v4: added return argument to DiffuseLikelihood* for outer product gradient [Marco]
git-svn-id: https://www.dynare.org/svn/dynare/dynare_v4@676 ac1d8469-bf42-47a9-8791-bf33cf982152time-shift
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@ -1,92 +1,91 @@
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function LIK = DiffuseLikelihood1(T,R,Q,Pinf,Pstar,Y,trend,start)
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% stephane.adjemian@cepremap.cnrs.fr [07-19-2004]
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%
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% Same as DiffuseLikelihoodH1 without measurement error.
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global bayestopt_ options_
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mf = bayestopt_.mf;
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smpl = size(Y,2);
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mm = size(T,2);
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pp = size(Y,1);
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a = zeros(mm,1);
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dF = 1;
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QQ = R*Q*transpose(R);
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t = 0;
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lik = zeros(smpl+1,1);
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LIK = Inf;
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lik(smpl+1) = smpl*pp*log(2*pi);
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notsteady = 1;
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crit = options_.kalman_tol;
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reste = 0;
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while rank(Pinf,crit) & t < smpl
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t = t+1;
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v = Y(:,t)-a(mf)-trend(:,t);
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Finf = Pinf(mf,mf);
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if rcond(Finf) < crit
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if ~all(abs(Finf(:)) < crit)
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return
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else
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iFstar = inv(Pstar(mf,mf));
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dFstar = det(Pstar(mf,mf));
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Kstar = Pstar(:,mf)*iFstar;
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lik(t) = log(dFstar) + transpose(v)*iFstar*v;
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Pinf = T*Pinf*transpose(T);
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Pstar = T*(Pstar-Pstar(:,mf)*transpose(Kstar))*transpose(T)+QQ;
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a = T*(a+Kstar*v);
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end
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else
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lik(t) = log(det(Finf));
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iFinf = inv(Finf);
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Kinf = Pinf(:,mf)*iFinf; %% premultiplication by the transition matrix T is removed (stephane)
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Fstar = Pstar(mf,mf);
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Kstar = (Pstar(:,mf)-Kinf*Fstar)*iFinf; %% premultiplication by the transition matrix T is removed (stephane)
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Pstar = T*(Pstar-Pstar(:,mf)*transpose(Kinf)-Pinf(:,mf)*transpose(Kstar))*transpose(T)+QQ;
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Pinf = T*(Pinf-Pinf(:,mf)*transpose(Kinf))*transpose(T);
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a = T*(a+Kinf*v);
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end
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end
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if t == smpl
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error(['There isn''t enough information to estimate the initial' ...
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' conditions of the nonstationary variables']);
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end
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F_singular = 1;
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while notsteady & t < smpl
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t = t+1;
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v = Y(:,t)-a(mf)-trend(:,t);
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F = Pstar(mf,mf);
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oldPstar = Pstar;
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dF = det(F);
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if rcond(F) < crit
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if ~all(abs(F(:))<crit)
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return
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else
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a = T*a;
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Pstar = T*Pstar*transpose(T)+QQ;
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end
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else
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F_singular = 0;
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iF = inv(F);
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lik(t) = log(dF)+transpose(v)*iF*v;
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K = Pstar(:,mf)*iF; %% premultiplication by the transition matrix T is removed (stephane)
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a = T*(a+K*v); %% --> factorization of the transition matrix...
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Pstar = T*(Pstar-Pstar(:,mf)*iF*Pstar(mf,:))*transpose(T)+QQ; %% ... idem (stephane)
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end
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notsteady = ~(max(max(abs(Pstar-oldPstar)))<crit);
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end
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if F_singular == 1
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error(['The variance of the forecast error remains singular until the' ...
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'end of the sample'])
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end
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reste = smpl-t;
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while t < smpl
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t = t+1;
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v = Y(:,t)-a(mf)-trend(:,t);
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a = T*(a+K*v);
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lik(t) = transpose(v)*iF*v;
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end
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lik(t) = lik(t) + reste*log(dF);
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LIK = .5*(sum(lik(start:end))-(start-1)*lik(smpl+1)/smpl);% Minus the
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% log-likelihood.
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function [LIK, lik] = DiffuseLikelihood1(T,R,Q,Pinf,Pstar,Y,trend,start)
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% M. Ratto added lik in output
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% stephane.adjemian@cepremap.cnrs.fr [07-19-2004]
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%
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% Same as DiffuseLikelihoodH1 without measurement error.
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global bayestopt_ options_
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mf = bayestopt_.mf;
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smpl = size(Y,2);
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mm = size(T,2);
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pp = size(Y,1);
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a = zeros(mm,1);
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dF = 1;
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QQ = R*Q*transpose(R);
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t = 0;
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lik = zeros(smpl+1,1);
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LIK = Inf;
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lik(smpl+1) = smpl*pp*log(2*pi);
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notsteady = 1;
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crit = options_.kalman_tol;
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reste = 0;
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while rank(Pinf,crit) & t < smpl
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t = t+1;
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v = Y(:,t)-a(mf)-trend(:,t);
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Finf = Pinf(mf,mf);
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if rcond(Finf) < crit
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if ~all(abs(Finf(:)) < crit)
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return
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else
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iFstar = inv(Pstar(mf,mf));
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dFstar = det(Pstar(mf,mf));
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Kstar = Pstar(:,mf)*iFstar;
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lik(t) = log(dFstar) + transpose(v)*iFstar*v;
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Pinf = T*Pinf*transpose(T);
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Pstar = T*(Pstar-Pstar(:,mf)*transpose(Kstar))*transpose(T)+QQ;
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a = T*(a+Kstar*v);
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end
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else
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lik(t) = log(det(Finf));
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iFinf = inv(Finf);
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Kinf = Pinf(:,mf)*iFinf; %% premultiplication by the transition matrix T is removed (stephane)
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Fstar = Pstar(mf,mf);
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Kstar = (Pstar(:,mf)-Kinf*Fstar)*iFinf; %% premultiplication by the transition matrix T is removed (stephane)
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Pstar = T*(Pstar-Pstar(:,mf)*transpose(Kinf)-Pinf(:,mf)*transpose(Kstar))*transpose(T)+QQ;
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Pinf = T*(Pinf-Pinf(:,mf)*transpose(Kinf))*transpose(T);
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a = T*(a+Kinf*v);
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end
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end
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if t == smpl
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error(['There isn''t enough information to estimate the initial' ...
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' conditions of the nonstationary variables']);
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end
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F_singular = 1;
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while notsteady & t < smpl
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t = t+1;
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v = Y(:,t)-a(mf)-trend(:,t);
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F = Pstar(mf,mf);
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oldPstar = Pstar;
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dF = det(F);
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if rcond(F) < crit
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if ~all(abs(F(:))<crit)
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return
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else
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a = T*a;
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Pstar = T*Pstar*transpose(T)+QQ;
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end
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else
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F_singular = 0;
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iF = inv(F);
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lik(t) = log(dF)+transpose(v)*iF*v;
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K = Pstar(:,mf)*iF; %% premultiplication by the transition matrix T is removed (stephane)
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a = T*(a+K*v); %% --> factorization of the transition matrix...
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Pstar = T*(Pstar-Pstar(:,mf)*iF*Pstar(mf,:))*transpose(T)+QQ; %% ... idem (stephane)
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end
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notsteady = ~(max(max(abs(Pstar-oldPstar)))<crit);
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end
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if F_singular == 1
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error(['The variance of the forecast error remains singular until the' ...
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'end of the sample'])
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end
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reste = smpl-t;
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while t < smpl
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t = t+1;
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v = Y(:,t)-a(mf)-trend(:,t);
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a = T*(a+K*v);
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lik(t) = transpose(v)*iF*v;
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end
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lik(t) = lik(t) + reste*log(dF);
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LIK = .5*(sum(lik(start:end))-(start-1)*lik(smpl+1)/smpl);% Minus the log-likelihood.
