Home > . > DiffuseKalmanSmootherH3.m

DiffuseKalmanSmootherH3

PURPOSE ^

Modified by M. Ratto

SYNOPSIS ^

function [alphahat,epsilonhat,etahat,a1, aK] = DiffuseKalmanSmootherH3(T,R,Q,H,Pinf1,Pstar1,Y,trend,pp,mm,smpl,mf)

DESCRIPTION ^

 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 global variable id_ 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 id_
 Li also stored during DKF iterations !!
 some bugs corrected in the DKF part of the smoother (Z matrix and
 alphahat)

 stephane.adjemian@cepremap.cnrs.fr [09-16-2004]
 
   See "Fast Filtering and Smoothing for Multivariate State Space
   Models", S.J. Koopman and J. Durbin (2000, in Journal of Time Series 
   Analysis, vol. 21(3), pp. 281-296).

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 function [alphahat,epsilonhat,etahat,a1, aK] = DiffuseKalmanSmootherH3(T,R,Q,H,Pinf1,Pstar1,Y,trend,pp,mm,smpl,mf)
0002 % Modified by M. Ratto
0003 % New output argument aK: 1-step to nk-stpe ahed predictions)
0004 % New input argument nk: max order of predictions in aK
0005 % New global variable id_ where the DKF stops (common with
0006 % diffuselikelihood3)
0007 % New icc variable to count number of iterations for Finf steps
0008 % Pstar % Pinf simmetric
0009 % New termination of DKF iterations based on id_
0010 % Li also stored during DKF iterations !!
0011 % some bugs corrected in the DKF part of the smoother (Z matrix and
0012 % alphahat)
0013 %
0014 % stephane.adjemian@cepremap.cnrs.fr [09-16-2004]
0015 %
0016 %   See "Fast Filtering and Smoothing for Multivariate State Space
0017 %   Models", S.J. Koopman and J. Durbin (2000, in Journal of Time Series
0018 %   Analysis, vol. 21(3), pp. 281-296).
0019 
0020 global options_
0021 
0022 nk = options_.nk;
0023 spinf       = size(Pinf1);
0024 spstar      = size(Pstar1);
0025 v           = zeros(pp,smpl);
0026 a           = zeros(mm,smpl+1);
0027 a1            = a;
0028 aK          = zeros(nk,mm,smpl+nk);
0029 Fstar       = zeros(pp,smpl);
0030 Finf        = zeros(pp,smpl);
0031 Ki           = zeros(mm,pp,smpl);
0032 Li          = zeros(mm,mm,pp,smpl);
0033 Linf        = zeros(mm,mm,pp,smpl);
0034 L0          = zeros(mm,mm,pp,smpl);
0035 Kstar       = zeros(mm,pp,smpl);
0036 P           = zeros(mm,mm,smpl+1);
0037 P1            = P;
0038 Pstar       = zeros(spstar(1),spstar(2),smpl+1); Pstar(:,:,1) = Pstar1;
0039 Pinf        = zeros(spinf(1),spinf(2),smpl+1); Pinf(:,:,1) = Pinf1;
0040 Pstar1         = Pstar;
0041 Pinf1          = Pinf;
0042 crit            = options_.kalman_tol;
0043 crit1       = 1.e-6;
0044 steady      = smpl;
0045 rr          = size(Q,1);
0046 QQ          = R*Q*transpose(R);
0047 QRt            = Q*transpose(R);
0048 alphahat       = zeros(mm,smpl);
0049 etahat           = zeros(rr,smpl);
