283 lines
9.5 KiB
Matlab
283 lines
9.5 KiB
Matlab
function [alphahat,epsilonhat,etahat,a,P,aK,PK,decomp] = missing_DiffuseKalmanSmootherH3_Z(T,Z,R,Q,H,Pinf1,Pstar1,Y,pp,mm,smpl,data_index,nk,kalman_tol,diffuse_kalman_tol,decomp_flag)
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% function [alphahat,epsilonhat,etahat,a1,P,aK,PK,d,decomp] = missing_DiffuseKalmanSmootherH3_Z(T,Z,R,Q,H,Pinf1,Pstar1,Y,pp,mm,smpl,data_index,nk,kalman_tol,decomp_flag)
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% Computes the diffuse kalman smoother without measurement error, in the case of a singular var-cov matrix.
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% Univariate treatment of multivariate time series.
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%
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% INPUTS
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% T: mm*mm matrix
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% Z: pp*mm matrix
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% R: mm*rr matrix
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% Q: rr*rr matrix
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% H: pp*1 vector of variance of measurement errors
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% Pinf1: mm*mm diagonal matrix with with q ones and m-q zeros
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% Pstar1: mm*mm variance-covariance matrix with stationary variables
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% Y: pp*1 vector
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% pp: number of observed variables
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% mm: number of state variables
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% smpl: sample size
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% data_index [cell] 1*smpl cell of column vectors of indices.
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% nk number of forecasting periods
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% kalman_tol tolerance for zero divider
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% diffuse_kalman_tol tolerance for zero divider
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% decomp_flag if true, compute filter decomposition
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%
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% OUTPUTS
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% alphahat: smoothed state variables (a_{t|T})
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% epsilonhat: measurement errors
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% etahat: smoothed shocks
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% a: matrix of updated variables (a_{t|t})
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% aK: 3D array of k step ahead filtered state variables (a_{t+k|t})
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% (meaningless for periods 1:d)
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% P: 3D array of one-step ahead forecast error variance
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% matrices
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% PK: 4D array of k-step ahead forecast error variance
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% matrices (meaningless for periods 1:d)
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% decomp: decomposition of the effect of shocks on filtered values
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%
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% SPECIAL REQUIREMENTS
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% See "Filtering and Smoothing of State Vector for Diffuse State Space
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% Models", S.J. Koopman and J. Durbin (2003, in Journal of Time Series
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% Analysis, vol. 24(1), pp. 85-98).
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% Copyright (C) 2004-2015 Dynare Team
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%
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% This file is part of Dynare.
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%
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% Dynare is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, either version 3 of the License, or
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% (at your option) any later version.
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%
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% Dynare is distributed in the hope that it will be useful,
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% but WITHOUT ANY WARRANTY; without even the implied warranty of
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% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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% GNU General Public License for more details.
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%
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% You should have received a copy of the GNU General Public License
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% along with Dynare. If not, see <http://www.gnu.org/licenses/>.
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% Modified by M. Ratto
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% New output argument aK: 1-step to nk-stpe ahed predictions)
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% New input argument nk: max order of predictions in aK
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d = 0;
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decomp = [];
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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);
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a1 = zeros(mm,smpl+1);
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aK = zeros(nk,mm,smpl+nk);
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Fstar = zeros(pp,smpl);
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Finf = zeros(pp,smpl);
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Fi = zeros(pp,smpl);
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Ki = zeros(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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PK = zeros(nk,mm,mm,smpl+nk);
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Pstar = zeros(spstar(1),spstar(2),smpl); Pstar(:,:,1) = Pstar1;
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Pinf = zeros(spinf(1),spinf(2),smpl); Pinf(:,:,1) = Pinf1;
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Pstar1 = Pstar;
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Pinf1 = Pinf;
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steady = smpl;
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rr = size(Q,1); % number of structural shocks
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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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epsilonhat = zeros(rr,smpl);
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r = zeros(mm,smpl);
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t = 0;
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icc=0;
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newRank = rank(Pinf(:,:,1),diffuse_kalman_tol);
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while newRank && t < smpl
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t = t+1;
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a(:,t) = a1(:,t);
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Pstar1(:,:,t) = Pstar(:,:,t);
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Pinf1(:,:,t) = Pinf(:,:,t);
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di = data_index{t}';
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for i=di
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Zi = Z(i,:);
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v(i,t) = Y(i,t)-Zi*a(:,t);
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Fstar(i,t) = Zi*Pstar(:,:,t)*Zi' +H(i);
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Finf(i,t) = Zi*Pinf(:,:,t)*Zi';
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Kstar(:,i,t) = Pstar(:,:,t)*Zi';
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if Finf(i,t) > kalman_tol && newRank
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icc=icc+1;
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Kinf(:,i,t) = Pinf(:,:,t)*Zi';
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Kinf_Finf = Kinf(:,i,t)/Finf(i,t);
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a(:,t) = a(:,t) + Kinf_Finf*v(i,t);
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Pstar(:,:,t) = Pstar(:,:,t) + ...