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@ -1,170 +1,222 @@
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function LIK = DiffuseLikelihood3(T,R,Q,Pinf,Pstar,Y,trend,start)%//Z,T,R,Q,Pinf,Pstar,Y)
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% stepane.adjemian@cepremap.cnrs.fr [07-19-2004]
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%
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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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%
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% Case where F_{\infty,t} is singular ==> Univariate treatment of multivariate
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% time series.
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%
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% THE PROBLEM:
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%
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% y_t = Z_t * \alpha_t + \varepsilon_t
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% \alpha_{t+1} = T_t * \alpha_t + R_t * \eta_t
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%
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% with:
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%
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% \alpha_1 = a + A*\delta + R_0*\eta_0
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%
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% m*q matrix A and m*(m-q) matrix R_0 are selection matrices (their
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% columns constitue all the columns of the m*m identity matrix) so that
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%
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% A'*R_0 = 0 and A'*\alpha_1 = \delta
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%
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% We assume that the vector \delta is distributed as a N(0,\kappa*I_q)
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% for a given \kappa > 0. So that the expectation of \alpha_1 is a and
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% its variance is P, with
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%
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% P = \kappa*P_{\infty} + P_{\star}
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%
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% P_{\infty} = A*A'
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% P_{\star} = R_0*Q_0*R_0'
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%
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% P_{\infty} is a m*m diagonal matrix with q ones and m-q zeros.
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%
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%
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% and where:
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%
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% y_t is a pp*1 vector
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% \alpha_t is a mm*1 vector
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% \varepsilon_t is a pp*1 multivariate random variable (iid N(0,H_t))
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% \eta_t is a rr*1 multivariate random variable (iid N(0,Q_t))
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% a_1 is a mm*1 vector
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%
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% Z_t is a pp*mm matrix
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% T_t is a mm*mm matrix
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% H_t is a pp*pp matrix
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% R_t is a mm*rr matrix
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% Q_t is a rr*rr matrix
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% P_1 is a mm*mm matrix
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%
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%
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% FILTERING EQUATIONS:
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%
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% v_t = y_t - Z_t* a_t
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% F_t = Z_t * P_t * Z_t' + H_t
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% K_t = T_t * P_t * Z_t' * F_t^{-1}
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% L_t = T_t - K_t * Z_t
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% a_{t+1} = T_t * a_t + K_t * v_t
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% P_{t+1} = T_t * P_t * L_t' + R_t*Q_t*R_t'
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%
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%
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% DIFFUSE FILTERING EQUATIONS:
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%
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% a_{t+1} = T_t*a_t + K_{\ast,t}v_t
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% P_{\infty,t+1} = T_t*P_{\infty,t}*T_t'
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% P_{\ast,t+1} = T_t*P_{\ast,t}*L_{\ast,t}' + R_t*Q_t*R_t'
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% K_{\ast,t} = T_t*P_{\ast,t}*Z_t'*F_{\ast,t}^{-1}
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% v_t = y_t - Z_t*a_t
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% L_{\ast,t} = T_t - K_{\ast,t}*Z_t
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% F_{\ast,t} = Z_t*P_{\ast,t}*Z_t' + H_t
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global bayestopt_ options_
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mf = bayestopt_.mf;
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pp = size(Y,1);
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mm = size(T,1);
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smpl = size(Y,2);
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a = zeros(mm,1);
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QQ = R*Q*transpose(R);
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t = 0;
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lik = zeros(smpl+1,1);
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lik(smpl+1) = smpl*pp*log(2*pi); %% the constant of minus two times the log-likelihood
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notsteady = 1;
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crit = options_.kalman_tol;
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newRank = rank(Pinf,crit);
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while newRank & t < smpl
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t = t+1;
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for i=1:pp
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v(i) = Y(i,t)-a(mf(i))-trend(i,t);
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Fstar = Pstar(mf(i),mf(i));
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Finf = Pinf(mf(i),mf(i));
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Kstar = Pstar(:,mf(i));
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if Finf > crit
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Kinf = Pinf(:,mf(i));
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a = a + Kinf*v(i)/Finf;
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Pstar = Pstar + Kinf*transpose(Kinf)*Fstar/(Finf*Finf) - ...
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(Kstar*transpose(Kinf)+Kinf*transpose(Kstar))/Finf;
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Pinf = Pinf - Kinf*transpose(Kinf)/Finf;
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lik(t) = lik(t) + log(Finf);
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elseif Fstar > crit %% 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. [stéphane,11-03-2004].
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if rank(Pinf,crit) == 0
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lik(t) = lik(t) + log(Fstar) + v(i)*v(i)/Fstar;
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end
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a = a + Kstar*v(i)/Fstar;
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Pstar = Pstar - Kstar*transpose(Kstar)/Fstar;
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else
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% disp(['zero F term in DKF for observed ',int2str(i),' ',num2str(Fi)])
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end
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end
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if all(abs(Pinf(:))<crit),
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oldRank = 0;
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else
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oldRank = rank(Pinf,crit);
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end
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a = T*a;
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Pstar = T*Pstar*transpose(T)+QQ;
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Pinf = T*Pinf*transpose(T);
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if all(abs(Pinf(:))<crit),
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newRank = 0;
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else
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newRank = rank(Pinf,crit);
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end
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if oldRank ~= newRank
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disp('DiffuseLiklihood3 :: T does influence the rank of Pinf!')
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end
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end
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if t == smpl
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error(['There isn''t enough information to estimate the initial' ...
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' conditions of the nonstationary variables']);
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end
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while notsteady & t < smpl
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t = t+1;
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oldP = Pstar;
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for i=1:pp
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v(i) = Y(i,t) - a(mf(i)) - trend(i,t);
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Fi = Pstar(mf(i),mf(i));
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if Fi > crit
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Ki = Pstar(:,mf(i));
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a = a + Ki*v(i)/Fi;
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Pstar = Pstar - Ki*transpose(Ki)/Fi;
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lik(t) = lik(t) + log(Fi) + v(i)*v(i)/Fi;
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else
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%disp(['zero F term for observed ',int2str(i),' ',num2str(Fi)])
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end
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end
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a = T*a;
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Pstar = T*Pstar*transpose(T) + QQ;
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notsteady = ~(max(max(abs(Pstar-oldP)))<crit);
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end
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while t < smpl
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t = t+1;
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Pstar = oldP;
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for i=1:pp
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v(i) = Y(i,t) - a(mf(i)) - trend(i,t);
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Fi = Pstar(mf(i),mf(i));
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if Fi > crit
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Ki = Pstar(:,mf(i));
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a = a + Ki*v(i)/Fi;
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Pstar = Pstar - Ki*transpose(Ki)/Fi;
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lik(t) = lik(t) + log(Fi) + v(i)*v(i)/Fi;
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else
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%disp(['zero F term for observed ',int2str(i),' ',num2str(Fi)])
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end
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end
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a = T*a;
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end
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LIK = .5*(sum(lik(start:end))-(start-1)*lik(smpl+1)/smpl);
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function [LIK, lik] = DiffuseLikelihood3(T,R,Q,Pinf,Pstar,Y,trend,start)%//Z,T,R,Q,Pinf,Pstar,Y)
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% M. Ratto added lik in output [October 2005]
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% changes by M. Ratto [April 2005]
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% introduced new options options_.diffuse_d for termination of DKF
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% new icc counter for Finf steps in DKF
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% new termination for DKF
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% likelihood terms for Fstar must be cumulated in DKF also when Pinf is non
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% zero.
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%
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% [4/5/2005] correctyed bug in the modified verson of Ratto for rank of Pinf
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% introduced a specific crit1 for the DKF termination
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%
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% stepane.adjemian@cepremap.cnrs.fr [07-19-2004]
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%
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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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%
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% Case where F_{\infty,t} is singular ==> Univariate treatment of multivariate
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% time series.
