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<title>Description of DiffuseKalmanSmootherH3corr</title>
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<div><a href="../index.html">Home</a> > <a href="index.html">.</a> > DiffuseKalmanSmootherH3corr.m</div>
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<!--<table width="100%"><tr><td align="left"><a href="../index.html"><img alt="<" border="0" src="../left.png"> Master index</a></td>
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<td align="right"><a href="index.html">Index for . <img alt=">" border="0" src="../right.png"></a></td></tr></table>-->
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<h1>DiffuseKalmanSmootherH3corr
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</h1>
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<h2><a name="_name"></a>PURPOSE <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
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<div class="box"><strong>stephane.adjemian@cepremap.cnrs.fr [09-16-2004]</strong></div>
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<h2><a name="_synopsis"></a>SYNOPSIS <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
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<div class="box"><strong>function [alphahat,epsilonhat,etahat,a1] = DiffuseKalmanSmootherH3corr(T,R,Q,H,Pinf1,Pstar1,Y,trend,pp,mm,smpl,mf) </strong></div>
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<h2><a name="_description"></a>DESCRIPTION <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
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<div class="fragment"><pre class="comment"> stephane.adjemian@cepremap.cnrs.fr [09-16-2004]
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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).</pre></div>
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<!-- crossreference -->
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<h2><a name="_cross"></a>CROSS-REFERENCE INFORMATION <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
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This function calls:
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<ul style="list-style-image:url(../matlabicon.gif)">
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<li><a href="steady.html" class="code" title="function steady(linear)">steady</a> Copyright (C) 2001 Michel Juillard</li></ul>
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This function is called by:
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<ul style="list-style-image:url(../matlabicon.gif)">
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</ul>
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<!-- crossreference -->
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<h2><a name="_source"></a>SOURCE CODE <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
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<div class="fragment"><pre>0001 <a name="_sub0" href="#_subfunctions" class="code">function [alphahat,epsilonhat,etahat,a1] = DiffuseKalmanSmootherH3corr(T,R,Q,H,Pinf1,Pstar1,Y,trend,pp,mm,smpl,mf)</a>
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0002 <span class="comment">% stephane.adjemian@cepremap.cnrs.fr [09-16-2004]</span>
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0003 <span class="comment">%</span>
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0004 <span class="comment">% See "Fast Filtering and Smoothing for Multivariate State Space</span>
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0005 <span class="comment">% Models", S.J. Koopman and J. Durbin (2000, in Journal of Time Series</span>
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0006 <span class="comment">% Analysis, vol. 21(3), pp. 281-296).</span>
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0007
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0008 <span class="keyword">global</span> options_;
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0009
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0010 rr = size(Q,1);
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0011 T = cat(1,cat(2,T,zeros(mm,pp)),zeros(pp,mm+pp));
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0012 R = cat(1,cat(2,R,zeros(mm,pp)),cat(2,zeros(pp,rr),eye(pp)));
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0013 Q = cat(1,cat(2,Q,zeros(rr,pp)),cat(2,zeros(pp,rr),H));
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0014 <span class="keyword">if</span> size(Pinf1,1) <span class="comment">% Otherwise Pinf = 0 (no unit root)</span>
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0015 Pinf1 = cat(1,cat(2,Pinf1,zeros(mm,pp)),zeros(pp,mm+pp));
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0016 <span class="keyword">end</span>
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0017 Pstar1 = cat(1,cat(2,Pstar1,zeros(mm,pp)),cat(2,zeros(pp,mm),H));
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0018 spinf = size(Pinf1);
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0019 spstar = size(Pstar1);
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0020 Pstar = zeros(spstar(1),spstar(2),smpl+1); Pstar(:,:,1) = Pstar1;
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0021 Pinf = zeros(spinf(1),spinf(2),smpl+1); Pinf(:,:,1) = Pinf1;
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0022 Pstar1 = Pstar;
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0023 Pinf1 = Pinf;
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0024 v = zeros(pp,smpl);
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0025 a = zeros(mm+pp,smpl+1);
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0026 a1 = a;
