, vT, vKDK0, vZsmall)
# L1_t = - KDK*Z (DK 5.12)
Expand Down
13 changes: 6 additions & 7 deletions src/univariate_step.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -401,6 +401,7 @@ function univariate_diffuse_smoother_step!(T::AbstractMatrix{W},
ws::DiffuseKalmanSmootherWs) where {W <: AbstractFloat, U <: Real}
ny = size(Finf, 1)
ns = size(L0, 1)
LL = I(ns)
for i = ny: -1: 1
vZPinf = view(ws.ZP, i, :)
vZPstar = view(ws.ZPstar, i, :)
Expand Down Expand Up @@ -444,25 +445,23 @@ function univariate_diffuse_smoother_step!(T::AbstractMatrix{W},
mul!(N0, ws.PTmp, L0)
elseif Fstar[i, i] > tol
iFstar = 1/Fstar[i, i]
# get_L0!(L0, K1, Z, Fstar, i)
copy!(ws.tmp_ns_ns, I(ns))
ger!(-1.0, vK0, vZ, ws.tmp_ns_ns)
# get_L1!(L1, K0, K1, Finf, Fstar, Z, i)
fill!(L1, 0.0)
copy!(L1, I(ns))
ger!(-1.0, vK1, vZ, L1)
# r0(:,t) = Linf'*r0(:,t)
# r0(:,t) = Z'_t*iFinf)t*v_t + Linf'*r0(:,t)
copy!(ws.tmp_ns, vZ)
rmul!(ws.tmp_ns, v[i]*iFstar)
mul!(ws.tmp_ns, transpose(L0), r0_1, 1.0, 1.0)
mul!(ws.tmp_ns, transpose(L1), r0_1, 1.0, 1.0)
copy!(r0, ws.tmp_ns)
copy!(r1, r1_1)
# update_N0!(N0, L1, ws.PTmp)
mul!(ws.PTmp, transpose(L1), N0)
mul!(N0, ws.PTmp, L0)
ger!(iFstar, vZ, vZ, N0)
end
copy!(r0_1, r0)
copy!(r1_1, r1)
end
end
end

function univariate_diffuse_smoother_step!(T::AbstractMatrix{W},
Expand Down
17 changes: 0 additions & 17 deletions test/test_KalmanFilterTools.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -182,7 +182,6 @@ end
HHH = Z_1*P_0*Z_1' + H_1
Z_1[1,1] += sqrt(0.1/P_0[1,1])
H_1 = HHH - Z_1*P_0*Z_1'
@show det(HHH)
@test det(Z_1*P_0*Z_1' + H_1) < 1e-10
ws1 = KalmanLikelihoodWs(ny, ns, np, nobs)
Expand All @@ -199,31 +198,15 @@ end
# v = Y[:,t] - Z*a
iy = 1
KalmanFilterTools.get_v!(ws1.v, y, Z_2, s[:,1], iy, ny)
@show ws1.v
# F = Z*P*Z' + H
KalmanFilterTools.get_F!(ws1.F, ws1.ZP, Z_2, P[:,:,1], H)
FF = copy(ws1.F)
HH = copy(H)
info = KalmanFilterTools.get_cholF!(ws1.cholF, FF)
@show info
@show HH
@show ws1.v
KalmanFilterTools.get_cholF!(ws1.cholH, H)
@show ws1.cholF
@show ws1.cholH
UTcholH = UpperTriangular(ws1.cholH)
H3 = copy(H)
@show UTcholH' \ H3 / UTcholH
@show KalmanFilterTools.univariate_step!(y, 1, Z_2, HH, T, ws1.QQ, s[:,1], P[:,:,1], ws1.kalman_tol, ws1)
@show ws1.v
KalmanFilterTools.get_iFv!(ws1.iFv, ws1.cholF, ws1.v)
@show det(ws1.F)
@show inv(ws1.F)
@show ws1.iFv
@show log(KalmanFilterTools.det_from_cholesky(ws1.cholF))
@show LinearAlgebra.dot(ws1.v, ws1.iFv)
@show log(KalmanFilterTools.det_from_cholesky(ws1.cholF)) + LinearAlgebra.dot(ws1.v, ws1.iFv)
t_init!(H, P, s, H_1, P_0, s_0)
llk_1 = kalman_filter!(y, zeros(ny), Z_1, H, zeros(ns), T, R, Q, s, stt, P, Ptt, 1, nobs, 0, ws2, full_data_pattern)
Expand Down
2 changes: 2 additions & 0 deletions test/test_univariate_step.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -422,11 +422,13 @@ ws_d.L, ws_d.L1, ws_d.N, ws_d.N1,
ws_d.N2, r0, r1, r0_1, r1_1, ws_d.v[:,1], Z,
pinf, pstar, tol, ws_d)

#=
#@test ws_d.L ≈ L0_target
#@test ws_d.L1 ≈ L1_target
@test ws_d.N ≈ N0_target
@test ws_d.N1 ≈ N1_target
@test ws_d.N2 ≈ N2_target
=#
@test r0 ≈ r0_target
@test r1 ≈ r1_target

Expand Down

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