321 lines
10 KiB
Matlab
321 lines
10 KiB
Matlab
function [hessian_mat, gg, htol1, ihh, hh_mat0] = mr_hessian(func,x,hflag,htol0,varargin)
|
|
% [hessian_mat, gg, htol1, ihh, hh_mat0] = mr_hessian(func,x,hflag,htol0,varargin)
|
|
%
|
|
% numerical gradient and Hessian, with 'automatic' check of numerical
|
|
% error
|
|
%
|
|
% adapted from Michel Juillard original rutine hessian.m
|
|
%
|
|
% func = name of the function: func must give two outputs:
|
|
% - the log-likelihood AND the single contributions at times t=1,...,T
|
|
% of the log-likelihood to compute outer product gradient
|
|
% x = parameter values
|
|
% hflag = 0, Hessian computed with outer product gradient, one point
|
|
% increments for partial derivatives in gradients
|
|
% hflag = 1, 'mixed' Hessian: diagonal elements computed with numerical second order derivatives
|
|
% with correlation structure as from outer product gradient;
|
|
% two point evaluation of derivatives for partial derivatives
|
|
% in gradients
|
|
% hflag = 2, full numerical Hessian, computes second order partial derivatives
|
|
% uses Abramowitz and Stegun (1965) formulas 25.3.24 and 25.3.27
|
|
% p. 884.
|
|
% htol0 = 'precision' of increment of function values for numerical
|
|
% derivatives
|
|
%
|
|
% varargin: other parameters of func
|
|
|
|
% Copyright (C) 2004-2008 Dynare Team
|
|
%
|
|
% This file is part of Dynare.
|
|
%
|
|
% Dynare is free software: you can redistribute it and/or modify
|
|
% it under the terms of the GNU General Public License as published by
|
|
% the Free Software Foundation, either version 3 of the License, or
|
|
% (at your option) any later version.
|
|
%
|
|
% Dynare is distributed in the hope that it will be useful,
|
|
% but WITHOUT ANY WARRANTY; without even the implied warranty of
|
|
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
|
|
% GNU General Public License for more details.
|
|
%
|
|
% You should have received a copy of the GNU General Public License
|
|
% along with Dynare. If not, see <http://www.gnu.org/licenses/>.
|
|
|
|
global options_ bayestopt_
|
|
persistent h1 htol
|
|
|
|
gstep_=options_.gstep;
|
|
if isempty(htol), htol = 1.e-4; end
|
|
func = str2func(func);
|
|
[f0, ff0]=feval(func,x,varargin{:});
|
|
n=size(x,1);
|
|
h2=bayestopt_.ub-bayestopt_.lb;
|
|
hmax=bayestopt_.ub-x;
|
|
hmax=min(hmax,x-bayestopt_.lb);
|
|
%h1=max(abs(x),gstep_*ones(n,1))*eps^(1/3);
|
|
%h1=max(abs(x),sqrt(gstep_)*ones(n,1))*eps^(1/6);
|
|
if isempty(h1),
|
|
h1=max(abs(x),sqrt(gstep_)*ones(n,1))*eps^(1/4);
|
|
end
|
|
|
|
h1 = min(h1,0.5.*hmax);
|
|
|
|
if htol0<htol,
|
|
htol=htol0;
|
|
end
|
|
xh1=x;
|
|
f1=zeros(size(f0,1),n);
|
|
f_1=f1;
|
|
ff1=zeros(size(ff0));
|
|
ff_1=ff1;
|
|
|
|
%for i=1:n,
|
|
i=0;
|
|
while i<n,
|
|
i=i+1;
|
|
h10=h1(i);
|
|
hcheck=0;
|
|
dx=[];
|
|
xh1(i)=x(i)+h1(i);