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0027 Fstar = zeros(pp,smpl);
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0028 Finf = zeros(pp,smpl);
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0029 Fi = zeros(pp,smpl);
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0030 Ki = zeros(mm+pp,pp,smpl);
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0031 Li = zeros(mm+pp,mm+pp,pp,smpl);
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0032 Linf = zeros(mm+pp,mm+pp,pp,smpl);
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0033 L0 = zeros(mm+pp,mm+pp,pp,smpl);
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0034 Kstar = zeros(mm+pp,pp,smpl);
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0035 Kinf = zeros(mm+pp,pp,smpl);
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0036 P = zeros(mm+pp,mm+pp,smpl+1);
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0037 P1 = zeros(mm+pp,mm+pp,smpl+1);
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0038 crit = options_.kalman_tol;
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0039 <a href="steady.html" class="code" title="function steady(linear)">steady</a> = smpl;
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0040 QQ = R*Q*transpose(R);
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0041 QRt = Q*transpose(R);
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0042 alphahat = zeros(mm+pp,smpl);
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0043 etahat = zeros(rr,smpl);
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0044 epsilonhat = zeros(pp,smpl);
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0045 r = zeros(mm+pp,smpl);
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0046 Z = zeros(pp,mm+pp);
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0047 <span class="keyword">for</span> i=1:pp;
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0048 Z(i,mf(i)) = 1;
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0049 Z(i,mm+i) = 1;
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0050 <span class="keyword">end</span>
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0051 <span class="comment">%% [1] Kalman filter...</span>
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0052 t = 0;
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0053 newRank = rank(Pinf(:,:,1),crit);
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0054 <span class="keyword">while</span> newRank & t < smpl
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0055 t = t+1;
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0056 a1(:,t) = a(:,t);
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0057 Pstar1(:,:,t) = Pstar(:,:,t);
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0058 Pinf1(:,:,t) = Pinf(:,:,t);
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0059 <span class="keyword">for</span> i=1:pp
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0060 v(i,t) = Y(i,t)-a(mf(i),t)-a(mm+i,t)-trend(i,t);
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0061 Fstar(i,t) = Pstar(mf(i),mf(i),t)+Pstar(mm+i,mm+i,t);
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0062 Finf(i,t) = Pinf(mf(i),mf(i),t);
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0063 Kstar(:,i,t) = Pstar(:,mf(i),t)+Pstar(:,mm+i,t);
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0064 <span class="keyword">if</span> Finf(i,t) > crit
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0065 Kinf(:,i,t) = Pinf(:,mf(i),t);
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0066 Linf(:,:,i,t) = eye(mm+pp) - Kinf(:,i,t)*Z(i,:)/Finf(i,t);
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0067 L0(:,:,i,t) = (Kinf(:,i,t)*Fstar(i,t)/Finf(i,t) - Kstar(:,i,t))*Z(i,:)/Finf(i,t);
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0068 a(:,t) = a(:,t) + Kinf(:,i,t)*v(i,t)/Finf(i,t);
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0069 Pstar(:,:,t) = Pstar(:,:,t) + <span class="keyword">...</span>
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0070 Kinf(:,i,t)*transpose(Kinf(:,i,t))*Fstar(i,t)/(Finf(i,t)*Finf(i,t)) - <span class="keyword">...</span>
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0071 (Kstar(:,i,t)*transpose(Kinf(:,i,t)) +<span class="keyword">...</span>
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0072 Kinf(:,i,t)*transpose(Kstar(:,i,t)))/Finf(i,t);
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0073 Pinf(:,:,t) = Pinf(:,:,t) - Kinf(:,i,t)*transpose(Kinf(:,i,t))/Finf(i,t);
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0074 <span class="keyword">else</span> <span class="comment">%% Note that : (1) rank(Pinf)=0 implies that Finf = 0, (2) outside this loop (when for some i and t the condition</span>
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0075 <span class="comment">%% rank(Pinf)=0 is satisfied we have P = Pstar and F = Fstar and (3) Finf = 0 does not imply that</span>
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0076 <span class="comment">%% rank(Pinf)=0. [stphane,11-03-2004].</span>
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0077 a(:,t) = a(:,t) + Kstar(:,i,t)*v(i,t)/Fstar(i,t);
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0078 Pstar(:,:,t) = Pstar(:,:,t) - Kstar(:,i,t)*transpose(Kstar(:,i,t))/Fstar(i,t);
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0079 <span class="keyword">end</span>
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0080 <span class="keyword">end</span>
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0081 a(:,t+1) = T*a(:,t);
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0082 Pstar(:,:,t+1) = T*Pstar(:,:,t)*transpose(T)+ QQ;
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0083 Pinf(:,:,t+1) = T*Pinf(:,:,t)*transpose(T);