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Kinf(:,i,t)*Kinf_Finf'*(Fstar(i,t)/Finf(i,t)) - ...
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Kstar(:,i,t)*Kinf_Finf' - ...
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Kinf_Finf*Kstar(:,i,t)';
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Pinf(:,:,t) = Pinf(:,:,t) - Kinf(:,i,t)*Kinf(:,i,t)'/Finf(i,t);
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elseif Fstar(i,t) > kalman_tol
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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)*Kstar(:,i,t)'/Fstar(i,t);
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end
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end
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if newRank
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oldRank = rank(Pinf(:,:,t),diffuse_kalman_tol);
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else
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oldRank = 0;
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end
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a1(:,t+1) = T*a(:,t);
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aK(1,:,t+1) = a1(:,t+1);
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for jnk=2:nk
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aK(jnk,:,t+jnk) = T*dynare_squeeze(aK(jnk-1,:,t+jnk-1));
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end
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Pstar(:,:,t+1) = T*Pstar(:,:,t)*T'+ QQ;
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Pinf(:,:,t+1) = T*Pinf(:,:,t)*T';
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P0=Pinf(:,:,t+1);
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if newRank,
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newRank = rank(Pinf(:,:,t+1),diffuse_kalman_tol);
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end
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if oldRank ~= newRank
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disp('univariate_diffuse_kalman_filter:: T does influence the rank of Pinf!')
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end
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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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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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a(:,t) = a1(:,t);
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P1(:,:,t) = P(:,:,t);
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di = data_index{t}';
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for i=di
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Zi = Z(i,:);
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v(i,t) = Y(i,t) - Zi*a(:,t);
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Fi(i,t) = Zi*P(:,:,t)*Zi' + H(i);
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Ki(:,i,t) = P(:,:,t)*Zi';
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if Fi(i,t) > kalman_tol
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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)*Ki(:,i,t)'/Fi(i,t);
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end
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end
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a1(:,t+1) = T*a(:,t);
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Pf = P(:,:,t);
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aK(1,:,t+1) = a1(:,t+1);
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for jnk=1:nk
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Pf = T*Pf*T' + QQ;
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PK(jnk,:,:,t+jnk) = Pf;
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if jnk>1
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aK(jnk,:,t+jnk) = T*dynare_squeeze(aK(jnk-1,:,t+jnk-1));
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end
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end
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P(:,:,t+1) = T*P(:,:,t)*T' + QQ;
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% notsteady = ~(max(max(abs(P(:,:,t+1)-P(:,:,t))))<kalman_tol);
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end
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% $$$ P_s=tril(P(:,:,t))+tril(P(:,:,t),-1)';
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% $$$ P1_s=tril(P1(:,:,t))+tril(P1(:,:,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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% $$$ P = cat(3,P(:,:,1:t),repmat(P_s,[1 1 smpl-t]));
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% $$$ P1 = cat(3,P1(:,:,1:t),repmat(P1_s,[1 1 smpl-t]));
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% $$$ Fi = cat(2,Fi(:,1:t),repmat(Fi_s,[1 1 smpl-t]));
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% $$$ Li = cat(4,Li(:,:,:,1:t),repmat(L_s,[1 1 smpl-t]));
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% $$$ Ki = cat(3,Ki(:,:,1:t),repmat(Ki_s,[1 1 smpl-t]));
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% $$$ end
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% $$$ while t<smpl
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% $$$ t=t+1;
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% $$$ a(:,t) = a1(:,t);
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% $$$ di = data_index{t}';
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% $$$ for i=di
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% $$$ Zi = Z(i,:);
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% $$$ v(i,t) = Y(i,t) - Zi*a(:,t);