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%
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% THE PROBLEM:
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%
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% y_t = Z_t * \alpha_t + \varepsilon_t
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% \alpha_{t+1} = T_t * \alpha_t + R_t * \eta_t
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%
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% with:
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%
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% \alpha_1 = a + A*\delta + R_0*\eta_0
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%
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% m*q matrix A and m*(m-q) matrix R_0 are selection matrices (their
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% columns constitue all the columns of the m*m identity matrix) so that
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%
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% A'*R_0 = 0 and A'*\alpha_1 = \delta
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%
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% We assume that the vector \delta is distributed as a N(0,\kappa*I_q)
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% for a given \kappa > 0. So that the expectation of \alpha_1 is a and
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% its variance is P, with
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%
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% P = \kappa*P_{\infty} + P_{\star}
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%
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% P_{\infty} = A*A'
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% P_{\star} = R_0*Q_0*R_0'
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%
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% P_{\infty} is a m*m diagonal matrix with q ones and m-q zeros.
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%
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%
|
||||
% and where:
|
||||
%
|
||||
% y_t is a pp*1 vector
|
||||
% \alpha_t is a mm*1 vector
|
||||
% \varepsilon_t is a pp*1 multivariate random variable (iid N(0,H_t))
|
||||
% \eta_t is a rr*1 multivariate random variable (iid N(0,Q_t))
|
||||
% a_1 is a mm*1 vector
|
||||
%
|
||||
% Z_t is a pp*mm matrix
|
||||
% T_t is a mm*mm matrix
|
||||
% H_t is a pp*pp matrix
|
||||
% R_t is a mm*rr matrix
|
||||
% Q_t is a rr*rr matrix
|
||||
% P_1 is a mm*mm matrix
|
||||
%
|
||||
%
|
||||
% FILTERING EQUATIONS:
|
||||
%
|
||||
% v_t = y_t - Z_t* a_t
|
||||
% F_t = Z_t * P_t * Z_t' + H_t
|
||||
% K_t = T_t * P_t * Z_t' * F_t^{-1}
|
||||
% L_t = T_t - K_t * Z_t
|
||||
% a_{t+1} = T_t * a_t + K_t * v_t
|
||||
% P_{t+1} = T_t * P_t * L_t' + R_t*Q_t*R_t'
|
||||
%
|
||||
%
|
||||
% DIFFUSE FILTERING EQUATIONS:
|
||||
%
|
||||
% a_{t+1} = T_t*a_t + K_{\ast,t}v_t
|
||||
% P_{\infty,t+1} = T_t*P_{\infty,t}*T_t'
|
||||
% P_{\ast,t+1} = T_t*P_{\ast,t}*L_{\ast,t}' + R_t*Q_t*R_t'
|
||||
% K_{\ast,t} = T_t*P_{\ast,t}*Z_t'*F_{\ast,t}^{-1}
|
||||
% v_t = y_t - Z_t*a_t
|
||||
% L_{\ast,t} = T_t - K_{\ast,t}*Z_t
|
||||
% F_{\ast,t} = Z_t*P_{\ast,t}*Z_t' + H_t
|
||||
global bayestopt_ options_
|
||||
|
||||
mf = bayestopt_.mf;
|
||||
pp = size(Y,1);
|
||||
mm = size(T,1);
|
||||
smpl = size(Y,2);
|
||||
a = zeros(mm,1);
|
||||
QQ = R*Q*transpose(R);
|
||||
t = 0;
|
||||
lik = zeros(smpl+1,1);
|
||||
lik(smpl+1) = smpl*pp*log(2*pi); %% the constant of minus two times the log-likelihood
|
||||
notsteady = 1;
|
||||
crit = options_.kalman_tol;
|
||||
crit1 = 1.e-6;
|
||||
newRank = rank(Pinf,crit1);
|
||||
icc=0;
|
||||
while newRank & t < smpl
|
||||
t = t+1;
|
||||
for i=1:pp
|
||||
v(i) = Y(i,t)-a(mf(i))-trend(i,t);
|
||||
Fstar = Pstar(mf(i),mf(i));
|
||||
Finf = Pinf(mf(i),mf(i));
|
||||
Kstar = Pstar(:,mf(i));
|
||||
if Finf > crit & newRank, %added newRank criterion
|
||||
icc=icc+1;
|
||||
Kinf = Pinf(:,mf(i));
|
||||
a = a + Kinf*v(i)/Finf;
|
||||
Pstar = Pstar + Kinf*transpose(Kinf)*Fstar/(Finf*Finf) - ...
|
||||
(Kstar*transpose(Kinf)+Kinf*transpose(Kstar))/Finf;
|
||||
Pinf = Pinf - Kinf*transpose(Kinf)/Finf;
|
||||
lik(t) = lik(t) + log(Finf);
|
||||
% start new termination criterion for DKF
|
||||
if ~isempty(options_.diffuse_d),
|
||||
newRank = (icc<options_.diffuse_d);
|
||||
%if newRank & any(diag(Pinf(mf,mf))>crit)==0; % M. Ratto this line is BUGGY
|
||||
if newRank & (any(diag(Pinf(mf,mf))>crit)==0 & rank(Pinf,crit1)==0);
|
||||
options_.diffuse_d = icc;
|
||||
newRank=0;
|
||||
disp('WARNING: Change in OPTIONS_.DIFFUSE_D in univariate DKF')
|
||||
disp(['new OPTIONS_.DIFFUSE_D = ',int2str(icc)])
|
||||
disp('You may have to reset the optimisation')
|
||||
end
|
||||
else
|
||||
%newRank = any(diag(Pinf(mf,mf))>crit); % M. Ratto this line is BUGGY
|
||||
newRank = (any(diag(Pinf(mf,mf))>crit) | rank(Pinf,crit1));
|
||||
if newRank==0,
|
||||
P0= T*Pinf*transpose(T);
|
||||
%newRank = any(diag(P0(mf,mf))>crit); % M. Ratto this line is BUGGY
|
||||
newRank = (any(diag(P0(mf,mf))>crit) | rank(P0,crit1)); % M. Ratto 11/10/2005
|
||||
if newRank==0,
|
||||
options_.diffuse_d = icc;
|
||||
end
|
||||
end
|
||||
end,
|
||||
% end new termination and checks for DKF
|
||||
elseif Fstar > crit
|
||||
%% Note that : (1) rank(Pinf)=0 implies that Finf = 0, (2) outside this loop (when for some i and t the condition
|
||||
%% rank(Pinf)=0 is satisfied we have P = Pstar and F = Fstar and (3) Finf = 0 does not imply that
|
||||
%% rank(Pinf)=0. [stéphane,11-03-2004].
|
||||
%if rank(Pinf,crit) == 0
|
||||
% the likelihood terms should alwasy be cumulated, not only
|
||||
% when Pinf=0, otherwise the lik would depend on the ordering
|
||||
% of observed variables
|
||||
% presample options can be used to ignore initial time points
|
||||
lik(t) = lik(t) + log(Fstar) + v(i)*v(i)/Fstar;
|
||||
%end
|
||||
a = a + Kstar*v(i)/Fstar;
|
||||
Pstar = Pstar - Kstar*transpose(Kstar)/Fstar;
|
||||
else
|
||||
%disp(['zero F term in DKF for observed ',int2str(i),' ',num2str(Fstar)])
|
||||
end
|
||||
end
|
||||
% if all(abs(Pinf(:))<crit),
|
||||
% oldRank = 0;
|
||||
% else
|
||||
% oldRank = rank(Pinf,crit);
|
||||
% end
|
||||
if newRank,
|
||||
oldRank = rank(Pinf,crit1);
|
||||
else
|
||||
oldRank = 0;
|
||||
end
|
||||
a = T*a;
|
||||
Pstar = T*Pstar*transpose(T)+QQ;
|
||||
Pinf = T*Pinf*transpose(T);
|
||||
% if all(abs(Pinf(:))<crit),
|
||||
% newRank = 0;
|
||||
% else
|
||||
% newRank = rank(Pinf,crit);
|
||||
% end
|
||||
if newRank,
|
||||
newRank = rank(Pinf,crit1); % new crit1 is used
|
||||
end
|
||||
if oldRank ~= newRank
|
||||
disp('DiffuseLiklihood3 :: T does influence the rank of Pinf!')
