223 lines
7.2 KiB
Matlab
223 lines
7.2 KiB
Matlab
function [LIK, lik] = DiffuseLikelihood3(T,R,Q,Pinf,Pstar,Y,trend,start)%//Z,T,R,Q,Pinf,Pstar,Y)
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% M. Ratto added lik in output [October 2005]
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% changes by M. Ratto [April 2005]
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% introduced new options options_.diffuse_d for termination of DKF
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% new icc counter for Finf steps in DKF
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% new termination for DKF
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% likelihood terms for Fstar must be cumulated in DKF also when Pinf is non
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% zero.
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%
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% [4/5/2005] correctyed bug in the modified verson of Ratto for rank of Pinf
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% introduced a specific crit1 for the DKF termination
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%
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% stepane.adjemian@cepremap.cnrs.fr [07-19-2004]
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%
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% See "Filtering and Smoothing of State Vector for Diffuse State Space
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% Models", S.J. Koopman and J. Durbin (2003, in Journal of Time Series
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% Analysis, vol. 24(1), pp. 85-98).
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%
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% Case where F_{\infty,t} is singular ==> Univariate treatment of multivariate
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% time series.
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%
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% THE PROBLEM:
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%
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% y_t = Z_t * \alpha_t + \varepsilon_t
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% \alpha_{t+1} = T_t * \alpha_t + R_t * \eta_t
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%
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% with:
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%
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% \alpha_1 = a + A*\delta + R_0*\eta_0
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%
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% m*q matrix A and m*(m-q) matrix R_0 are selection matrices (their
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% columns constitue all the columns of the m*m identity matrix) so that
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%
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% A'*R_0 = 0 and A'*\alpha_1 = \delta
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%
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% We assume that the vector \delta is distributed as a N(0,\kappa*I_q)
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% for a given \kappa > 0. So that the expectation of \alpha_1 is a and
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% its variance is P, with
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%
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% P = \kappa*P_{\infty} + P_{\star}
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%
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% P_{\infty} = A*A'
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% P_{\star} = R_0*Q_0*R_0'
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%
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% P_{\infty} is a m*m diagonal matrix with q ones and m-q zeros.
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%
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%
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% and where:
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%
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% y_t is a pp*1 vector
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% \alpha_t is a mm*1 vector
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% \varepsilon_t is a pp*1 multivariate random variable (iid N(0,H_t))
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% \eta_t is a rr*1 multivariate random variable (iid N(0,Q_t))
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% a_1 is a mm*1 vector
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%
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% Z_t is a pp*mm matrix
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% T_t is a mm*mm matrix
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% H_t is a pp*pp matrix
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% R_t is a mm*rr matrix
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% Q_t is a rr*rr matrix
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% P_1 is a mm*mm matrix
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%
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%
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% FILTERING EQUATIONS:
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%
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% v_t = y_t - Z_t* a_t
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% F_t = Z_t * P_t * Z_t' + H_t
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% K_t = T_t * P_t * Z_t' * F_t^{-1}
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% L_t = T_t - K_t * Z_t
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% a_{t+1} = T_t * a_t + K_t * v_t
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% P_{t+1} = T_t * P_t * L_t' + R_t*Q_t*R_t'
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%
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%
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% DIFFUSE FILTERING EQUATIONS:
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%
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% a_{t+1} = T_t*a_t + K_{\ast,t}v_t
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% P_{\infty,t+1} = T_t*P_{\infty,t}*T_t'
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% P_{\ast,t+1} = T_t*P_{\ast,t}*L_{\ast,t}' + R_t*Q_t*R_t'
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% K_{\ast,t} = T_t*P_{\ast,t}*Z_t'*F_{\ast,t}^{-1}
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% v_t = y_t - Z_t*a_t
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% L_{\ast,t} = T_t - K_{\ast,t}*Z_t
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% F_{\ast,t} = Z_t*P_{\ast,t}*Z_t' + H_t
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global bayestopt_ options_
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mf = bayestopt_.mf;
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pp = size(Y,1);
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mm = size(T,1);
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smpl = size(Y,2);
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a = zeros(mm,1);
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QQ = R*Q*transpose(R);
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t = 0;
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lik = zeros(smpl+1,1);
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lik(smpl+1) = smpl*pp*log(2*pi); %% the constant of minus two times the log-likelihood
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notsteady = 1;
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crit = options_.kalman_tol;
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crit1 = 1.e-6;
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newRank = rank(Pinf,crit1);
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icc=0;
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while newRank & t < smpl
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t = t+1;
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for i=1:pp
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v(i) = Y(i,t)-a(mf(i))-trend(i,t);
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Fstar = Pstar(mf(i),mf(i));
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Finf = Pinf(mf(i),mf(i));
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Kstar = Pstar(:,mf(i));
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if Finf > crit & newRank, %added newRank criterion
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icc=icc+1;
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Kinf = Pinf(:,mf(i));
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a = a + Kinf*v(i)/Finf;
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Pstar = Pstar + Kinf*transpose(Kinf)*Fstar/(Finf*Finf) - ...
