0001 function [alphahat,etahat,a, aK] = DiffuseKalmanSmoother1(T,R,Q,Pinf1,Pstar1,Y,trend,pp,mm,smpl,mf)
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0014 global options_
0015
0016 nk = options_.nk;
0017 spinf = size(Pinf1);
0018 spstar = size(Pstar1);
0019 v = zeros(pp,smpl);
0020 a = zeros(mm,smpl+1);
0021 aK = zeros(nk,mm,smpl+1);
0022 iF = zeros(pp,pp,smpl);
0023 Fstar = zeros(pp,pp,smpl);
0024 iFinf = zeros(pp,pp,smpl);
0025 K = zeros(mm,pp,smpl);
0026 L = zeros(mm,mm,smpl);
0027 Linf = zeros(mm,mm,smpl);
0028 Kstar = zeros(mm,pp,smpl);
0029 P = zeros(mm,mm,smpl+1);
0030 Pstar = zeros(spstar(1),spstar(2),smpl+1); Pstar(:,:,1) = Pstar1;
0031 Pinf = zeros(spinf(1),spinf(2),smpl+1); Pinf(:,:,1) = Pinf1;
0032 crit = options_.kalman_tol;
0033 crit1 = 1.e-8;
0034 steady = smpl;
0035 rr = size(Q,1);
0036 QQ = R*Q*transpose(R);
0037 QRt = Q*transpose(R);
0038 alphahat = zeros(mm,smpl);
0039 etahat = zeros(rr,smpl);
0040 r = zeros(mm,smpl);
0041
0042 Z = zeros(pp,mm);
0043 for i=1:pp;
0044 Z(i,mf(i)) = 1;
0045 end
0046
0047 t = 0;
0048 while rank(Pinf(:,:,t+1),crit1) & t<smpl
0049 t = t+1;
0050 v(:,t) = Y(:,t) - a(mf,t) - trend(:,t);
0051 if rcond(Pinf(mf,mf,t)) < crit
0052 return
0053 end
0054 iFinf(:,:,t) = inv(Pinf(mf,mf,t));
0055 Kinf(:,:,t) = T*Pinf(:,mf,t)*iFinf(:,:,t);
0056 a(:,t+1) = T*a(:,t) + Kinf(:,:,t)*v(:,t);
0057 aK(1,:,t+1) = a(:,t+1);
0058 for jnk=2:nk,
0059 aK(jnk,:,t+jnk) = T^(jnk-1)*a(:,t+1);
0060 end
0061 Linf(:,:,t) = T - Kinf(:,:,t)*Z;
0062 Fstar(:,:,t) = Pstar(mf,mf,t);
0063 Kstar(:,:,t) = (T*Pstar(:,mf,t)-Kinf(:,:,t)*Fstar(:,:,t))*iFinf(:,:,t);
0064 Pstar(:,:,t+1) = T*Pstar(:,:,t)*transpose(T)-T*Pstar(:,mf,t)*transpose(Kinf(:,:,t))-Kinf(:,:,t)*Pinf(mf,mf,t)*transpose(Kstar(:,:,t)) + QQ;
0065 Pinf(:,:,t+1) = T*Pinf(:,:,t)*transpose(T)-T*Pinf(:,mf,t)*transpose(Kinf(:,:,t));
0066 end
0067 d = t;
0068 P(:,:,d+1) = Pstar(:,:,d+1);
0069 iFinf = iFinf(:,:,1:d);
0070 Linf = Linf(:,:,1:d);
0071 Fstar = Fstar(:,:,1:d);
0072 Kstar = Kstar(:,:,1:d);
0073 Pstar = Pstar(:,:,1:d);
0074 Pinf = Pinf(:,:,1:d);
0075 notsteady = 1;
0076 while notsteady & t<smpl
0077 t = t+1;
0078 v(:,t) = Y(:,t) - a(mf,t) - trend(:,t);
