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DiffuseLikelihood3.m 7.18 KiB
function [LIK, lik] = DiffuseLikelihood3(T,R,Q,Pinf,Pstar,Y,trend,start)%//Z,T,R,Q,Pinf,Pstar,Y)
% M. Ratto added lik in output [October 2005]
% changes by M. Ratto [April 2005]
% introduced new options options_.diffuse_d for termination of DKF
% new icc counter for Finf steps in DKF
% new termination for DKF
% likelihood terms for Fstar must be cumulated in DKF also when Pinf is non
% zero.
%
% [4/5/2005] correctyed bug in the modified verson of Ratto for rank of Pinf
% introduced a specific crit1 for the DKF termination
%
% stepane.adjemian@cepremap.cnrs.fr [07-19-2004]
%
% 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).
%
% Case where F_{\infty,t} is singular ==> Univariate treatment of multivariate
% time series.
%
% THE PROBLEM:
%
% y_t = Z_t * \alpha_t + \varepsilon_t
% \alpha_{t+1} = T_t * \alpha_t + R_t * \eta_t
%
% with:
%
% \alpha_1 = a + A*\delta + R_0*\eta_0
%
% m*q matrix A and m*(m-q) matrix R_0 are selection matrices (their
% columns constitue all the columns of the m*m identity matrix) so that
%
% A'*R_0 = 0 and A'*\alpha_1 = \delta
%
% We assume that the vector \delta is distributed as a N(0,\kappa*I_q)
% for a given \kappa > 0. So that the expectation of \alpha_1 is a and
% its variance is P, with
%
% P = \kappa*P_{\infty} + P_{\star}
%
% P_{\infty} = A*A'
% P_{\star} = R_0*Q_0*R_0'
%
% P_{\infty} is a m*m diagonal matrix with q ones and m-q zeros.
%
%
% and where:
%
% y_t is a pp*1 vector
% \alpha_t is a mm*1 vector
% \varepsilon_t is a pp*1 multivariate random variable (iid N(0,H_t))
% \eta_t is a rr*1 multivariate random variable (iid N(0,Q_t))
% a_1 is a mm*1 vector
%
% Z_t is a pp*mm matrix
% T_t is a mm*mm matrix
% H_t is a pp*pp matrix
% R_t is a mm*rr matrix
% Q_t is a rr*rr matrix
% P_1 is a mm*mm matrix
%
%
% FILTERING EQUATIONS:
%
% v_t = y_t - Z_t* a_t
% F_t = Z_t * P_t * Z_t' + H_t
% K_t = T_t * P_t * Z_t' * F_t^{-1}
% L_t = T_t - K_t * Z_t
% a_{t+1} = T_t * a_t + K_t * v_t% P_{t+1} = T_t * P_t * L_t' + R_t*Q_t*R_t'
%
%
% DIFFUSE FILTERING EQUATIONS:
%
% a_{t+1} = T_t*a_t + K_{\ast,t}v_t
% P_{\infty,t+1} = T_t*P_{\infty,t}*T_t'
% P_{\ast,t+1} = T_t*P_{\ast,t}*L_{\ast,t}' + R_t*Q_t*R_t'
% K_{\ast,t} = T_t*P_{\ast,t}*Z_t'*F_{\ast,t}^{-1}
% v_t = y_t - Z_t*a_t
% L_{\ast,t} = T_t - K_{\ast,t}*Z_t
% F_{\ast,t} = Z_t*P_{\ast,t}*Z_t' + H_t
global bayestopt_ options_
mf = bayestopt_.mf;
pp = size(Y,1);
mm = size(T,1);
smpl = size(Y,2);
a = zeros(mm,1);
QQ = R*Q*transpose(R);
t = 0;
lik = zeros(smpl+1,1);
lik(smpl+1) = smpl*pp*log(2*pi); %% the constant of minus two times the log-likelihood
notsteady = 1;
crit = options_.kalman_tol;
crit1 = 1.e-6;
newRank = rank(Pinf,crit1);
icc=0;
while newRank & t < smpl
t = t+1;
for i=1:pp
v(i) = Y(i,t)-a(mf(i))-trend(i,t);
Fstar = Pstar(mf(i),mf(i));
Finf = Pinf(mf(i),mf(i));
Kstar = Pstar(:,mf(i));
if Finf > crit & newRank, %added newRank criterion
icc=icc+1;
Kinf = Pinf(:,mf(i));
a = a + Kinf*v(i)/Finf;
Pstar = Pstar + Kinf*transpose(Kinf)*Fstar/(Finf*Finf) - ...
