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_t0001 function [LIK, lik] = DiffuseLikelihood3(T,R,Q,Pinf,Pstar,Y,trend,start)%//Z,T,R,Q,Pinf,Pstar,Y) 0002 % M. Ratto added lik in output [October 2005] 0003 % changes by M. Ratto [April 2005] 0004 % introduced new options options_.diffuse_d for termination of DKF 0005 % new icc counter for Finf steps in DKF 0006 % new termination for DKF 0007 % likelihood terms for Fstar must be cumulated in DKF also when Pinf is non 0008 % zero. 0009 % 0010 % [4/5/2005] correctyed bug in the modified verson of Ratto for rank of Pinf 0011 % introduced a specific crit1 for the DKF termination 0012 % 0013 % stepane.adjemian@cepremap.cnrs.fr [07-19-2004] 0014 % 0015 % See "Filtering and Smoothing of State Vector for Diffuse State Space 0016 % Models", S.J. Koopman and J. Durbin (2003, in Journal of Time Series 0017 % Analysis, vol. 24(1), pp. 85-98). 0018 % 0019 % Case where F_{\infty,t} is singular ==> Univariate treatment of multivariate 0020 % time series. 0021 % 0022 % THE PROBLEM: 0023 % 0024 % y_t = Z_t * \alpha_t + \varepsilon_t 0025 % \alpha_{t+1} = T_t * \alpha_t + R_t * \eta_t 0026 % 0027 % with: 0028 % 0029 % \alpha_1 = a + A*\delta + R_0*\eta_0 0030 % 0031 % m*q matrix A and m*(m-q) matrix R_0 are selection matrices (their 0032 % columns constitue all the columns of the m*m identity matrix) so that 0033 % 0034 % A'*R_0 = 0 and A'*\alpha_1 = \delta 0035 % 0036 % We assume that the vector \delta is distributed as a N(0,\kappa*I_q) 0037 % for a given \kappa > 0. So that the expectation of \alpha_1 is a and 0038 % its variance is P, with 0039 % 0040 % P = \kappa*P_{\infty} + P_{\star} 0041 % 0042 % P_{\infty} = A*A' 0043 % P_{\star} = R_0*Q_0*R_0' 0044 % 0045 % P_{\infty} is a m*m diagonal matrix with q ones and m-q zeros. 0046 % 0047 % 0048 % and where: 0049 % 0050 % y_t is a pp*1 vector 0051 % \alpha_t is a mm*1 vector 0052 % \varepsilon_t is a pp*1 multivariate random variable (iid N(0,H_t)) 0053 % \eta_t is a rr*1 multivariate random variable (iid N(0,Q_t)) 0054 % a_1 is a mm*1 vector 0055 % 0056 % Z_t is a pp*mm matrix 0057 % T_t is a mm*mm matrix 0058 % H_t is a pp*pp matrix 0059 % R_t is a mm*rr matrix 0060 % Q_t is a rr*rr matrix 0061 % P_1 is a mm*mm matrix 0062 % 0063 % 0064 % FILTERING EQUATIONS: 0065 % 0066 % v_t = y_t - Z_t* a_t 0067 % F_t = Z_t * P_t * Z_t' + H_t 0068 % K_t = T_t * P_t * Z_t' * F_t^{-1} 0069 % L_t = T_t - K_t * Z_t 0070 % a_{t+1} = T_t * a_t + K_t * v_t 0071 % P_{t+1} = T_t * P_t * L_t' + R_t*Q_t*R_t' 0072 % 0073 % 0074 % DIFFUSE FILTERING EQUATIONS: 0075 % 0076 % a_{t+1} = T_t*a_t + K_{\ast,t}v_t 0077 % P_{\infty,t+1} = T_t*P_{\infty,t}*T_t' 0078 % P_{\ast,t+1} = T_t*P_{\ast,t}*L_{\ast,t}' + R_t*Q_t*R_t' 0079 % K_{\ast,t} = T_t*P_{\ast,t}*Z_t'*F_{\ast,t}^{-1} 0080 % v_t = y_t - Z_t*a_t 0081 % L_{\ast,t} = T_t - K_{\ast,t}*Z_t 0082 % F_{\ast,t} = Z_t*P_{\ast,t}*Z_t' + H_t 0083 global bayestopt_ options_ 0084 0085 mf = bayestopt_.mf; 0086 pp = size(Y,1); 0087 mm = size(T,1); 0088 smpl = size(Y,2); 0089 a = zeros(mm,1); 0090 QQ = R*Q*transpose(R); 0091 t = 0; 0092 lik = zeros(smpl+1,1); 0093 lik(smpl+1) = smpl*pp*log(2*pi); %% the constant of minus two times the log-likelihood 0094 notsteady = 1; 0095 crit = options_.kalman_tol; 0096 crit1 = 1.e-6; 0097 newRank = rank(Pinf,crit1); 0098 icc=0; 0099 while newRank & t < smpl 0100 t = t+1; 0101 for i=1:pp 0102 v(i) = Y(i,t)-a(mf(i))-trend(i,t); 0103 Fstar = Pstar(mf(i),mf(i)); 0104 Finf = Pinf(mf(i),mf(i)); 0105 Kstar = Pstar(:,mf(i)); 0106 if Finf > crit & newRank, %added newRank criterion 0107 icc=icc+1; 0108 Kinf = Pinf(:,mf(i)); 0109 a = a + Kinf*v(i)/Finf; 0110 Pstar = Pstar + Kinf*transpose(Kinf)*Fstar/(Finf*Finf) - ... 