0050 epsilonhat      = zeros(size(Y));
0051 r                = zeros(mm,smpl);
0052 
0053 Z = zeros(pp,mm);
0054 for i=1:pp;
0055     Z(i,mf(i)) = 1;
0056 end
0057 
0058 t = 0;
0059 icc=0;
0060 newRank      = rank(Pinf(:,:,1),crit1);
0061 while newRank & t < smpl
0062   t = t+1;
0063   a1(:,t) = a(:,t);
0064   Pstar(:,:,t)=tril(Pstar(:,:,t))+transpose(tril(Pstar(:,:,t),-1));
0065   Pinf(:,:,t)=tril(Pinf(:,:,t))+transpose(tril(Pinf(:,:,t),-1));
0066   Pstar1(:,:,t) = Pstar(:,:,t);
0067   Pinf1(:,:,t) = Pinf(:,:,t);
0068   for i=1:pp
0069     v(i,t)     = Y(i,t)-a(mf(i),t)-trend(i,t);
0070     Fstar(i,t)     = Pstar(mf(i),mf(i),t) + H(i,i);
0071     Finf(i,t)    = Pinf(mf(i),mf(i),t);
0072     Kstar(:,i,t)     = Pstar(:,mf(i),t);
0073     if Finf(i,t) > crit & newRank,  % original MJ: if Finf(i,t) > crit
0074       icc=icc+1;
0075       Kinf(:,i,t)    = Pinf(:,mf(i),t);
0076       Linf(:,:,i,t)      = eye(mm) - Kinf(:,i,t)*Z(i,:)/Finf(i,t);
0077       L0(:,:,i,t)      = (Kinf(:,i,t)*Fstar(i,t)/Finf(i,t) - Kstar(:,i,t))*Z(i,:)/Finf(i,t);
0078       a(:,t)        = a(:,t) + Kinf(:,i,t)*v(i,t)/Finf(i,t);
0079       Pstar(:,:,t)    = Pstar(:,:,t) + ...
0080       Kinf(:,i,t)*transpose(Kinf(:,i,t))*Fstar(i,t)/(Finf(i,t)*Finf(i,t)) - ...
0081       (Kstar(:,i,t)*transpose(Kinf(:,i,t)) +...
0082        Kinf(:,i,t)*transpose(Kstar(:,i,t)))/Finf(i,t);
0083       Pinf(:,:,t)    = Pinf(:,:,t) - Kinf(:,i,t)*transpose(Kinf(:,i,t))/Finf(i,t);
0084       Pstar(:,:,t)=tril(Pstar(:,:,t))+transpose(tril(Pstar(:,:,t),-1));
0085       Pinf(:,:,t)=tril(Pinf(:,:,t))+transpose(tril(Pinf(:,:,t),-1));
0086       % new terminiation criteria by M. Ratto
0087       P0=Pinf(:,:,t);
0088       %             newRank = any(diag(P0(mf,mf))>crit);
0089       %             if newRank==0, options_.diffuse_d = i; end,
0090       if ~isempty(options_.diffuse_d),  
0091     newRank = (icc<options_.diffuse_d);  
0092     %if newRank & any(diag(P0(mf,mf))>crit)==0;
0093     if newRank & (any(diag(P0(mf,mf))>crit)==0 & rank(P0,crit1)==0); 
0094       disp('WARNING!! Change in OPTIONS_.DIFFUSE_D in univariate DKF')
0095       options_.diffuse_d = icc;
0096       newRank=0;
0097     end
0098       else
0099     %newRank = any(diag(P0(mf,mf))>crit);
0100     newRank = (any(diag(P0(mf,mf))>crit) | rank(P0,crit1));                 
0101     if newRank==0, 
0102       options_.diffuse_d = icc;
0103     end                    
0104       end,
0105 %       if newRank==0,
0106 %     options_.diffuse_d=i;   % this line is buggy!
0107 %       end
0108       % end new terminiation criteria by M. Ratto
0109     else 
0110       %% Note that : (1) rank(Pinf)=0 implies that Finf = 0, (2) outside this loop (when for some i and t the condition