|
||||
end
|
||||
end
|
||||
if t == smpl
|
||||
error(['There isn''t enough information to estimate the initial' ...
|
||||
' conditions of the nonstationary variables']);
|
||||
end
|
||||
while notsteady & t < smpl
|
||||
t = t+1;
|
||||
oldP = Pstar;
|
||||
for i=1:pp
|
||||
v(i) = Y(i,t) - a(mf(i)) - trend(i,t);
|
||||
Fi = Pstar(mf(i),mf(i));
|
||||
if Fi > crit
|
||||
Ki = Pstar(:,mf(i));
|
||||
a = a + Ki*v(i)/Fi;
|
||||
Pstar = Pstar - Ki*transpose(Ki)/Fi;
|
||||
lik(t) = lik(t) + log(Fi) + v(i)*v(i)/Fi;
|
||||
else
|
||||
%disp(['zero F term for observed ',int2str(i),' ',num2str(Fi)])
|
||||
end
|
||||
end
|
||||
a = T*a;
|
||||
Pstar = T*Pstar*transpose(T) + QQ;
|
||||
notsteady = ~(max(max(abs(Pstar-oldP)))<crit);
|
||||
end
|
||||
while t < smpl
|
||||
t = t+1;
|
||||
Pstar = oldP;
|
||||
for i=1:pp
|
||||
v(i) = Y(i,t) - a(mf(i)) - trend(i,t);
|
||||
Fi = Pstar(mf(i),mf(i));
|
||||
if Fi > crit
|
||||
Ki = Pstar(:,mf(i));
|
||||
a = a + Ki*v(i)/Fi;
|
||||
Pstar = Pstar - Ki*transpose(Ki)/Fi;
|
||||
lik(t) = lik(t) + log(Fi) + v(i)*v(i)/Fi;
|
||||
else
|
||||
%disp(['zero F term for observed ',int2str(i),' ',num2str(Fi)])
|
||||
end
|
||||
end
|
||||
a = T*a;
|
||||
end
|
||||
|
||||
LIK = .5*(sum(lik(start:end))-(start-1)*lik(smpl+1)/smpl);
|
||||
|
||||
|
|
|
@ -1,159 +1,162 @@
|
|||
function LIK = DiffuseLikelihoodH1(T,R,Q,H,Pinf,Pstar,Y,trend,start)
|
||||
% stephane.adjemian@cepremap.cnrs.fr [07-19-2004]
|
||||
%
|
||||
% See "Filtering and Smoothing of State Vector for Diffuse State Space
|
||||
% Models", S.J. Koopman and J. Durbin (2003, in Journal of Time Series
|
||||
% Analysis, vol. 24(1), pp. 85-98).
|
||||
%
|
||||
% THE PROBLEM:
|
||||
%
|
||||
% y_t = Z_t * \alpha_t + \varepsilon_t
|
||||
% \alpha_{t+1} = T_t * \alpha_t + R_t * \eta_t
|
||||
%
|
||||
% with:
|
||||
%
|
||||
% \alpha_1 = a + A*\delta + R_0*\eta_0
|
||||
%
|
||||
% m*q matrix A and m*(m-q) matrix R_0 are selection matrices (their
|
||||
% columns constitue all the columns of the m*m identity matrix) so that
|
||||
%
|
||||
% A'*R_0 = 0 and A'*\alpha_1 = \delta
|
||||
%
|
||||
% We assume that the vector \delta is distributed as a N(0,\kappa*I_q)
|
||||
% for a given \kappa > 0. So that the expectation of \alpha_1 is a and
|
||||
% its variance is P, with
|
||||
%
|
||||
% P = \kappa*P_{\infty} + P_{\star}
|
||||
%
|
||||
% P_{\infty} = A*A'
|
||||
% P_{\star} = R_0*Q_0*R_0'
|
||||
%
|
||||
% P_{\infty} is a m*m diagonal matrix with q ones and m-q zeros.
|
||||
%
|
||||
%
|
||||
% and where:
|
||||
%
|
||||
% y_t is a pp*1 vector
|
||||
% \alpha_t is a mm*1 vector
|
||||
% \varepsilon_t is a pp*1 multivariate random variable (iid N(0,H_t))
|
||||
% \eta_t is a rr*1 multivariate random variable (iid N(0,Q_t))
|
||||
% a_1 is a mm*1 vector
|
||||
%
|
||||
% Z_t is a pp*mm matrix
|
||||
% T_t is a mm*mm matrix
|
||||
% H_t is a pp*pp matrix
|
||||
% R_t is a mm*rr matrix
|
||||
% Q_t is a rr*rr matrix
|
||||
% P_1 is a mm*mm matrix
|
||||
%
|
||||
%
|
||||
% FILTERING EQUATIONS:
|
||||
%
|
||||
% v_t = y_t - Z_t* a_t
|
||||
% F_t = Z_t * P_t * Z_t' + H_t
|
||||
% K_t = T_t * P_t * Z_t' * F_t^{-1}
|
||||
% L_t = T_t - K_t * Z_t
|
||||
% a_{t+1} = T_t * a_t + K_t * v_t
|
||||
% P_{t+1} = T_t * P_t * L_t' + R_t*Q_t*R_t'
|
||||
%
|
||||
%
|
||||
% DIFFUSE FILTERING EQUATIONS:
|
||||
%
|
||||
% a_{t+1} = T_t*a_t + K_{\infty,t}v_t
|
||||
% P_{\infty,t+1} = T_t*P_{\infty,t}*L_{\infty,t}'
|
||||
% P_{\ast,t+1} = T_t*P_{\ast,t}*L_{\infty,t}' - K_{\infty,t}*F_{\infty,t}*K_{\ast,t}' + R_t*Q_t*R_t'
|
||||
% K_{\infty,t} = T_t*P_{\infty,t}*Z_t'*F_{\infty,t}^{-1}
|
||||
% v_t = y_t - Z_t*a_t
|
||||
% L_{\infty,t} = T_t - K_{\infty,t}*Z_t
|
||||
% F_{\infty,t} = Z_t*P_{\infty,t}*Z_t'
|
||||
% K_{\ast,t} = (T_t*P_{\ast,t}*Z_t' + K_{\infty,t}*F_{\ast,t})*F_{\infty,t}^{-1}
|
||||
% F_{\ast,t} = Z_t*P_{\ast,t}*Z_t' + H_t
|
||||
%
|
||||
% Matrix Finf is assumed to be non singular. If this is not the case we have
|
||||
% to switch to another algorithm (NewAlg=3).
|
||||
%
|
||||
% start = options_.presample
|
||||
global bayestopt_ options_
|
||||
|
||||
mf = bayestopt_.mf;
|
||||
smpl = size(Y,2);
|
||||
mm = size(T,2);
|
||||
pp = size(Y,1);
|
||||
a = zeros(mm,1);
|
||||
dF = 1;
|
||||
QQ = R*Q*transpose(R);
|
||||
t = 0;
|
||||
lik = zeros(smpl+1,1);
|
||||
LIK = Inf;
|
||||
lik(smpl+1) = smpl*pp*log(2*pi);
|
||||
notsteady = 1;
|
||||
crit = options_.kalman_tol;
|
||||
reste = 0;
|
||||
while rank(Pinf,crit) & t < smpl
|
||||
t = t+1;
|
||||
v = Y(:,t)-a(mf)-trend(:,t);
|
||||
Finf = Pinf(mf,mf);
|
||||
if rcond(Finf) < crit
|
||||
if ~all(abs(Finf(:))<crit)
|
||||
return
|
||||
else
|
||||
iFstar = inv(Pstar(mf,mf)+H);
|
||||
dFstar = det(Pstar(mf,mf)+H);
|
||||
Kstar = Pstar(:,mf)*iFstar;
|
||||
lik(t) = log(dFstar) + transpose(v)*iFstar*v;
|
||||
Pinf = T*Pinf*transpose(T);
|
||||
Pstar = T*(Pstar-Pstar(:,mf)*transpose(Kstar))*transpose(T)+QQ;
|
||||
a = T*(a+Kstar*v);
|
||||
end
|
||||
else
|
||||
lik(t) = log(det(Finf));
|
||||
iFinf = inv(Finf);
|
||||
Kinf = Pinf(:,mf)*iFinf; %% premultiplication by the transition matrix T is removed (stephane)
|
||||
Fstar = Pstar(mf,mf)+H;
|
||||
Kstar = (Pstar(:,mf)-Kinf*Fstar)*iFinf; %% premultiplication by the transition matrix T is removed (stephane)
|
||||
Pstar = T*(Pstar-Pstar(:,mf)*transpose(Kinf)-Pinf(:,mf)*transpose(Kstar))*transpose(T)+QQ;
|
||||
Pinf = T*(Pinf-Pinf(:,mf)*transpose(Kinf))*transpose(T);
|
||||
a = T*(a+Kinf*v);
|
||||
end
|
||||
end
|
||||
if t == smpl
|
||||
error(['There isn''t enough information to estimate the initial' ...
