185 lines
5.7 KiB
Matlab
185 lines
5.7 KiB
Matlab
function [alphahat,epsilonhat,etahat,a1] = DiffuseKalmanSmootherH3(T,R,Q,H,Pinf1,Pstar1,Y,trend,pp,mm,smpl,mf)
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% stephane.adjemian@cepremap.cnrs.fr [09-16-2004]
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%
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% See "Fast Filtering and Smoothing for Multivariate State Space
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% Models", S.J. Koopman and J. Durbin (2000, in Journal of Time Series
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% Analysis, vol. 21(3), pp. 281-296).
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global options_
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spinf = size(Pinf1);
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spstar = size(Pstar1);
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v = zeros(pp,smpl);
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a = zeros(mm,smpl+1);
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a1 = a;
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Fstar = zeros(pp,smpl);
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Finf = zeros(pp,smpl);
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Ki = zeros(mm,pp,smpl);
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Li = zeros(mm,mm,pp,smpl);
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Linf = zeros(mm,mm,pp,smpl);
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L0 = zeros(mm,mm,pp,smpl);
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Kstar = zeros(mm,pp,smpl);
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P = zeros(mm,mm,smpl+1);
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P1 = P;
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Pstar = zeros(spstar(1),spstar(2),smpl+1); Pstar(:,:,1) = Pstar1;
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Pinf = zeros(spinf(1),spinf(2),smpl+1); Pinf(:,:,1) = Pinf1;
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Pstar1 = Pstar;
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Pinf1 = Pinf;
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crit = options_.kalman_tol;
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steady = smpl;
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rr = size(Q,1);
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QQ = R*Q*transpose(R);
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QRt = Q*transpose(R);
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alphahat = zeros(mm,smpl);
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etahat = zeros(rr,smpl);
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r = zeros(mm,smpl);
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Z = zeros(pp,mm);
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for i=1:pp;
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Z(i,mf(i)) = 1;
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end
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t = 0;
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newRank = rank(Pinf(:,:,1),crit);
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while newRank & t < smpl
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t = t+1;
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a1(:,t) = a(:,t);
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Pstar1(:,:,t) = Pstar(:,:,t);
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Pinf1(:,:,t) = Pinf(:,:,t);
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for i=1:pp
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v(i,t) = Y(i,t)-a(mf(i),t)-trend(i,t);
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Fstar(i,t) = Pstar(mf(i),mf(i),t)+H(i,i);
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Finf(i,t) = Pinf(mf(i),mf(i),t);
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Kstar(:,i,t) = Pstar(:,mf(i),t);
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if Finf(i,t) > crit
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Kinf(:,i,t) = Pinf(:,mf(i),t);
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Linf(:,:,i,t) = eye(mm) - Kinf(:,i,t)*Z(i,:)/Finf(i,t);
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L0(:,:,i,t) = (Kinf(:,i,t)*Fstar(i,t)/Finf(i,t) - Kstar(:,i,t))*Z(i,:)/Finf(i,t);
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a(:,t) = a(:,t) + Kinf(:,i,t)*v(i,t)/Finf(i,t);
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Pstar(:,:,t) = Pstar(:,:,t) + ...
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Kinf(:,i,t)*transpose(Kinf(:,i,t))*Fstar(i,t)/(Finf(i,t)*Finf(i,t)) - ...
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(Kstar(:,i,t)*transpose(Kinf(:,i,t)) +...
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Kinf(:,i,t)*transpose(Kstar(:,i,t)))/Finf(i,t);
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Pinf(:,:,t) = Pinf(:,:,t) - Kinf(:,i,t)*transpose(Kinf(:,i,t))/Finf(i,t);
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else %% 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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a(:,t) = a(:,t) + Kstar(:,i,t)*v(i,t)/Fstar(i,t);
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Pstar(:,:,t) = Pstar(:,:,t) - Kstar(:,i,t)*transpose(Kstar(:,i,t))/Fstar(i,t);
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end
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end
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a(:,t+1) = T*a(:,t);
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Pstar(:,:,t+1) = T*Pstar(:,:,t)*transpose(T)+ QQ;
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Pinf(:,:,t+1) = T*Pinf(:,:,t)*transpose(T);
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P0=Pinf(:,:,t+1);
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newRank = ~all(abs(P0(:))<crit);
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end
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d = t;
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P(:,:,d+1) = Pstar(:,:,d+1);
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Linf = Linf(:,:,:,1:d);
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L0 = L0(:,:,:,1:d);
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Fstar = Fstar(:,1:d);
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Finf = Finf(:,1:d);
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Kstar = Kstar(:,:,1:d);
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Pstar = Pstar(:,:,1:d);
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Pinf = Pinf(:,:,1:d);
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Pstar1 = Pstar1(:,:,1:d);
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Pinf1 = Pinf1(:,:,1:d);
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notsteady = 1;
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while notsteady & t<smpl
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t = t+1;
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a1(:,t) = a(:,t);
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P(:,:,t)=tril(P(:,:,t))+transpose(tril(P(:,:,t),-1));
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P1(:,:,t) = P(:,:,t);
