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function [alphahat,etahat,a,P,aK,PK,d,decomp] = missing_DiffuseKalmanSmoother3(T,R,Q,Pinf1,Pstar1,Y,trend,pp,mm,smpl,mf,data_index)
% function [alphahat,etahat,a1, aK] = missing_DiffuseKalmanSmoother3(T,R,Q,Pinf1,Pstar1,Y,trend,pp,mm,smpl,mf,data_index)
% Computes the diffuse kalman smoother without measurement error, in the case of a singular var-cov matrix.
% Univariate treatment of multivariate time series.
%
% INPUTS
% T: mm*mm matrix
% R: mm*rr matrix
% Q: rr*rr matrix
% Pinf1: mm*mm diagonal matrix with with q ones and m-q zeros
% Pstar1: mm*mm variance-covariance matrix with stationary variables
% Y: pp*1 vector
% trend
% pp: number of observed variables
% mm: number of state variables
% smpl: sample size
% mf: observed variables index in the state vector
% data_index [cell] 1*smpl cell of column vectors of indices.
%
% OUTPUTS
% alphahat: smoothed state variables (a_{t|T})
% etahat: smoothed shocks
% a: matrix of updated variables (a_{t|t})
% aK: 3D array of k step ahead filtered state variables (a_{t+k|t})
%
% SPECIAL REQUIREMENTS
% See "Filtering and Smoothing of State Vector for Diffuse State Space
% Models", S.J. Koopman and J. Durbin (2003, in Journal of Time Series
% Analysis, vol. 24(1), pp. 85-98).
% Copyright (C) 2004-2008 Dynare Team
%
% This file is part of Dynare.
%
% Dynare is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% Dynare is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
% GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with Dynare. If not, see <http://www.gnu.org/licenses/>.
% Modified by M. Ratto
% New output argument aK: 1-step to nk-stpe ahed predictions)
% New input argument nk: max order of predictions in aK
% New option options_.diffuse_d where the DKF stops (common with
% diffuselikelihood3)
% New icc variable to count number of iterations for Finf steps
% Pstar % Pinf simmetric
% New termination of DKF iterations based on options_.diffuse_d
% Li also stored during DKF iterations !!
% some bugs corrected in the DKF part of the smoother (Z matrix and
% alphahat)
global options_
d=0;
decomp=[];
nk = options_.nk;
spinf = size(Pinf1);
spstar = size(Pstar1);
v = zeros(pp,smpl);
a = zeros(mm,smpl);
a1 = zeros(mm,smpl+1);
aK = zeros(nk,mm,smpl+nk);
PK = zeros(nk,mm,mm,smpl+nk);
if isempty(options_.diffuse_d),
smpl_diff = 1;
else
smpl_diff=rank(Pinf1);
end
Fstar = zeros(pp,smpl_diff);
Finf = zeros(pp,smpl_diff);
Fi = zeros(pp,smpl_diff);
Ki = zeros(mm,pp,smpl);
Li = zeros(mm,mm,pp,smpl);
Linf = zeros(mm,mm,pp,smpl_diff);
L0 = zeros(mm,mm,pp,smpl_diff);
Kstar = zeros(mm,pp,smpl_diff);
P = zeros(mm,mm,smpl+1);
P1 = P;
Pstar = zeros(spstar(1),spstar(2),smpl_diff+1); Pstar(:,:,1) = Pstar1;
Pinf = zeros(spinf(1),spinf(2),smpl_diff+1); Pinf(:,:,1) = Pinf1;
Pstar1 = Pstar;
Pinf1 = Pinf;
crit = options_.kalman_tol;
crit1 = 1.e-6;
steady = smpl;
rr = size(Q,1);
QQ = R*Q*transpose(R);
QRt = Q*transpose(R);
alphahat = zeros(mm,smpl);
etahat = zeros(rr,smpl);
r = zeros(mm,smpl+1);
Z = zeros(pp,mm);
for i=1:pp;
Z(i,mf(i)) = 1;
end
t = 0;
icc=0;
newRank = rank(Pinf(:,:,1),crit1);
while newRank & t < smpl
t = t+1;
a(:,t) = a1(:,t);
Pstar(:,:,t)=tril(Pstar(:,:,t))+transpose(tril(Pstar(:,:,t),-1));
Pinf(:,:,t)=tril(Pinf(:,:,t))+transpose(tril(Pinf(:,:,t),-1));
Pstar1(:,:,t) = Pstar(:,:,t);
Pinf1(:,:,t) = Pinf(:,:,t);
di = data_index{t}';
for i=di
v(i,t) = Y(i,t)-a(mf(i),t)-trend(i,t);
Fstar(i,t) = Pstar(mf(i),mf(i),t);
Finf(i,t) = Pinf(mf(i),mf(i),t);
Kstar(:,i,t) = Pstar(:,mf(i),t);
if Finf(i,t) > crit & newRank, % original MJ: if Finf(i,t) > crit
icc=icc+1;
Kinf(:,i,t) = Pinf(:,mf(i),t);
Linf(:,:,i,t) = eye(mm) - Kinf(:,i,t)*Z(i,:)/Finf(i,t);
L0(:,:,i,t) = (Kinf(:,i,t)*Fstar(i,t)/Finf(i,t) - Kstar(:,i,t))*Z(i,:)/Finf(i,t);
a(:,t) = a(:,t) + Kinf(:,i,t)*v(i,t)/Finf(i,t);
Pstar(:,:,t) = Pstar(:,:,t) + ...