|
||||
' conditions of the nonstationary variables']);
|
||||
end
|
||||
F_singular = 1;
|
||||
while notsteady & t < smpl
|
||||
t = t+1;
|
||||
v = Y(:,t)-a(mf)-trend(:,t);
|
||||
F = Pstar(mf,mf)+H;
|
||||
oldPstar = Pstar;
|
||||
dF = det(F);
|
||||
if rcond(F) < crit
|
||||
if ~all(abs(F(:))<crit)
|
||||
return
|
||||
else
|
||||
a = T*a;
|
||||
Pstar = T*Pstar*transpose(T)+QQ;
|
||||
end
|
||||
else
|
||||
F_singular = 0;
|
||||
iF = inv(F);
|
||||
lik(t) = log(dF)+transpose(v)*iF*v;
|
||||
K = Pstar(:,mf)*iF; %% premultiplication by the transition matrix T is removed (stephane)
|
||||
a = T*(a+K*v); %% --> factorization of the transition matrix...
|
||||
Pstar = T*(Pstar-Pstar(:,mf)*iF*Pstar(mf,:))*transpose(T)+QQ; %% ... idem (stephane)
|
||||
end
|
||||
notsteady = ~(max(max(abs(Pstar-oldPstar)))<crit);
|
||||
end
|
||||
if F_singular == 1
|
||||
error(['The variance of the forecast error remains singular until the' ...
|
||||
'end of the sample'])
|
||||
end
|
||||
reste = smpl-t;
|
||||
while t < smpl
|
||||
t = t+1;
|
||||
v = Y(:,t)-a(mf)-trend(:,t);
|
||||
a = T*(a+K*v);
|
||||
lik(t) = transpose(v)*iF*v;
|
||||
end
|
||||
lik(t) = lik(t) + reste*log(dF);
|
||||
LIK = .5*(sum(lik(start:end))-(start-1)*lik(smpl+1)/smpl);% Minus the log-likelihood.
|
||||
function [LIK, lik] = DiffuseLikelihoodH1(T,R,Q,H,Pinf,Pstar,Y,trend,start)
|
||||
% M. Ratto added lik in output
|
||||
% stephane.adjemian@cepremap.cnrs.fr [07-19-2004]
|
||||
%
|
||||
% See "Filtering and Smoothing of State Vector for Diffuse State Space
|
||||
% Models", S.J. Koopman and J. Durbin (2003, in Journal of Time Series
|
||||
% Analysis, vol. 24(1), pp. 85-98).
|
||||
%
|
||||
% THE PROBLEM:
|
||||
%
|
||||
% y_t = Z_t * \alpha_t + \varepsilon_t
|
||||
% \alpha_{t+1} = T_t * \alpha_t + R_t * \eta_t
|
||||
%
|
||||
% with:
|
||||
%
|
||||
% \alpha_1 = a + A*\delta + R_0*\eta_0
|
||||
%
|
||||
% m*q matrix A and m*(m-q) matrix R_0 are selection matrices (their
|
||||
% columns constitue all the columns of the m*m identity matrix) so that
|
||||
%
|
||||
% A'*R_0 = 0 and A'*\alpha_1 = \delta
|
||||
%
|
||||
% We assume that the vector \delta is distributed as a N(0,\kappa*I_q)
|
||||
% for a given \kappa > 0. So that the expectation of \alpha_1 is a and
|
||||
% its variance is P, with
|
||||
%
|
||||
% P = \kappa*P_{\infty} + P_{\star}
|
||||
%
|
||||
% P_{\infty} = A*A'
|
||||
% P_{\star} = R_0*Q_0*R_0'
|
||||
%
|
||||
% P_{\infty} is a m*m diagonal matrix with q ones and m-q zeros.
|
||||
%
|
||||
%
|
||||
% and where:
|
||||
%
|
||||
% y_t is a pp*1 vector
|
||||
% \alpha_t is a mm*1 vector
|
||||
% \varepsilon_t is a pp*1 multivariate random variable (iid N(0,H_t))
|
||||
% \eta_t is a rr*1 multivariate random variable (iid N(0,Q_t))
|
||||
% a_1 is a mm*1 vector
|
||||
%
|
||||
% Z_t is a pp*mm matrix
|
||||
% T_t is a mm*mm matrix
|
||||
% H_t is a pp*pp matrix
|
||||
% R_t is a mm*rr matrix
|
||||
% Q_t is a rr*rr matrix
|
||||
% P_1 is a mm*mm matrix
|
||||
%
|
||||
%
|
||||
% FILTERING EQUATIONS:
|
||||
%
|
||||
% v_t = y_t - Z_t* a_t
|
||||
% F_t = Z_t * P_t * Z_t' + H_t
|
||||
% K_t = T_t * P_t * Z_t' * F_t^{-1}
|
||||
% L_t = T_t - K_t * Z_t
|
||||
% a_{t+1} = T_t * a_t + K_t * v_t
|
||||
% P_{t+1} = T_t * P_t * L_t' + R_t*Q_t*R_t'
|
||||
%
|
||||
%
|
||||
% DIFFUSE FILTERING EQUATIONS:
|
||||
%
|
||||
% a_{t+1} = T_t*a_t + K_{\infty,t}v_t
|
||||
% P_{\infty,t+1} = T_t*P_{\infty,t}*L_{\infty,t}'
|
||||
% P_{\ast,t+1} = T_t*P_{\ast,t}*L_{\infty,t}' - K_{\infty,t}*F_{\infty,t}*K_{\ast,t}' + R_t*Q_t*R_t'
|
||||
% K_{\infty,t} = T_t*P_{\infty,t}*Z_t'*F_{\infty,t}^{-1}
|
||||
% v_t = y_t - Z_t*a_t
|
||||
% L_{\infty,t} = T_t - K_{\infty,t}*Z_t
|
||||
% F_{\infty,t} = Z_t*P_{\infty,t}*Z_t'
|
||||
% K_{\ast,t} = (T_t*P_{\ast,t}*Z_t' + K_{\infty,t}*F_{\ast,t})*F_{\infty,t}^{-1}
|
||||
% F_{\ast,t} = Z_t*P_{\ast,t}*Z_t' + H_t
|
||||
%
|
||||
% Matrix Finf is assumed to be non singular. If this is not the case we have
|
||||
% to switch to another algorithm (NewAlg=3).