0079 P(:,:,t)=tril(P(:,:,t))+transpose(tril(P(:,:,t),-1));
0080 if rcond(P(mf,mf,t)) < crit
0081 return
0082 end
0083 iF(:,:,t) = inv(P(mf,mf,t));
0084 K(:,:,t) = T*P(:,mf,t)*iF(:,:,t);
0085 L(:,:,t) = T-K(:,:,t)*Z;
0086 a(:,t+1) = T*a(:,t) + K(:,:,t)*v(:,t);
0087 aK(1,:,t+1) = a(:,t+1);
0088 for jnk=2:nk,
0089 aK(jnk,:,t+jnk) = T^(jnk-1)*a(:,t+1);
0090 end
0091 P(:,:,t+1) = T*P(:,:,t)*transpose(T)-T*P(:,mf,t)*transpose(K(:,:,t)) + QQ;
0092 notsteady = ~(max(max(abs(P(:,:,t+1)-P(:,:,t))))<crit);
0093 end
0094 K_s = K(:,:,t);
0095 iF_s = iF(:,:,t);
0096 P_s = P(:,:,t+1);
0097 if t<smpl
0098 t_steady = t+1;
0099 P = cat(3,P(:,:,1:t),repmat(P(:,:,t),[1 1 smpl-t_steady+1]));
0100 iF = cat(3,iF(:,:,1:t),repmat(inv(P_s(mf,mf)),[1 1 smpl-t_steady+1]));
0101 L = cat(3,L(:,:,1:t),repmat(T-K_s*Z,[1 1 smpl-t_steady+1]));
0102 K = cat(3,K(:,:,1:t),repmat(T*P_s(:,mf)*iF_s,[1 1 smpl-t_steady+1]));
0103 end
0104 while t<smpl
0105 t=t+1;
0106 v(:,t) = Y(:,t) - a(mf,t) - trend(:,t);
0107 a(:,t+1) = T*a(:,t) + K_s*v(:,t);
0108 aK(1,:,t+1) = a(:,t+1);
0109 for jnk=2:nk,
0110 aK(jnk,:,t+jnk) = T^(jnk-1)*a(:,t+1);
0111 end
0112 end
0113 t = smpl+1;
0114 while t>d+1 & t>2
0115 t = t-1;
0116 r(:,t-1) = transpose(Z)*iF(:,:,t)*v(:,t) + transpose(L(:,:,t))*r(:,t);
0117 alphahat(:,t) = a(:,t) + P(:,:,t)*r(:,t-1);
0118 etahat(:,t) = QRt*r(:,t);
0119 end
0120 if d
0121 r0 = zeros(mm,d); r0(:,d) = r(:,d);
0122 r1 = zeros(mm,d);
0123 for t = d:-1:2
0124 r0(:,t-1) = transpose(Linf(:,:,t))*r0(:,t);
0125 r1(:,t-1) = transpose(Z)*(iFinf(:,:,t)*v(:,t)-transpose(Kstar(:,:,t))*r0(:,t)) + transpose(Linf(:,:,t))*r1(:,t);
0126 alphahat(:,t) = a(:,t) + Pstar(:,:,t)*r0(:,t-1) + Pinf(:,:,t)*r1(:,t-1);
0127 etahat(:,t) = QRt*r0(:,t);
0128 end
0129 r0_0 = transpose(Linf(:,:,1))*r0(:,1);
0130 r1_0 = transpose(Z)*(iFinf(:,:,1)*v(:,1)-transpose(Kstar(:,:,1))*r0(:,1)) + transpose(Linf(:,:,1))*r1(:,1);
0131 alphahat(:,1) = a(:,1) + Pstar(:,:,1)*r0_0 + Pinf(:,:,1)*r1_0;
0132 etahat(:,1) = QRt*r0(:,1);
0133 else
0134 r0 = transpose(Z)*iF(:,:,1)*v(:,1) + transpose(L(:,:,1))*r(:,1);
0135 alphahat(:,1) = a(:,1) + P(:,:,1)*r0;
0136 etahat(:,1) = QRt*r(:,1);
0137 end