0111       %% rank(Pinf)=0 is satisfied we have P = Pstar and F = Fstar and (3) Finf = 0 does not imply that
0112       %% rank(Pinf)=0. [stphane,11-03-2004].
0113       Li(:,:,i,t)    = eye(mm)-Kstar(:,i,t)*Z(i,:)/Fstar(i,t);  % we need to store Li for DKF smoother
0114       a(:,t)         = a(:,t) + Kstar(:,i,t)*v(i,t)/Fstar(i,t);
0115       Pstar(:,:,t)    = Pstar(:,:,t) - Kstar(:,i,t)*transpose(Kstar(:,i,t))/Fstar(i,t);
0116       Pstar(:,:,t)=tril(Pstar(:,:,t))+transpose(tril(Pstar(:,:,t),-1));
0117     end
0118   end
0119   a(:,t+1)          = T*a(:,t);
0120   for jnk=1:nk,
0121     aK(jnk,:,t+jnk)          = T^jnk*a(:,t);
0122   end
0123   Pstar(:,:,t+1)    = T*Pstar(:,:,t)*transpose(T)+ QQ;
0124   Pinf(:,:,t+1)    = T*Pinf(:,:,t)*transpose(T);
0125   P0=Pinf(:,:,t+1);
0126   if newRank,
0127     %newRank = any(diag(P0(mf,mf))>crit);
0128     newRank      = rank(P0,crit1);
0129   end
0130 end
0131 
0132 
0133 d = t;
0134 P(:,:,d+1) = Pstar(:,:,d+1);
0135 Linf  = Linf(:,:,:,1:d);
0136 L0  = L0(:,:,:,1:d);
0137 Fstar = Fstar(:,1:d);
0138 Finf = Finf(:,1:d);
0139 Kstar = Kstar(:,:,1:d);
0140 Pstar = Pstar(:,:,1:d);
0141 Pinf  = Pinf(:,:,1:d);
0142 Pstar1 = Pstar1(:,:,1:d);
0143 Pinf1  = Pinf1(:,:,1:d);
0144 notsteady = 1;
0145 while notsteady & t<smpl
0146   t = t+1;
0147   a1(:,t) = a(:,t);
0148   P(:,:,t)=tril(P(:,:,t))+transpose(tril(P(:,:,t),-1));
0149   P1(:,:,t) = P(:,:,t);
0150   for i=1:pp
0151     v(i,t)  = Y(i,t) - a(mf(i),t) - trend(i,t);
0152     Fi(i,t) = P(mf(i),mf(i),t);
0153     Ki(:,i,t) = P(:,mf(i),t) + H(i,i);
0154     if Fi(i,t) > crit
0155       Li(:,:,i,t)    = eye(mm)-Ki(:,i,t)*Z(i,:)/Fi(i,t);
0156       a(:,t) = a(:,t) + Ki(:,i,t)*v(i,t)/Fi(i,t);
0157       P(:,:,t) = P(:,:,t) - Ki(:,i,t)*transpose(Ki(:,i,t))/Fi(i,t);
0158       P(:,:,t)=tril(P(:,:,t))+transpose(tril(P(:,:,t),-1));
0159     end
0160   end
0161   a(:,t+1) = T*a(:,t);
0162   for jnk=1:nk,
0163     aK(jnk,:,t+jnk)          = T^jnk*a(:,t);
0164   end
0165   P(:,:,t+1) = T*P(:,:,t)*transpose(T) + QQ;
0166   notsteady   = ~(max(max(abs(P(:,:,t+1)-P(:,:,t))))<crit);
0167 end
0168 P_s=tril(P(:,:,t))+transpose(tril(P(:,:,t),-1));
0169 Fi_s = Fi(:,t);
0170 Ki_s = Ki(:,:,t);
0171 L_s  =Li(:,:,:,t);
0172 if t<smpl
0173   t_steady = t+1;
0174   P  = cat(3,P(:,:,1:t),repmat(P(:,:,t),[1 1 smpl-t_steady+1]));
0175   Fi = cat(2,Fi(:,1:t),repmat(Fi_s,[1 1 smpl-t_steady+1]));
0176   Li  = cat(4,Li(:,:,:,1:t),repmat(L_s,[1 1 smpl-t_steady+1]));
0177   Ki  = cat(3,Ki(:,:,1:t),repmat(Ki_s,[1 1 smpl-t_steady+1]));
0178 end
0179 while t<smpl
0180   t=t+1;
0181   a1(:,t) = a(:,t);
0182   for i=1:pp
0183     v(i,t)      = Y(i,t) - a(mf(i),t) - trend(i,t);
0184     if Fi_s(i) > crit
0185       a(:,t) = a(:,t) + Ki_s(:,i)*v(i,t)/Fi_s(i);