Kinf(:,i,t)*transpose(Kinf(:,i,t))*Fstar(i,t)/(Finf(i,t)*Finf(i,t)) - ...
(Kstar(:,i,t)*transpose(Kinf(:,i,t)) +...
Kinf(:,i,t)*transpose(Kstar(:,i,t)))/Finf(i,t);
Pinf(:,:,t) = Pinf(:,:,t) - Kinf(:,i,t)*transpose(Kinf(:,i,t))/Finf(i,t);
Pstar(:,:,t)=tril(Pstar(:,:,t))+transpose(tril(Pstar(:,:,t),-1));
Pinf(:,:,t)=tril(Pinf(:,:,t))+transpose(tril(Pinf(:,:,t),-1));
% new terminiation criteria by M. Ratto
P0=Pinf(:,:,t);
% newRank = any(diag(P0(mf,mf))>crit);
% if newRank==0, id = i; end,
if ~isempty(options_.diffuse_d),
newRank = (icc<options_.diffuse_d);
%if newRank & any(diag(P0(mf,mf))>crit)==0;
if newRank & (any(diag(P0(mf,mf))>crit)==0 & rank(P0,crit1)==0);
disp('WARNING!! Change in OPTIONS_.DIFFUSE_D in univariate DKF')
options_.diffuse_d = icc;
newRank=0;
end
else
%newRank = any(diag(P0(mf,mf))>crit);
newRank = (any(diag(P0(mf,mf))>crit) | rank(P0,crit1));
if newRank==0,
options_.diffuse_d = icc;
end
end,
% if newRank==0,
% options_.diffuse_d=i; %this is buggy
% end
% end new terminiation criteria by M. Ratto
elseif Fstar(i,t) > crit
%% Note that : (1) rank(Pinf)=0 implies that Finf = 0, (2) outside this loop (when for some i and t the condition
%% rank(Pinf)=0 is satisfied we have P = Pstar and F = Fstar and (3) Finf = 0 does not imply that
%% rank(Pinf)=0. [st<73>phane,11-03-2004].
Li(:,:,i,t) = eye(mm)-Kstar(:,i,t)*Z(i,:)/Fstar(i,t); % we need to store Li for DKF smoother
a(:,t) = a(:,t) + Kstar(:,i,t)*v(i,t)/Fstar(i,t);
Pstar(:,:,t) = Pstar(:,:,t) - Kstar(:,i,t)*transpose(Kstar(:,i,t))/Fstar(i,t);
Pstar(:,:,t)=tril(Pstar(:,:,t))+transpose(tril(Pstar(:,:,t),-1));
end
end
a1(:,t+1) = T*a(:,t);
for jnk=1:nk,
aK(jnk,:,t+jnk) = T^jnk*a(:,t);
end
Pstar(:,:,t+1) = T*Pstar(:,:,t)*transpose(T)+ QQ;
Pinf(:,:,t+1) = T*Pinf(:,:,t)*transpose(T);
P0=Pinf(:,:,t+1);
if newRank,
%newRank = any(diag(P0(mf,mf))>crit);
newRank = rank(P0,crit1);
end
end
d = t;
P(:,:,d+1) = Pstar(:,:,d+1);
Linf = Linf(:,:,:,1:d);
L0 = L0(:,:,:,1:d);
Fstar = Fstar(:,1:d);
Finf = Finf(:,1:d);
Kstar = Kstar(:,:,1:d);
Pstar = Pstar(:,:,1:d);
Pinf = Pinf(:,:,1:d);
Pstar1 = Pstar1(:,:,1:d);
Pinf1 = Pinf1(:,:,1:d);
notsteady = 1;
while notsteady & t<smpl
t = t+1;
a(:,t) = a1(:,t);
P(:,:,t)=tril(P(:,:,t))+transpose(tril(P(:,:,t),-1));
P1(:,:,t) = P(:,:,t);
di = data_index{t}';
for i=di
v(i,t) = Y(i,t) - a(mf(i),t) - trend(i,t);
Fi(i,t) = P(mf(i),mf(i),t);
Ki(:,i,t) = P(:,mf(i),t);
if Fi(i,t) > crit
Li(:,:,i,t) = eye(mm)-Ki(:,i,t)*Z(i,:)/Fi(i,t);
a(:,t) = a(:,t) + Ki(:,i,t)*v(i,t)/Fi(i,t);