|
||||
%
|
||||
% start = options_.presample
|
||||
global bayestopt_ options_
|
||||
|
||||
mf = bayestopt_.mf;
|
||||
smpl = size(Y,2);
|
||||
mm = size(T,2);
|
||||
pp = size(Y,1);
|
||||
a = zeros(mm,1);
|
||||
dF = 1;
|
||||
QQ = R*Q*transpose(R);
|
||||
t = 0;
|
||||
lik = zeros(smpl+1,1);
|
||||
LIK = Inf;
|
||||
lik(smpl+1) = smpl*pp*log(2*pi);
|
||||
notsteady = 1;
|
||||
crit = options_.kalman_tol;
|
||||
reste = 0;
|
||||
while rank(Pinf,crit) & t < smpl
|
||||
t = t+1;
|
||||
v = Y(:,t)-a(mf)-trend(:,t);
|
||||
Finf = Pinf(mf,mf);
|
||||
if rcond(Finf) < crit
|
||||
if ~all(abs(Finf(:))<crit)
|
||||
return
|
||||
else
|
||||
iFstar = inv(Pstar(mf,mf)+H);
|
||||
dFstar = det(Pstar(mf,mf)+H);
|
||||
Kstar = Pstar(:,mf)*iFstar;
|
||||
lik(t) = log(dFstar) + transpose(v)*iFstar*v;
|
||||
Pinf = T*Pinf*transpose(T);
|
||||
Pstar = T*(Pstar-Pstar(:,mf)*transpose(Kstar))*transpose(T)+QQ;
|
||||
a = T*(a+Kstar*v);
|
||||
end
|
||||
else
|
||||
lik(t) = log(det(Finf));
|
||||
iFinf = inv(Finf);
|
||||
Kinf = Pinf(:,mf)*iFinf; %% premultiplication by the transition matrix T is removed (stephane)
|
||||
Fstar = Pstar(mf,mf)+H;
|
||||
Kstar = (Pstar(:,mf)-Kinf*Fstar)*iFinf; %% premultiplication by the transition matrix T is removed (stephane)
|
||||
Pstar = T*(Pstar-Pstar(:,mf)*transpose(Kinf)-Pinf(:,mf)*transpose(Kstar))*transpose(T)+QQ;
|
||||
Pinf = T*(Pinf-Pinf(:,mf)*transpose(Kinf))*transpose(T);
|
||||
a = T*(a+Kinf*v);
|
||||
end
|
||||
end
|
||||
if t == smpl
|
||||
error(['There isn''t enough information to estimate the initial' ...
|
||||
' conditions of the nonstationary variables']);
|
||||
end
|
||||
F_singular = 1;
|
||||
while notsteady & t < smpl
|
||||
t = t+1;
|
||||
v = Y(:,t)-a(mf)-trend(:,t);
|
||||
F = Pstar(mf,mf)+H;
|
||||
oldPstar = Pstar;
|
||||
dF = det(F);
|
||||
if rcond(F) < crit
|
||||
if ~all(abs(F(:))<crit)
|
||||
return
|
||||
else
|
||||
a = T*a;
|
||||
Pstar = T*Pstar*transpose(T)+QQ;
|
||||
end
|
||||
else
|
||||
F_singular = 0;
|
||||
iF = inv(F);
|
||||
lik(t) = log(dF)+transpose(v)*iF*v;
|
||||
K = Pstar(:,mf)*iF; %% premultiplication by the transition matrix T is removed (stephane)
|
||||
a = T*(a+K*v); %% --> factorization of the transition matrix...
|
||||
Pstar = T*(Pstar-Pstar(:,mf)*iF*Pstar(mf,:))*transpose(T)+QQ; %% ... idem (stephane)
|
||||
end
|
||||
notsteady = ~(max(max(abs(Pstar-oldPstar)))<crit);
|
||||
end
|
||||
if F_singular == 1
|
||||
error(['The variance of the forecast error remains singular until the' ...
|
||||
'end of the sample'])
|
||||
end
|
||||
reste = smpl-t;
|
||||
while t < smpl
|
||||
t = t+1;
|
||||
v = Y(:,t)-a(mf)-trend(:,t);
|
||||
a = T*(a+K*v);
|
||||
lik(t) = transpose(v)*iF*v;
|
||||
end
|
||||
lik(t) = lik(t) + reste*log(dF);
|
||||
LIK = .5*(sum(lik(start:end))-(start-1)*lik(smpl+1)/smpl);% Minus the
|
||||
% log-likelihood.
|
||||
|
||||
|
|
|
@ -1,156 +1,204 @@
|
|||
function LIK = DiffuseLikelihoodH3(T,R,Q,H,Pinf,Pstar,Y,trend,start)
|
||||
% stephane.adjemian@cepremap.cnrs.fr [07-19-2004]
|
||||
%
|
||||
% See "Filtering and Smoothing of State Vector for Diffuse State Space
|
||||
% Models", S.J. Koopman and J. Durbin (2003, in Journal of Time Series
|
||||
% Analysis, vol. 24(1), pp. 85-98).
|
||||
%
|
||||
% Case where F_{\infty,t} is singular ==> Univariate treatment of multivariate
|
||||
% time series.
|
||||
%
|
||||
% THE PROBLEM:
|
||||
%
|
||||
% y_t = Z_t * \alpha_t + \varepsilon_t
|
||||
% \alpha_{t+1} = T_t * \alpha_t + R_t * \eta_t
|
||||
%
|
||||
% with:
|
||||
%
|
||||
% \alpha_1 = a + A*\delta + R_0*\eta_0
|
||||
%
|
||||
% m*q matrix A and m*(m-q) matrix R_0 are selection matrices (their
|
||||
% columns constitue all the columns of the m*m identity matrix) so that
|
||||
%
|
||||
% A'*R_0 = 0 and A'*\alpha_1 = \delta
|
||||
%
|
||||
% We assume that the vector \delta is distributed as a N(0,\kappa*I_q)
|
||||
% for a given \kappa > 0. So that the expectation of \alpha_1 is a and
|
||||
% its variance is P, with
|
||||
%
|
||||
% P = \kappa*P_{\infty} + P_{\star}
|
||||
%
|
||||
% P_{\infty} = A*A'
|
||||
% P_{\star} = R_0*Q_0*R_0'
|
||||
%
|
||||
% P_{\infty} is a m*m diagonal matrix with q ones and m-q zeros.
|
||||
%
|
||||
%
|
||||
% and where:
|
||||
%
|
||||
% y_t is a pp*1 vector
|
||||
% \alpha_t is a mm*1 vector
|
||||
% \varepsilon_t is a pp*1 multivariate random variable (iid N(0,H_t))
|
||||
% \eta_t is a rr*1 multivariate random variable (iid N(0,Q_t))
|
||||
% a_1 is a mm*1 vector
|
||||
%
|
||||
% Z_t is a pp*mm matrix
|
||||
% T_t is a mm*mm matrix
|
||||
% H_t is a pp*pp matrix
|
||||
% R_t is a mm*rr matrix
|
||||
% Q_t is a rr*rr matrix
|
||||
% P_1 is a mm*mm matrix
|
||||
%
|
||||
%
|
||||
% FILTERING EQUATIONS:
|
||||
%
|
||||
% v_t = y_t - Z_t* a_t
|
||||
% F_t = Z_t * P_t * Z_t' + H_t
|
||||
% K_t = T_t * P_t * Z_t' * F_t^{-1}
|
||||
% L_t = T_t - K_t * Z_t
|
||||
% a_{t+1} = T_t * a_t + K_t * v_t
|
||||
% P_{t+1} = T_t * P_t * L_t' + R_t*Q_t*R_t'
|
||||
%
|
||||
%
|
||||
% DIFFUSE FILTERING EQUATIONS:
|
||||
%
|
||||
% a_{t+1} = T_t*a_t + K_{\ast,t}v_t
|
||||
% P_{\infty,t+1} = T_t*P_{\infty,t}*T_t'
|
||||
% P_{\ast,t+1} = T_t*P_{\ast,t}*L_{\ast,t}' + R_t*Q_t*R_t'
|
||||
% K_{\ast,t} = T_t*P_{\ast,t}*Z_t'*F_{\ast,t}^{-1}
|
||||
% v_t = y_t - Z_t*a_t
|
||||
% L_{\ast,t} = T_t - K_{\ast,t}*Z_t
|
||||
% F_{\ast,t} = Z_t*P_{\ast,t}*Z_t' + H_t
|
||||
global bayestopt_ options_
|
||||
|
||||
mf = bayestopt_.mf;
|
||||
NewAlg = 0;
|
||||
pp = size(Y,1);
|
||||
mm = size(T,1);
|
||||
smpl = size(Y,2);
|
||||
a = zeros(mm,1);
|
||||
QQ = R*Q*transpose(R);
|
||||
t = 0;
|
||||
lik = zeros(smpl+1,1);
|
||||
lik(smpl+1) = smpl*pp*log(2*pi); %% the constant of minus two times the log-likelihood
|
||||
notsteady = 1;
|
||||
crit = options_.kalman_tol;
|
||||
newRank = rank(Pinf,crit);
|
||||
|
||||
while rank(Pinf,crit) & t < smpl %% Matrix Finf is assumed to be zero
|
||||
t = t+1;
|
||||
for i=1:pp
|
||||
v(i) = Y(i,t)-a(mf(i))-trend(i,t);
|
||||
Fstar = Pstar(mf(i),mf(i))+H(i,i);
|
||||
Finf = Pinf(mf(i),mf(i));
|
||||
Kstar = Pstar(:,mf(i));
|
||||
if Finf > crit
|
||||
Kinf = Pinf(:,mf(i));
|
||||
a = a + Kinf*v(i)/Finf;
|
||||
Pstar = Pstar + Kinf*transpose(Kinf)*Fstar/(Finf*Finf) - ...