(Kstar*transpose(Kinf)+Kinf*transpose(Kstar))/Finf;
Pinf = Pinf - Kinf*transpose(Kinf)/Finf;
lik(t) = lik(t) + log(Finf);
% start new termination criterion for DKF
if ~isempty(options_.diffuse_d),
newRank = (icc<options_.diffuse_d);
%if newRank & any(diag(Pinf(mf,mf))>crit)==0; % M. Ratto this line is BUGGY
if newRank & (any(diag(Pinf(mf,mf))>crit)==0 & rank(Pinf,crit1)==0);
options_.diffuse_d = icc;
newRank=0;
disp('WARNING: Change in OPTIONS_.DIFFUSE_D in univariate DKF')
disp(['new OPTIONS_.DIFFUSE_D = ',int2str(icc)])
disp('You may have to reset the optimisation')
end
else
%newRank = any(diag(Pinf(mf,mf))>crit); % M. Ratto this line is BUGGY
newRank = (any(diag(Pinf(mf,mf))>crit) | rank(Pinf,crit1));
if newRank==0,
P0= T*Pinf*transpose(T);
%newRank = any(diag(P0(mf,mf))>crit); % M. Ratto this line is BUGGY
newRank = (any(diag(P0(mf,mf))>crit) | rank(P0,crit1)); % M. Ratto 11/10/2005
if newRank==0,
options_.diffuse_d = icc;
end
end
end,
% end new termination and checks for DKF
elseif Fstar > 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. [stphane,11-03-2004].
%if rank(Pinf,crit) == 0
% the likelihood terms should alwasy be cumulated, not only
% when Pinf=0, otherwise the lik would depend on the ordering
% of observed variables
% presample options can be used to ignore initial time points
lik(t) = lik(t) + log(Fstar) + v(i)*v(i)/Fstar;
%end
a = a + Kstar*v(i)/Fstar;
Pstar = Pstar - Kstar*transpose(Kstar)/Fstar;
else
%disp(['zero F term in DKF for observed ',int2str(i),' ',num2str(Fstar)])
end
end
% if all(abs(Pinf(:))<crit),
% oldRank = 0;
% else
% oldRank = rank(Pinf,crit);
% end
if newRank,
oldRank = rank(Pinf,crit1);
else
oldRank = 0;
end
a = T*a;
Pstar = T*Pstar*transpose(T)+QQ;
Pinf = T*Pinf*transpose(T);
% if all(abs(Pinf(:))<crit),
% newRank = 0;
% else
% newRank = rank(Pinf,crit);
% end
if newRank,
newRank = rank(Pinf,crit1); % new crit1 is used
end
if oldRank ~= newRank
disp('DiffuseLiklihood3 :: T does influence the rank of Pinf!')
end
end
if t == smpl
error(['There isn''t enough information to estimate the initial' ...
' conditions of the nonstationary variables']);
end
while notsteady & t < smpl
t = t+1;
oldP = Pstar;
for i=1:pp
v(i) = Y(i,t) - a(mf(i)) - trend(i,t);
Fi = Pstar(mf(i),mf(i));
if Fi > crit
Ki = Pstar(:,mf(i));
a = a + Ki*v(i)/Fi;
Pstar = Pstar - Ki*transpose(Ki)/Fi;
lik(t) = lik(t) + log(Fi) + v(i)*v(i)/Fi;
else
%disp(['zero F term for observed ',int2str(i),' ',num2str(Fi)])
end
end
a = T*a;
Pstar = T*Pstar*transpose(T) + QQ;
notsteady = ~(max(max(abs(Pstar-oldP)))<crit);
end
while t < smpl
t = t+1;
Pstar = oldP;
for i=1:pp
v(i) = Y(i,t) - a(mf(i)) - trend(i,t);
Fi = Pstar(mf(i),mf(i));
if Fi > crit
Ki = Pstar(:,mf(i)); a = a + Ki*v(i)/Fi;
Pstar = Pstar - Ki*transpose(Ki)/Fi;
lik(t) = lik(t) + log(Fi) + v(i)*v(i)/Fi;
else
%disp(['zero F term for observed ',int2str(i),' ',num2str(Fi)])
end
end
a = T*a;
end
LIK = .5*(sum(lik(start:end))-(start-1)*lik(smpl+1)/smpl);