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(Kstar*transpose(Kinf)+Kinf*transpose(Kstar))/Finf;
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Pinf = Pinf - Kinf*transpose(Kinf)/Finf;
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lik(t) = lik(t) + log(Finf);
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% start new termination criterion for DKF
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if ~isempty(options_.diffuse_d),
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newRank = (icc<options_.diffuse_d);
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%if newRank & any(diag(Pinf(mf,mf))>crit)==0; % M. Ratto this line is BUGGY
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if newRank & (any(diag(Pinf(mf,mf))>crit)==0 & rank(Pinf,crit1)==0);
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options_.diffuse_d = icc;
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newRank=0;
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disp('WARNING: Change in OPTIONS_.DIFFUSE_D in univariate DKF')
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disp(['new OPTIONS_.DIFFUSE_D = ',int2str(icc)])
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disp('You may have to reset the optimisation')
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end
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else
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%newRank = any(diag(Pinf(mf,mf))>crit); % M. Ratto this line is BUGGY
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newRank = (any(diag(Pinf(mf,mf))>crit) | rank(Pinf,crit1));
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if newRank==0,
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P0= T*Pinf*transpose(T);
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%newRank = any(diag(P0(mf,mf))>crit); % M. Ratto this line is BUGGY
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newRank = (any(diag(P0(mf,mf))>crit) | rank(P0,crit1)); % M. Ratto 11/10/2005
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if newRank==0,
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options_.diffuse_d = icc;
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end
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end
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end,
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% end new termination and checks for DKF
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elseif Fstar > crit
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%% Note that : (1) rank(Pinf)=0 implies that Finf = 0, (2) outside this loop (when for some i and t the condition
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%% rank(Pinf)=0 is satisfied we have P = Pstar and F = Fstar and (3) Finf = 0 does not imply that
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%% rank(Pinf)=0. [st<73>phane,11-03-2004].
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%if rank(Pinf,crit) == 0
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% the likelihood terms should alwasy be cumulated, not only
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% when Pinf=0, otherwise the lik would depend on the ordering
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% of observed variables
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% presample options can be used to ignore initial time points
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lik(t) = lik(t) + log(Fstar) + v(i)*v(i)/Fstar;
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%end
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a = a + Kstar*v(i)/Fstar;
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Pstar = Pstar - Kstar*transpose(Kstar)/Fstar;
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else
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%disp(['zero F term in DKF for observed ',int2str(i),' ',num2str(Fstar)])
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end
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end
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% if all(abs(Pinf(:))<crit),
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% oldRank = 0;
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% else
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% oldRank = rank(Pinf,crit);
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% end
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if newRank,
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oldRank = rank(Pinf,crit1);
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else
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oldRank = 0;
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end
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a = T*a;
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Pstar = T*Pstar*transpose(T)+QQ;
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Pinf = T*Pinf*transpose(T);
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% if all(abs(Pinf(:))<crit),
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% newRank = 0;
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% else
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% newRank = rank(Pinf,crit);
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% end
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if newRank,
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newRank = rank(Pinf,crit1); % new crit1 is used
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end
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if oldRank ~= newRank
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disp('DiffuseLiklihood3 :: T does influence the rank of Pinf!')
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end
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end
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if t == smpl
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error(['There isn''t enough information to estimate the initial' ...
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' conditions of the nonstationary variables']);
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end
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while notsteady & t < smpl
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t = t+1;
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oldP = Pstar;
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for i=1:pp
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v(i) = Y(i,t) - a(mf(i)) - trend(i,t);
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Fi = Pstar(mf(i),mf(i));
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if Fi > crit
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Ki = Pstar(:,mf(i));
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a = a + Ki*v(i)/Fi;
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Pstar = Pstar - Ki*transpose(Ki)/Fi;
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lik(t) = lik(t) + log(Fi) + v(i)*v(i)/Fi;
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else
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%disp(['zero F term for observed ',int2str(i),' ',num2str(Fi)])
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end
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end
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a = T*a;
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Pstar = T*Pstar*transpose(T) + QQ;
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notsteady = ~(max(max(abs(Pstar-oldP)))<crit);
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end
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while t < smpl
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t = t+1;
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Pstar = oldP;
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for i=1:pp
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v(i) = Y(i,t) - a(mf(i)) - trend(i,t);
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Fi = Pstar(mf(i),mf(i));
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if Fi > crit
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Ki = Pstar(:,mf(i));
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a = a + Ki*v(i)/Fi;
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Pstar = Pstar - Ki*transpose(Ki)/Fi;
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lik(t) = lik(t) + log(Fi) + v(i)*v(i)/Fi;
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else
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%disp(['zero F term for observed ',int2str(i),' ',num2str(Fi)])
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end
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end
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a = T*a;
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end
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LIK = .5*(sum(lik(start:end))-(start-1)*lik(smpl+1)/smpl);
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