0186     end
0187   end
0188   a(:,t+1) = T*a(:,t);
0189   for jnk=1:nk,
0190     aK(jnk,:,t+jnk)    = T^jnk*a(:,t);
0191   end
0192 end
0193 a1(:,t+1) = a(:,t+1);
0194 ri=r;
0195 t = smpl+1;
0196 while t>d+1 & t>2,
0197   t = t-1;
0198   for i=pp:-1:1
0199     if Fi(i,t) > crit
0200       ri(:,t)=transpose(Z(i,:))/Fi(i,t)*v(i,t)+transpose(Li(:,:,i,t))*ri(:,t);
0201     end
0202   end
0203   r(:,t-1) = ri(:,t);
0204   alphahat(:,t)    = a1(:,t) + P1(:,:,t)*r(:,t-1);
0205   etahat(:,t)        = QRt*r(:,t);
0206   ri(:,t-1) = transpose(T)*ri(:,t);
0207 end
0208 if d
0209   r0 = zeros(mm,d); r0(:,d) = ri(:,d);
0210   r1 = zeros(mm,d);
0211   for t = d:-1:2
0212     for i=pp:-1:1
0213       if Finf(i,t) > crit & ~(t==d & i>options_.diffuse_d),  % use of options_.diffuse_d to be sure of DKF termination
0214                          %r1(:,t) = transpose(Z)*v(:,t)/Finf(i,t) + ... BUG HERE in transpose(Z)
0215                          r1(:,t) = transpose(Z(i,:))*v(i,t)/Finf(i,t) + ...
0216                                transpose(L0(:,:,i,t))*r0(:,t) + transpose(Linf(:,:,i,t))*r1(:,t);
0217                          r0(:,t) = transpose(Linf(:,:,i,t))*r0(:,t);
0218       elseif Fstar(i,t) > crit % step needed whe Finf == 0
0219     r0(:,t)=transpose(Z(i,:))/Fstar(i,t)*v(i,t)+Li(:,:,i,t)'*r0(:,t);
0220       end
0221     end
0222     alphahat(:,t)    = a1(:,t) + Pstar1(:,:,t)*r0(:,t) + Pinf1(:,:,t)*r1(:,t);
0223     r(:,t-1)        = r0(:,t);
0224     etahat(:,t)        = QRt*r(:,t);
0225     r0(:,t-1) = transpose(T)*r0(:,t);
0226     r1(:,t-1) = transpose(T)*r1(:,t);
0227   end
0228   r0_0 = r0(:,1);
0229   r1_0 = r1(:,1);
0230   for i=pp:-1:1
0231     if Finf(i,1) > crit,
0232       %r1_0 = transpose(Z)*v(:,1)/Finf(i,1) + ... %bug with Z here
0233       r1_0 = transpose(Z(i,:))*v(i,1)/Finf(i,1) + ...
0234          transpose(L0(:,:,i,1))*r0_0 + transpose(Linf(:,:,i,1))*r1_0;
0235       r0_0 = transpose(Linf(:,:,i,1))*r0_0;
0236     elseif Fstar(i,1) > crit, % step needed when Finf=0
0237       r0_0=transpose(Z(i,:))/Fstar(i,1)*v(i,1)+Li(:,:,i,1)'*r0_0;
0238     end
0239   end
0240   %alphahat(:,1)      = a(:,1) + Pstar(:,:,1)*r0_0 + Pinf(:,:,1)*r1_0; %this line is buggy
0241   alphahat(:,1)      = a1(:,1) + Pstar1(:,:,1)*r0_0 + Pinf1(:,:,1)*r1_0;
0242   etahat(:,1)        = QRt*r(:,1);
0243 else
0244   r0 = ri(:,1);
0245   for i=pp:-1:1
0246     if Fi(i,1) > crit
0247       r0=transpose(Z(i,:))/Fi(i,1)*v(i,1)+transpose(Li(:,:,i,1))*r0;
0248     end
0249   end 
0250   %alphahat(:,1)    = a(:,1) + P(:,:,1)*r0;  % this line is buggy
0251   alphahat(:,1)    = a1(:,1) + P1(:,:,1)*r0;
0252   etahat(:,1)    = QRt*r(:,1);
0253 end
0254 epsilonhat = Y-alphahat(mf,:)-trend;
0255
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