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% $$$ if Fi_s(i) > kalman_tol
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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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% $$$ end
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% $$$ a1(:,t+1) = T*a(:,t);
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% $$$ Pf = P(:,:,t);
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% $$$ for jnk=1:nk,
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% $$$ Pf = T*Pf*T' + QQ;
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% $$$ aK(jnk,:,t+jnk) = T^jnk*a(:,t);
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% $$$ PK(jnk,:,:,t+jnk) = Pf;
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% $$$ end
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% $$$ end
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ri=zeros(mm,1);
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t = smpl+1;
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while t > d+1
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t = t-1;
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di = flipud(data_index{t})';
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for i = di
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if Fi(i,t) > kalman_tol
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ri = Z(i,:)'/Fi(i,t)*v(i,t)+ri-Ki(:,i,t)'*ri/Fi(i,t)*Z(i,:)';
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end
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end
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r(:,t) = ri;
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alphahat(:,t) = a1(:,t) + P1(:,:,t)*r(:,t);
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etahat(:,t) = QRt*r(:,t);
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ri = T'*ri;
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end
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if d
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r0 = zeros(mm,d);
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r0(:,d) = ri;
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r1 = zeros(mm,d);
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for t = d:-1:1
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di = flipud(data_index{t})';
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for i = di
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if Finf(i,t) > kalman_tol
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r1(:,t) = Z(i,:)'*v(i,t)/Finf(i,t) + ...
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(Kinf(:,i,t)'*Fstar(i,t)/Finf(i,t)-Kstar(:,i,t)')*r0(:,t)/Finf(i,t)*Z(i,:)' + ...
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r1(:,t)-Kinf(:,i,t)'*r1(:,t)/Finf(i,t)*Z(i,:)';
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r0(:,t) = r0(:,t)-Kinf(:,i,t)'*r0(:,t)/Finf(i,t)*Z(i,:)';
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elseif Fstar(i,t) > kalman_tol % step needed whe Finf == 0
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r0(:,t) = Z(i,:)'/Fstar(i,t)*v(i,t)+r0(:,t)-(Kstar(:,i,t)'*r0(:,t))/Fstar(i,t)*Z(i,:)';
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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) = r0(:,t);
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etahat(:,t) = QRt*r(:,t);
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if t > 1
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r0(:,t-1) = T'*r0(:,t);
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r1(:,t-1) = T'*r1(:,t);
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end
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end
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end
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if decomp_flag
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decomp = zeros(nk,mm,rr,smpl+nk);
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ZRQinv = inv(Z*QQ*Z');
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for t = max(d,1):smpl
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ri_d = zeros(mm,1);
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di = flipud(data_index{t})';
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for i = di
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if Fi(i,t) > kalman_tol
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ri_d = Z(i,:)'/Fi(i,t)*v(i,t)+ri_d-Ki(:,i,t)'*ri_d/Fi(i,t)*Z(i,:)';
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end
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end
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% calculate eta_tm1t
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eta_tm1t = QRt*ri_d;
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% calculate decomposition
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Ttok = eye(mm,mm);
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AAA = P1(:,:,t)*Z'*ZRQinv*Z*R;
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for h = 1:nk
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BBB = Ttok*AAA;
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for j=1:rr
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decomp(h,:,j,t+h) = eta_tm1t(j)*BBB(:,j);
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end
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Ttok = T*Ttok;
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end
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end
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end
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epsilonhat = Y - Z*alphahat;
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