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0084 P0=Pinf(:,:,t+1);
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0085 newRank = ~all(abs(P0(:))<crit);
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0086 <span class="keyword">end</span>
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0087 d = t;
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0088 P(:,:,d+1) = Pstar(:,:,d+1);
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0089 Linf = Linf(:,:,:,1:d);
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0090 L0 = L0(:,:,:,1:d);
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0091 Fstar = Fstar(:,1:d);
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0092 Finf = Finf(:,1:d);
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0093 Kstar = Kstar(:,:,1:d);
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0094 Pstar = Pstar(:,:,1:d);
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0095 Pinf = Pinf(:,:,1:d);
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0096 Pstar1 = Pstar1(:,:,1:d);
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0097 Pinf1 = Pinf1(:,:,1:d);
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0098 notsteady = 1;
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0099 <span class="keyword">while</span> notsteady & t<smpl
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0100 t = t+1;
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0101 a1(:,t) = a(:,t);
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0102 P(:,:,t)=tril(P(:,:,t))+transpose(tril(P(:,:,t),-1));
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0103 P1(:,:,t) = P(:,:,t);
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0104 <span class="keyword">for</span> i=1:pp
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0105 v(i,t) = Y(i,t) - a(mf(i),t) - a(mm+i,t) - trend(i,t);
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0106 Fi(i,t) = P(mf(i),mf(i),t)+P(mm+i,mm+i,t);
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0107 Ki(:,i,t) = P(:,mf(i),t)+P(:,mm+i,t);
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0108 <span class="keyword">if</span> Fi(i,t) > crit
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0109 Li(:,:,i,t) = eye(mm+pp)-Ki(:,i,t)*Z(i,:)/Fi(i,t);
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0110 a(:,t) = a(:,t) + Ki(:,i,t)*v(i,t)/Fi(i,t);
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0111 P(:,:,t) = P(:,:,t) - Ki(:,i,t)*transpose(Ki(:,i,t))/Fi(i,t);
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0112 P(:,:,t)=tril(P(:,:,t))+transpose(tril(P(:,:,t),-1));
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0113 <span class="keyword">end</span>
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0114 <span class="keyword">end</span>
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0115 a(:,t+1) = T*a(:,t);
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0116 P(:,:,t+1) = T*P(:,:,t)*transpose(T) + QQ;
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0117 notsteady = ~(max(max(abs(P(:,:,t+1)-P(:,:,t))))<crit);
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0118 <span class="keyword">end</span>
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0119 P_s=tril(P(:,:,t))+transpose(tril(P(:,:,t),-1));
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0120 Fi_s = Fi(:,t);
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0121 Ki_s = Ki(:,:,t);
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0122 L_s =Li(:,:,:,t);
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0123 <span class="keyword">if</span> t<smpl
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0124 t_steady = t+1;
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0125 P = cat(3,P(:,:,1:t),repmat(P(:,:,t),[1 1 smpl-t_steady+1]));
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0126 Fi = cat(2,Fi(:,1:t),repmat(Fi_s,[1 1 smpl-t_steady+1]));
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0127 Li = cat(4,Li(:,:,:,1:t),repmat(L_s,[1 1 smpl-t_steady+1]));
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0128 Ki = cat(3,Ki(:,:,1:t),repmat(Ki_s,[1 1 smpl-t_steady+1]));
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0129 <span class="keyword">end</span>
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0130 <span class="keyword">while</span> t<smpl
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0131 t=t+1;
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0132 a1(:,t) = a(:,t);
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0133 <span class="keyword">for</span> i=1:pp
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0134 v(i,t) = Y(i,t) - a(mf(i),t) - a(mm+i,t) - trend(i,t);
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0135 <span class="keyword">if</span> Fi_s(i) > crit
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0136 a(:,t) = a(:,t) + Ki_s(:,i)*v(i,t)/Fi_s(i);
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0137 <span class="keyword">end</span>
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0138 <span class="keyword">end</span>
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0139 a(:,t+1) = T*a(:,t);
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0140 <span class="keyword">end</span>
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0141 a1(:,t+1) = a(:,t+1);
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0142 <span class="comment">%% [2] Kalman smoother...</span>
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0143 ri=r;
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0144 t = smpl+1;
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0145 <span class="keyword">while</span> t>d+1 & t>2,
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0146 t = t-1;