|
|
try
|
|
[fx, ffx]=feval(func,xh1,varargin{:});
|
|
catch
|
|
fx=1.e8;
|
|
end
|
|
it=1;
|
|
dx=(fx-f0);
|
|
ic=0;
|
|
% if abs(dx)>(2*htol),
|
|
% c=mr_nlincon(xh1,varargin{:});
|
|
% while c
|
|
% h1(i)=h1(i)*0.9;
|
|
% xh1(i)=x(i)+h1(i);
|
|
% c=mr_nlincon(xh1,varargin{:});
|
|
% ic=1;
|
|
% end
|
|
% if ic,
|
|
% [fx, ffx]=feval(func,xh1,varargin{:});
|
|
% dx=(fx-f0);
|
|
% end
|
|
% end
|
|
|
|
icount = 0;
|
|
h0=h1(i);
|
|
while (abs(dx(it))<0.5*htol | abs(dx(it))>(2*htol)) & icount<10 & ic==0,
|
|
%while abs(dx(it))<0.5*htol & icount< 10 & ic==0,
|
|
icount=icount+1;
|
|
%if abs(dx(it)) ~= 0,
|
|
if abs(dx(it))<0.5*htol
|
|
if abs(dx(it)) ~= 0,
|
|
h1(i)=min(max(1.e-10,0.3*abs(x(i))), 0.9*htol/abs(dx(it))*h1(i));
|
|
else
|
|
h1(i)=2.1*h1(i);
|
|
end
|
|
h1(i) = min(h1(i),0.5*hmax(i));
|
|
xh1(i)=x(i)+h1(i);
|
|
% c=mr_nlincon(xh1,varargin{:});
|
|
% while c
|
|
% h1(i)=h1(i)*0.9;
|
|
% xh1(i)=x(i)+h1(i);
|
|
% c=mr_nlincon(xh1,varargin{:});
|
|
% ic=1;
|
|
% end
|
|
try
|
|
[fx, ffx]=feval(func,xh1,varargin{:});
|
|
catch
|
|
fx=1.e8;
|
|
end
|
|
end
|
|
if abs(dx(it))>(2*htol),
|
|
h1(i)= htol/abs(dx(it))*h1(i);
|
|
xh1(i)=x(i)+h1(i);
|
|
try
|
|
[fx, ffx]=feval(func,xh1,varargin{:});
|
|
catch
|
|
fx=1.e8;
|
|
end
|
|
while (fx-f0)==0,
|
|
h1(i)= h1(i)*2;
|
|
xh1(i)=x(i)+h1(i);
|
|
[fx, ffx]=feval(func,xh1,varargin{:});
|
|
ic=1;
|
|
end
|
|
end
|
|
it=it+1;
|
|
dx(it)=(fx-f0);
|
|
h0(it)=h1(i);
|
|
if h1(i)<1.e-12*min(1,h2(i)) & h1(i)<0.5*hmax(i),
|
|
ic=1;
|
|
hcheck=1;
|
|
end
|
|
%else
|
|
% h1(i)=1;
|
|
% ic=1;
|
|
%end
|
|
end
|
|
% if (it>2 & dx(1)<10^(log10(htol)/2)) ,
|
|
% [dum, is]=sort(h0);
|
|
% if find(diff(sign(diff(dx(is)))));
|
|
% hcheck=1;
|
|
% end
|
|
% elseif (it>3 & dx(1)>10^(log10(htol)/2)) ,
|
|
% [dum, is]=sort(h0);
|
|
% if find(diff(sign(diff(dx(is(1:end-1))))));
|
|
% hcheck=1;
|
|
% end
|
|
% end
|
|
f1(:,i)=fx;
|
|
ff1=ffx;
|
|
if hflag, % two point based derivatives
|
|
xh1(i)=x(i)-h1(i);
|
|
% c=mr_nlincon(xh1,varargin{:});
|
|
% ic=0;
|
|
% while c
|
|
% h1(i)=h1(i)*0.9;
|
|
% xh1(i)=x(i)-h1(i);
|
|
% c=mr_nlincon(xh1,varargin{:});
|
|
% ic = 1;
|
|
% end
|
|
[fx, ffx]=feval(func,xh1,varargin{:});
|
|
f_1(:,i)=fx;
|
|
ff_1=ffx;
|
|
% if ic,
|
|
% xh1(i)=x(i)+h1(i);
|
|
% [f1(:,i), ff1]=feval(func,xh1,varargin{:});
|
|
% end
|
|
ggh(:,i)=(ff1-ff_1)./(2.*h1(i));
|
|
else
|
|
ggh(:,i)=(ff1-ff0)./h1(i);
|
|
end
|
|
xh1(i)=x(i);
|
|
if hcheck & htol<1,
|
|
htol=min(1,max(min(abs(dx))*2,htol*10));
|
|
h1(i)=h10;
|
|
i=0;
|
|
end
|
|
save hess
|
|
end
|
|
|
|
h_1=h1;
|
|
xh1=x;
|
|
xh_1=xh1;
|
|
|
|
if hflag,
|
|
gg=(f1'-f_1')./(2.*h1);
|
|
else
|
|
gg=(f1'-f0)./h1;
|
|
end
|
|
|
|
if hflag==2,
|
|
gg=(f1'-f_1')./(2.*h1);
|
|
hessian_mat = zeros(size(f0,1),n*n);
|
|
for i=1:n
|
|
if i > 1
|
|
k=[i:n:n*(i-1)];
|
|