0111 (Kstar*transpose(Kinf)+Kinf*transpose(Kstar))/Finf; 0112 Pinf = Pinf - Kinf*transpose(Kinf)/Finf; 0113 lik(t) = lik(t) + log(Finf); 0114 % start new termination criterion for DKF 0115 if ~isempty(options_.diffuse_d), 0116 newRank = (icc<options_.diffuse_d); 0117 %if newRank & any(diag(Pinf(mf,mf))>crit)==0; % M. Ratto this line is BUGGY 0118 if newRank & (any(diag(Pinf(mf,mf))>crit)==0 & rank(Pinf,crit1)==0); 0119 options_.diffuse_d = icc; 0120 newRank=0; 0121 disp('WARNING: Change in OPTIONS_.DIFFUSE_D in univariate DKF') 0122 disp(['new OPTIONS_.DIFFUSE_D = ',int2str(icc)]) 0123 disp('You may have to reset the optimisation') 0124 end 0125 else 0126 %newRank = any(diag(Pinf(mf,mf))>crit); % M. Ratto this line is BUGGY 0127 newRank = (any(diag(Pinf(mf,mf))>crit) | rank(Pinf,crit1)); 0128 if newRank==0, 0129 P0= T*Pinf*transpose(T); 0130 %newRank = any(diag(P0(mf,mf))>crit); % M. Ratto this line is BUGGY 0131 newRank = (any(diag(P0(mf,mf))>crit) | rank(P0,crit1)); % M. Ratto 11/10/2005 0132 if newRank==0, 0133 options_.diffuse_d = icc; 0134 end 0135 end 0136 end, 0137 % end new termination and checks for DKF 0138 elseif Fstar > crit 0139 %% Note that : (1) rank(Pinf)=0 implies that Finf = 0, (2) outside this loop (when for some i and t the condition 0140 %% rank(Pinf)=0 is satisfied we have P = Pstar and F = Fstar and (3) Finf = 0 does not imply that 0141 %% rank(Pinf)=0. [stphane,11-03-2004]. 0142 %if rank(Pinf,crit) == 0 0143 % the likelihood terms should alwasy be cumulated, not only 0144 % when Pinf=0, otherwise the lik would depend on the ordering 0145 % of observed variables 0146 % presample options can be used to ignore initial time points 0147 lik(t) = lik(t) + log(Fstar) + v(i)*v(i)/Fstar; 0148 %end 0149 a = a + Kstar*v(i)/Fstar; 0150 Pstar = Pstar - Kstar*transpose(Kstar)/Fstar; 0151 else 0152 %disp(['zero F term in DKF for observed ',int2str(i),' ',num2str(Fstar)]) 0153 end 0154 end 0155 % if all(abs(Pinf(:))<crit), 0156 % oldRank = 0; 0157 % else 0158 % oldRank = rank(Pinf,crit); 0159 % end 0160 if newRank, 0161 oldRank = rank(Pinf,crit1); 0162 else 0163 oldRank = 0; 0164 end 0165 a = T*a; 0166 Pstar = T*Pstar*transpose(T)+QQ; 0167 Pinf = T*Pinf*transpose(T); 0168 % if all(abs(Pinf(:))<crit), 0169 % newRank = 0; 0170 % else 0171 % newRank = rank(Pinf,crit); 0172 % end 0173 if newRank, 0174 newRank = rank(Pinf,crit1); % new crit1 is used 0175 end 0176 if oldRank ~= newRank 0177 disp('DiffuseLiklihood3 :: T does influence the rank of Pinf!') 0178 end 0179 end 0180 if t == smpl 0181 error(['There isn''t enough information to estimate the initial' ... 0182 ' conditions of the nonstationary variables']); 0183 end 0184 while notsteady & t < smpl 0185 t = t+1; 0186 oldP = Pstar; 0187 for i=1:pp 0188 v(i) = Y(i,t) - a(mf(i)) - trend(i,t); 0189 Fi = Pstar(mf(i),mf(i)); 0190 if Fi > crit 0191 Ki = Pstar(:,mf(i)); 0192 a = a + Ki*v(i)/Fi; 0193 Pstar = Pstar - Ki*transpose(Ki)/Fi; 0194 lik(t) = lik(t) + log(Fi) + v(i)*v(i)/Fi; 0195 else 0196 %disp(['zero F term for observed ',int2str(i),' ',num2str(Fi)]) 0197 end 0198 end 0199 a = T*a; 0200 Pstar = T*Pstar*transpose(T) + QQ; 0201 notsteady = ~(max(max(abs(Pstar-oldP)))<crit); 0202 end 0203 while t < smpl 0204 t = t+1; 0205 Pstar = oldP; 0206 for i=1:pp 0207 v(i) = Y(i,t) - a(mf(i)) - trend(i,t); 0208 Fi = Pstar(mf(i),mf(i)); 0209 if Fi > crit 0210 Ki = Pstar(:,mf(i)); 0211 a = a + Ki*v(i)/Fi; 0212 Pstar = Pstar - Ki*transpose(Ki)/Fi; 0213 lik(t) = lik(t) + log(Fi) + v(i)*v(i)/Fi; 0214 else 0215 %disp(['zero F term for observed ',int2str(i),' ',num2str(Fi)]) 0216 end 0217 end 0218 a = T*a; 0219 end 0220 0221 LIK = .5*(sum(lik(start:end))-(start-1)*lik(smpl+1)/smpl); 0222