function LIK = DiffuseLikelihood3(T,R,Q,Pinf,Pstar,Y,trend,start)%//Z,T,R,Q,Pinf,Pstar,Y)
% 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;
newRank	 	= rank(Pinf,crit);
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
			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);
        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. [stéphane,11-03-2004].	  
			if rank(Pinf,crit) == 0
				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(Fi)])
    	end
	end
    if all(abs(Pinf(:))<crit),
        oldRank = 0;
    else
        oldRank = rank(Pinf,crit);
    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 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);