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0147 <span class="keyword">for</span> i=pp:-1:1
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0148 <span class="keyword">if</span> Fi(i,t) > crit
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0149 ri(:,t)=transpose(Z(i,:))/Fi(i,t)*v(i,t)+transpose(Li(:,:,i,t))*ri(:,t);
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0150 <span class="keyword">end</span>
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0151 <span class="keyword">end</span>
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0152 r(:,t-1) = ri(:,t);
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0153 alphahat(:,t) = a1(:,t) + P1(:,:,t)*r(:,t-1);
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0154 tmp = QRt*r(:,t);
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0155 etahat(:,t) = tmp(1:rr);
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0156 epsilonhat(:,t) = tmp(rr+1:rr+pp);
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0157 ri(:,t-1) = transpose(T)*ri(:,t);
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0158 <span class="keyword">end</span>
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0159 <span class="keyword">if</span> d
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0160 r0 = zeros(mm+pp,d); r0(:,d) = ri(:,d);
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0161 r1 = zeros(mm+pp,d);
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0162 <span class="keyword">for</span> t = d:-1:2
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0163 <span class="keyword">for</span> i=pp:-1:1
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0164 <span class="keyword">if</span> Finf(i,t) > crit
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0165 r1(:,t) = transpose(Z)*v(:,t)/Finf(i,t) + <span class="keyword">...</span>
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0166 transpose(L0(:,:,i,t))*r0(:,t) + transpose(Linf(:,:,i,t))*r1(:,t);
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0167 r0(:,t) = transpose(Linf(:,:,i,t))*r0(:,t);
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0168 <span class="keyword">end</span>
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0169 <span class="keyword">end</span>
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0170 alphahat(:,t) = a1(:,t) + Pstar1(:,:,t)*r0(:,t) + Pinf1(:,:,t)*r1(:,t);
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0171 r(:,t-1) = r0(:,t);
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0172 tmp = QRt*r(:,t);
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0173 etahat(:,t) = tmp(1:rr);
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0174 epsilonhat(:,t) = tmp(rr+1:rr+pp);
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0175 r0(:,t-1) = transpose(T)*r0(:,t);
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0176 r1(:,t-1) = transpose(T)*r1(:,t);
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0177 <span class="keyword">end</span>
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0178 r0_0 = r0(:,1);
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0179 r1_0 = r1(:,1);
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0180 <span class="keyword">for</span> i=pp:-1:1
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0181 <span class="keyword">if</span> Finf(i,1) > crit,
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0182 r1_0 = transpose(Z)*v(:,1)/Finf(i,1) + <span class="keyword">...</span>
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0183 transpose(L0(:,:,i,1))*r0_0 + transpose(Linf(:,:,i,1))*r1_0;
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0184 r0_0 = transpose(Linf(:,:,i,1))*r0_0;
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0185 <span class="keyword">end</span>
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0186 <span class="keyword">end</span>
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0187 alphahat(:,1) = a(:,1) + Pstar(:,:,1)*r0_0 + Pinf(:,:,1)*r1_0;
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0188 tmp = QRt*r(:,1);
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0189 etahat(:,1) = tmp(1:rr);
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0190 epsilonhat(:,1) = tmp(rr+1:rr+pp);
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0191 <span class="keyword">else</span>
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0192 r0 = ri(:,1);
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0193 <span class="keyword">for</span> i=pp:-1:1
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0194 <span class="keyword">if</span> Fi(i,1) > crit
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0195 r0=transpose(Z(i,:))/Fi(i,1)*v(i,1)+transpose(Li(:,:,i,1))*r0;
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0196 <span class="keyword">end</span>
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0197 <span class="keyword">end</span>
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0198 alphahat(:,1) = a(:,1) + P(:,:,1)*r0;
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0199 tmp = QRt*r(:,1);
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0200 etahat(:,1) = tmp(1:rr);
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0201 epsilonhat(:,1) = tmp(rr+1:rr+pp);
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0202 <span class="keyword">end</span>
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0203 alphahat = alphahat(1:mm,:);</pre></div>
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<hr><address>Generated on Fri 16-Jun-2006 09:09:06 by <strong><a href="http://www.artefact.tk/software/matlab/m2html/">m2html</a></strong> © 2003</address>
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