|
||||
(Kstar*transpose(Kinf)+Kinf*transpose(Kstar))/Finf;
|
||||
Pinf = Pinf - Kinf*transpose(Kinf)/Finf;
|
||||
lik(t) = lik(t) + log(Finf);
|
||||
else %% Note that : (1) rank(Pinf)=0 implies that Finf = 0, (2) outside this loop (when for some i and t the condition
|
||||
%% rank(Pinf)=0 is satisfied we have P = Pstar and F = Fstar and (3) Finf = 0 does not imply that
|
||||
%% rank(Pinf)=0. [stéphane,11-03-2004].
|
||||
if rank(Pinf) == 0
|
||||
lik(t) = lik(t) + log(Fstar) + v(i)*v(i)/Fstar;
|
||||
end
|
||||
a = a + Kstar*v(i)/Fstar;
|
||||
Pstar = Pstar - Kstar*transpose(Kstar)/Fstar;
|
||||
end
|
||||
oldRank = rank(Pinf,crit);
|
||||
a = T*a;
|
||||
Pstar = T*Pstar*transpose(T)+QQ;
|
||||
Pinf = T*Pinf*transpose(T);
|
||||
newRank = rank(Pinf,crit);
|
||||
if oldRank ~= newRank
|
||||
disp('DiffuseLiklihoodH3 :: T does influence the rank of Pinf!')
|
||||
end
|
||||
end
|
||||
end
|
||||
if t == smpl
|
||||
error(['There isn''t enough information to estimate the initial' ...
|
||||
' conditions of the nonstationary variables']);
|
||||
end
|
||||
while notsteady & t < smpl
|
||||
t = t+1;
|
||||
for i=1:pp
|
||||
v(i) = Y(i,t) - a(mf(i)) - trend(i,t);
|
||||
Fi = Pstar(mf(i),mf(i))+H(i,i);
|
||||
if Fi > crit
|
||||
Ki = Pstar(:,mf(i));
|
||||
a = a + Ki*v(i)/Fi;
|
||||
Pstar = Pstar - Ki*transpose(Ki)/Fi;
|
||||
lik(t) = lik(t) + log(Fi) + v(i)*v(i)/Fi;
|
||||
end
|
||||
end
|
||||
oldP = Pstar;
|
||||
a = T*a;
|
||||
Pstar = T*Pstar*transpose(T) + QQ;
|
||||
notsteady = ~(max(max(abs(Pstar-oldP)))<crit);
|
||||
end
|
||||
while t < smpl
|
||||
t = t+1;
|
||||
for i=1:pp
|
||||
v(i) = Y(i,t) - a(mf(i)) - trend(i,t);
|
||||
Fi = Pstar(mf(i),mf(i))+H(i,i);
|
||||
if Fi > crit
|
||||
Ki = Pstar(:,mf(i));
|
||||
a = a + Ki*v(i)/Fi;
|
||||
Pstar = Pstar - Ki*transpose(Ki)/Fi;
|
||||
lik(t) = lik(t) + log(Fi) + v(i)*v(i)/Fi;
|
||||
end
|
||||
end
|
||||
a = T*a;
|
||||
end
|
||||
LIK = .5*(sum(lik(start:end))-(start-1)*lik(smpl+1)/smpl);
|
||||
function [LIK, lik] = DiffuseLikelihoodH3(T,R,Q,H,Pinf,Pstar,Y,trend,start)
|
||||
% M. Ratto added lik in output [October 2005]
|
||||
% changes by M. Ratto
|
||||
% introduced new global variable id_ for termination of DKF
|
||||
% introduced a persistent fmax, in order to keep track the max order of
|
||||
% magnitude of the 'zero' values in Pinf at DKF termination
|
||||
% new icc counter for Finf steps in DKF
|
||||
% new termination for DKF
|
||||
% likelihood terms for Fstar must be cumulated in DKF also when Pinf is non
|
||||
% zero. this bug is fixed.
|
||||
%
|
||||
% stephane.adjemian@cepremap.cnrs.fr [07-19-2004]
|
||||
%
|
||||
% See "Filtering and Smoothing of State Vector for Diffuse State Space
|
||||
% Models", S.J. Koopman and J. Durbin (2003, in Journal of Time Series
|
||||
% Analysis, vol. 24(1), pp. 85-98).
|
||||
%
|
||||
% Case where F_{\infty,t} is singular ==> Univariate treatment of multivariate
|
||||
% time series.
|
||||
%
|
||||
% THE PROBLEM:
|
||||
%
|
||||
% y_t = Z_t * \alpha_t + \varepsilon_t
|
||||
% \alpha_{t+1} = T_t * \alpha_t + R_t * \eta_t
|
||||
%
|
||||
% with:
|
||||
%
|
||||
% \alpha_1 = a + A*\delta + R_0*\eta_0
|
||||
%
|
||||
% m*q matrix A and m*(m-q) matrix R_0 are selection matrices (their
|
||||
% columns constitue all the columns of the m*m identity matrix) so that
|
||||
%
|
||||
% A'*R_0 = 0 and A'*\alpha_1 = \delta
|
||||
%
|
||||
% We assume that the vector \delta is distributed as a N(0,\kappa*I_q)
|
||||
% for a given \kappa > 0. So that the expectation of \alpha_1 is a and
|
||||
% its variance is P, with
|
||||
%
|
||||
% P = \kappa*P_{\infty} + P_{\star}
|
||||
%
|
||||
% P_{\infty} = A*A'
|
||||
% P_{\star} = R_0*Q_0*R_0'
|
||||
%
|
||||
% P_{\infty} is a m*m diagonal matrix with q ones and m-q zeros.