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for i=1:pp
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v(i,t) = Y(i,t) - a(mf(i),t) - trend(i,t);
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Fi(i,t) = P(mf(i),mf(i),t)+H(i,i);
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Ki(:,i,t) = P(:,mf(i),t);
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if Fi(i,t) > crit
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Li(:,:,i,t) = eye(mm)-Ki(:,i,t)*Z(i,:)/Fi(i,t);
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a(:,t) = a(:,t) + Ki(:,i,t)*v(i,t)/Fi(i,t);
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P(:,:,t) = P(:,:,t) - Ki(:,i,t)*transpose(Ki(:,i,t))/Fi(i,t);
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P(:,:,t)=tril(P(:,:,t))+transpose(tril(P(:,:,t),-1));
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end
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end
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a(:,t+1) = T*a(:,t);
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P(:,:,t+1) = T*P(:,:,t)*transpose(T) + QQ;
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notsteady = ~(max(max(abs(P(:,:,t+1)-P(:,:,t))))<crit);
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end
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P_s=tril(P(:,:,t))+transpose(tril(P(:,:,t),-1));
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Fi_s = Fi(:,t);
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Ki_s = Ki(:,:,t);
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L_s =Li(:,:,:,t);
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if t<smpl
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t_steady = t+1;
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P = cat(3,P(:,:,1:t),repmat(P(:,:,t),[1 1 smpl-t_steady+1]));
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Fi = cat(2,Fi(:,1:t),repmat(Fi_s,[1 1 smpl-t_steady+1]));
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Li = cat(4,Li(:,:,:,1:t),repmat(L_s,[1 1 smpl-t_steady+1]));
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Ki = cat(3,Ki(:,:,1:t),repmat(Ki_s,[1 1 smpl-t_steady+1]));
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end
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while t<smpl
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t=t+1;
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a1(:,t) = a(:,t);
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for i=1:pp
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v(i,t) = Y(i,t) - a(mf(i),t) - trend(i,t);
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if Fi_s(i) > crit
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a(:,t) = a(:,t) + Ki_s(:,i)*v(i,t)/Fi_s(i);
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end
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end
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a(:,t+1) = T*a(:,t);
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end
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a1(:,t+1) = a(:,t+1);
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ri=r;
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t = smpl+1;
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while t>d+1 & t>2,
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t = t-1;
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for i=pp:-1:1
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if Fi(i,t) > crit
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ri(:,t)=transpose(Z(i,:))/Fi(i,t)*v(i,t)+transpose(Li(:,:,i,t))*ri(:,t);
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end
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end
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r(:,t-1) = ri(:,t);
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alphahat(:,t) = a1(:,t) + P1(:,:,t)*r(:,t-1);
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etahat(:,t) = QRt*r(:,t);
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ri(:,t-1) = transpose(T)*ri(:,t);
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end
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if d
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r0 = zeros(mm,d); r0(:,d) = ri(:,d);
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r1 = zeros(mm,d);
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for t = d:-1:2
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for i=pp:-1:1
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if Finf(i,t) > crit,
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r1(:,t) = transpose(Z)*v(:,t)/Finf(i,t) + ...
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transpose(L0(:,:,i,t))*r0(:,t) + transpose(Linf(:,:,i,t))*r1(:,t);
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r0(:,t) = transpose(Linf(:,:,i,t))*r0(:,t);
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end
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end
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alphahat(:,t) = a1(:,t) + Pstar1(:,:,t)*r0(:,t) + Pinf1(:,:,t)*r1(:,t);
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r(:,t-1) = r0(:,t);
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etahat(:,t) = QRt*r(:,t);
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r0(:,t-1) = transpose(T)*r0(:,t);
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r1(:,t-1) = transpose(T)*r1(:,t);
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end
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r0_0 = r0(:,1);
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r1_0 = r1(:,1);
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for i=pp:-1:1
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if Finf(i,1) > crit,
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r1_0 = transpose(Z)*v(:,1)/Finf(i,1) + ...
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transpose(L0(:,:,i,1))*r0_0 + transpose(Linf(:,:,i,1))*r1_0;
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r0_0 = transpose(Linf(:,:,i,1))*r0_0;
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end
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end
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alphahat(:,1) = a(:,1) + Pstar(:,:,1)*r0_0 + Pinf(:,:,1)*r1_0;
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etahat(:,1) = QRt*r(:,1);
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else
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r0 = ri(:,1);
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for i=pp:-1:1
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if Fi(i,1) > crit
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r0=transpose(Z(i,:))/Fi(i,1)*v(i,1)+transpose(Li(:,:,i,1))*r0;
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end
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end
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alphahat(:,1) = a(:,1) + P(:,:,1)*r0;
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etahat(:,1) = QRt*r(:,1);
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end
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epsilonhat = Y-alphahat(mf,:)-trend; |