P(:,:,t) = P(:,:,t) - Ki(:,i,t)*transpose(Ki(:,i,t))/Fi(i,t);
P(:,:,t)=tril(P(:,:,t))+transpose(tril(P(:,:,t),-1));
end
end
a1(:,t+1) = T*a(:,t);
Pf = P(:,:,t);
for jnk=1:nk,
Pf = T*Pf*T' + QQ;
aK(jnk,:,t+jnk) = T^jnk*a1(:,t);
PK(jnk,:,:,t+jnk) = Pf;
end
P(:,:,t+1) = T*P(:,:,t)*transpose(T) + QQ;
% notsteady = ~(max(max(abs(P(:,:,t+1)-P(:,:,t))))<crit);
end
% $$$ P_s=tril(P(:,:,t))+transpose(tril(P(:,:,t),-1));
% $$$ Fi_s = Fi(:,t);
% $$$ Ki_s = Ki(:,:,t);
% $$$ L_s =Li(:,:,:,t);
% $$$ if t<smpl
% $$$ t_steady = t+1;
% $$$ P = cat(3,P(:,:,1:t),repmat(P(:,:,t),[1 1 smpl-t_steady+1]));
% $$$ Fi = cat(2,Fi(:,1:t),repmat(Fi_s,[1 1 smpl-t_steady+1]));
% $$$ Li = cat(4,Li(:,:,:,1:t),repmat(L_s,[1 1 smpl-t_steady+1]));
% $$$ Ki = cat(3,Ki(:,:,1:t),repmat(Ki_s,[1 1 smpl-t_steady+1]));
% $$$ end
% $$$ while t<smpl
% $$$ t=t+1;
% $$$ a(:,t) = a1(:,t);
% $$$ di = data_index{t}';
% $$$ for i=di
% $$$ v(i,t) = Y(i,t) - a(mf(i),t) - trend(i,t);
% $$$ if Fi_s(i) > crit
% $$$ a(:,t) = a(:,t) + Ki_s(:,i)*v(i,t)/Fi_s(i);
% $$$ end
% $$$ end
% $$$ a1(:,t+1) = T*a(:,t);
% $$$ for jnk=1:nk,
% $$$ aK(jnk,:,t+jnk) = T^jnk*a(:,t);
% $$$ end
% $$$ end
ri=zeros(mm,1);
t = smpl+1;
while t>d+1
t = t-1;
di = flipud(data_index{t})';
for i = di
if Fi(i,t) > crit
ri = Z(i,:)'/Fi(i,t)*v(i,t)+Li(:,:,i,t)'*ri;
end
end
r(:,t) = ri;
alphahat(:,t) = a1(:,t) + P1(:,:,t)*r(:,t);
etahat(:,t) = QRt*r(:,t);
ri = T'*ri;
end
if d
r0 = zeros(mm,d);
r0(:,d) = ri;
r1 = zeros(mm,d);
for t = d:-1:1
di = flipud(data_index{t})';
for i = di
if Finf(i,t) > crit & ~(t==d & i>options_.diffuse_d),
% use of options_.diffuse_d to be sure of DKF termination
%r1(:,t) = transpose(Z)*v(:,t)/Finf(i,t) + ... BUG HERE in transpose(Z)
r1(:,t) = Z(i,:)'*v(i,t)/Finf(i,t) + ...
L0(:,:,i,t)'*r0(:,t) + Linf(:,:,i,t)'*r1(:,t);
r0(:,t) = Linf(:,:,i,t)'*r0(:,t);
elseif Fstar(i,t) > crit % step needed whe Finf == 0
r0(:,t) = Z(i,:)'/Fstar(i,t)*v(i,t)+Li(:,:,i,t)'*r0(:,t);
end
end
alphahat(:,t) = a1(:,t) + Pstar1(:,:,t)*r0(:,t) + Pinf1(:,:,t)*r1(:,t);
r(:,t) = r0(:,t);
etahat(:,t) = QRt*r(:,t);
if t > 1
r0(:,t-1) = T'*r0(:,t);
r1(:,t-1) = T'*r1(:,t);
end
end
end
if nargout > 7
decomp = zeros(nk,mm,rr,smpl+nk);
ZRQinv = inv(Z*QQ*Z');
for t = max(d,1):smpl
ri_d = Z'*iF(:,:,t)*v(:,t);
% calculate eta_tm1t
eta_tm1t = QRt*ri_d;
% calculate decomposition
Ttok = eye(mm,mm);
for h = 1:nk
for j=1:rr
eta=zeros(rr,1);
eta(j) = eta_tm1t(j);
decomp(h,:,j,t+h) = T^(h-1)*P(:,:,t)*Z'*ZRQinv*Z*R*eta;
end
end
end
end
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