|
||||
%
|
||||
%
|
||||
% and where:
|
||||
%
|
||||
% y_t is a pp*1 vector
|
||||
% \alpha_t is a mm*1 vector
|
||||
% \varepsilon_t is a pp*1 multivariate random variable (iid N(0,H_t))
|
||||
% \eta_t is a rr*1 multivariate random variable (iid N(0,Q_t))
|
||||
% a_1 is a mm*1 vector
|
||||
%
|
||||
% Z_t is a pp*mm matrix
|
||||
% T_t is a mm*mm matrix
|
||||
% H_t is a pp*pp matrix
|
||||
% R_t is a mm*rr matrix
|
||||
% Q_t is a rr*rr matrix
|
||||
% P_1 is a mm*mm matrix
|
||||
%
|
||||
%
|
||||
% FILTERING EQUATIONS:
|
||||
%
|
||||
% v_t = y_t - Z_t* a_t
|
||||
% F_t = Z_t * P_t * Z_t' + H_t
|
||||
% K_t = T_t * P_t * Z_t' * F_t^{-1}
|
||||
% L_t = T_t - K_t * Z_t
|
||||
% a_{t+1} = T_t * a_t + K_t * v_t
|
||||
% P_{t+1} = T_t * P_t * L_t' + R_t*Q_t*R_t'
|
||||
%
|
||||
%
|
||||
% DIFFUSE FILTERING EQUATIONS:
|
||||
%
|
||||
% a_{t+1} = T_t*a_t + K_{\ast,t}v_t
|
||||
% P_{\infty,t+1} = T_t*P_{\infty,t}*T_t'
|
||||
% P_{\ast,t+1} = T_t*P_{\ast,t}*L_{\ast,t}' + R_t*Q_t*R_t'
|
||||
% K_{\ast,t} = T_t*P_{\ast,t}*Z_t'*F_{\ast,t}^{-1}
|
||||
% v_t = y_t - Z_t*a_t
|
||||
% L_{\ast,t} = T_t - K_{\ast,t}*Z_t
|
||||
% F_{\ast,t} = Z_t*P_{\ast,t}*Z_t' + H_t
|
||||
global bayestopt_ options_
|
||||
|
||||
mf = bayestopt_.mf;
|
||||
pp = size(Y,1);
|
||||
mm = size(T,1);
|
||||
smpl = size(Y,2);
|
||||
a = zeros(mm,1);
|
||||
QQ = R*Q*transpose(R);
|
||||
t = 0;
|
||||
lik = zeros(smpl+1,1);
|
||||
lik(smpl+1) = smpl*pp*log(2*pi); %% the constant of minus two times the log-likelihood
|
||||
notsteady = 1;
|
||||
crit = options_.kalman_tol;
|
||||
crit1 = 1.e-6;
|
||||
newRank = rank(Pinf,crit1);
|
||||
icc = 0;
|
||||
while newRank & t < smpl %% Matrix Finf is assumed to be zero
|
||||
t = t+1;
|
||||
for i=1:pp
|
||||
v(i) = Y(i,t)-a(mf(i))-trend(i,t);
|
||||
Fstar = Pstar(mf(i),mf(i))+H(i,i);
|
||||
Finf = Pinf(mf(i),mf(i));
|
||||
Kstar = Pstar(:,mf(i));
|
||||
if Finf > crit & newRank
|
||||
icc = icc + 1;
|
||||
Kinf = Pinf(:,mf(i));
|
||||
a = a + Kinf*v(i)/Finf;
|
||||
Pstar = Pstar + Kinf*transpose(Kinf)*Fstar/(Finf*Finf) - ...
|
||||
(Kstar*transpose(Kinf)+Kinf*transpose(Kstar))/Finf;
|
||||
Pinf = Pinf - Kinf*transpose(Kinf)/Finf;
|
||||
lik(t) = lik(t) + log(Finf);
|
||||
% start new termination criterion for DKF
|
||||
if ~isempty(options_.diffuse_d),
|
||||
newRank = (icc<options_.diffuse_d);
|
||||
%if newRank & any(diag(Pinf(mf,mf))>crit)==0; % M. Ratto this line is BUGGY
|
||||
if newRank & (any(diag(Pinf(mf,mf))>crit)==0 & rank(Pinf,crit1)==0);
|
||||
options_.diffuse_d = icc;
|
||||
newRank=0;
|
||||
disp('WARNING: Change in OPTIONS_.DIFFUSE_D in univariate DKF')
|
||||
disp(['new OPTIONS_.DIFFUSE_D = ',int2str(icc)])
|
||||
disp('You may have to reset the optimisation')
|
||||
end
|
||||
else
|
||||
%newRank = any(diag(Pinf(mf,mf))>crit); % M. Ratto this line is BUGGY
|
||||
newRank = (any(diag(Pinf(mf,mf))>crit) | rank(Pinf,crit1));
|
||||
if newRank==0,
|
||||
P0= T*Pinf*transpose(T);
|
||||
%newRank = any(diag(P0(mf,mf))>crit); % M. Ratto this line is BUGGY
|
||||
newRank = (any(diag(P0(mf,mf))>crit) | rank(P0,crit1)); % M. Ratto 10 Oct 2005
|
||||
if newRank==0,
|
||||
options_.diffuse_d = icc;
|
||||
end
|
||||
end
|
||||
end,
|
||||
% end new termination and checks for DKF and fmax
|
||||
elseif Finf > crit
|
||||
%% Note that : (1) rank(Pinf)=0 implies that Finf = 0, (2) outside this loop (when for some i and t the condition
|
||||
%% rank(Pinf)=0 is satisfied we have P = Pstar and F = Fstar and (3) Finf = 0 does not imply that
|
||||
%% rank(Pinf)=0. [stéphane,11-03-2004].
|
||||
%if rank(Pinf) == 0
|
||||
% the likelihood terms should alwasy be cumulated, not only
|
||||
% when Pinf=0, otherwise the lik would depend on the ordering
|
||||
% of observed variables
|
||||
lik(t) = lik(t) + log(Fstar) + v(i)*v(i)/Fstar;
|
||||
%end
|
||||
a = a + Kstar*v(i)/Fstar;
|
||||
Pstar = Pstar - Kstar*transpose(Kstar)/Fstar;
|
||||
else
|
||||
% disp(['zero F term in DKF for observed ',int2str(i),' ',num2str(Fi)])
|
||||
end
|
||||
end
|
||||
if newRank
|
||||
oldRank = rank(Pinf,crit1);
|
||||
else
|
||||
oldRank = 0;
|
||||
end
|
||||
a = T*a;
|
||||
Pstar = T*Pstar*transpose(T)+QQ;
|
||||
Pinf = T*Pinf*transpose(T);
|
||||
if newRank
|
||||
newRank = rank(Pinf,crit1);
|
||||
end
|
||||
if oldRank ~= newRank
|
||||
disp('DiffuseLiklihoodH3 :: T does influence the rank of Pinf!')
|
||||
end
|
||||
end
|
||||
if t == smpl
|
||||
error(['There isn''t enough information to estimate the initial' ...
|
||||
' conditions of the nonstationary variables']);
|
||||
end
|
||||
while notsteady & t < smpl
|
||||
t = t+1;
|
||||
for i=1:pp
|
||||
v(i) = Y(i,t) - a(mf(i)) - trend(i,t);
|
||||
Fi = Pstar(mf(i),mf(i))+H(i,i);
|
||||
if Fi > crit
|
||||
Ki = Pstar(:,mf(i));
|
||||
a = a + Ki*v(i)/Fi;
|
||||
Pstar = Pstar - Ki*transpose(Ki)/Fi;
|
||||
lik(t) = lik(t) + log(Fi) + v(i)*v(i)/Fi;
|
||||
end
|
||||
end
|
||||
oldP = Pstar;
|
||||
a = T*a;
|
||||
Pstar = T*Pstar*transpose(T) + QQ;
|
||||
notsteady = ~(max(max(abs(Pstar-oldP)))<crit);
|
||||
end
|
||||
while t < smpl
|
||||
t = t+1;
|
||||
for i=1:pp
|
||||
v(i) = Y(i,t) - a(mf(i)) - trend(i,t);
|
||||
Fi = Pstar(mf(i),mf(i))+H(i,i);
|
||||
if Fi > crit
|
||||
Ki = Pstar(:,mf(i));
|
||||
a = a + Ki*v(i)/Fi;
|
||||
Pstar = Pstar - Ki*transpose(Ki)/Fi;
|
||||
lik(t) = lik(t) + log(Fi) + v(i)*v(i)/Fi;
|
||||
end
|
||||
end
|
||||
a = T*a;
|
||||
end
|
||||
LIK = .5*(sum(lik(start:end))-(start-1)*lik(smpl+1)/smpl);
|
||||
|
||||
|
|
Loading…
Reference in New Issue