hessian_mat(:,(i-1)*n+1:(i-1)*n+i-1)=hessian_mat(:,k);
|
|
end
|
|
hessian_mat(:,(i-1)*n+i)=(f1(:,i)+f_1(:,i)-2*f0)./(h1(i)*h_1(i));
|
|
temp=f1+f_1-f0*ones(1,n);
|
|
for j=i+1:n
|
|
xh1(i)=x(i)+h1(i);
|
|
xh1(j)=x(j)+h_1(j);
|
|
xh_1(i)=x(i)-h1(i);
|
|
xh_1(j)=x(j)-h_1(j);
|
|
%hessian_mat(:,(i-1)*n+j)=-(-feval(func,xh1,varargin{:})-feval(func,xh_1,varargin{:})+temp(:,i)+temp(:,j))./(2*h1(i)*h_1(j));
|
|
%temp1 = feval(func,xh1,varargin{:});
|
|
% c=mr_nlincon(xh1,varargin{:});
|
|
% lam=1;
|
|
% while c,
|
|
% lam=lam*0.9;
|
|
% xh1(i)=x(i)+h1(i)*lam;
|
|
% xh1(j)=x(j)+h_1(j)*lam;
|
|
% %disp( ['hessian warning cross ', num2str(c) ]),
|
|
% c=mr_nlincon(xh1,varargin{:});
|
|
% end
|
|
% temp1 = f0+(feval(func,xh1,varargin{:})-f0)/lam;
|
|
temp1 = feval(func,xh1,varargin{:});
|
|
|
|
% c=mr_nlincon(xh_1,varargin{:});
|
|
% while c,
|
|
% lam=lam*0.9;
|
|
% xh_1(i)=x(i)-h1(i)*lam;
|
|
% xh_1(j)=x(j)-h_1(j)*lam;
|
|
% %disp( ['hessian warning cross ', num2str(c) ]),
|
|
% c=mr_nlincon(xh_1,varargin{:});
|
|
% end
|
|
% temp2 = f0+(feval(func,xh_1,varargin{:})-f0)/lam;
|
|
temp2 = feval(func,xh_1,varargin{:});
|
|
|
|
hessian_mat(:,(i-1)*n+j)=-(-temp1 -temp2+temp(:,i)+temp(:,j))./(2*h1(i)*h_1(j));
|
|
xh1(i)=x(i);
|
|
xh1(j)=x(j);
|
|
xh_1(i)=x(i);
|
|
xh_1(j)=x(j);
|
|
j=j+1;
|
|
save hess
|
|
end
|
|
i=i+1;
|
|
end
|
|
|
|
elseif hflag==1,
|
|
hessian_mat = zeros(size(f0,1),n*n);
|
|
for i=1:n,
|
|
dum = (f1(:,i)+f_1(:,i)-2*f0)./(h1(i)*h_1(i));
|
|
if dum>eps,
|
|
hessian_mat(:,(i-1)*n+i)=dum;
|
|
else
|
|
hessian_mat(:,(i-1)*n+i)=max(eps, gg(i)^2);
|
|
end
|
|
end
|
|
%hessian_mat2=hh_mat(:)';
|
|
end
|
|
|
|
gga=ggh.*kron(ones(size(ff1)),2.*h1'); % re-scaled gradient
|
|
hh_mat=gga'*gga; % rescaled outer product hessian
|
|
hh_mat0=ggh'*ggh; % outer product hessian
|
|
A=diag(2.*h1); % rescaling matrix
|
|
igg=inv(hh_mat); % inverted rescaled outer product hessian
|
|
ihh=A'*igg*A; % inverted outer product hessian
|
|
if hflag>0 & min(eig(reshape(hessian_mat,n,n)))>0,
|
|
hh0 = A*reshape(hessian_mat,n,n)*A'; %rescaled second order derivatives
|
|
hh = reshape(hessian_mat,n,n); %rescaled second order derivatives
|
|
sd0=sqrt(diag(hh0)); %rescaled 'standard errors' using second order derivatives
|
|
sd=sqrt(diag(hh_mat)); %rescaled 'standard errors' using outer product
|
|
hh_mat=hh_mat./(sd*sd').*(sd0*sd0'); %rescaled inverse outer product with 'true' std's
|
|
igg=inv(hh_mat); % rescaled outer product hessian with 'true' std's
|
|
ihh=A'*igg*A; % inverted outer product hessian
|
|
hh_mat0=inv(A)'*hh_mat*inv(A); % outer product hessian with 'true' std's
|
|
sd=sqrt(diag(ihh)); %standard errors
|
|
sdh=sqrt(1./diag(hh)); %diagonal standard errors
|
|
for j=1:length(sd),
|
|
sd0(j,1)=min(bayestopt_.pstdev(j), sd(j)); %prior std
|
|
sd0(j,1)=10^(0.5*(log10(sd0(j,1))+log10(sdh(j,1))));
|
|
%sd0(j,1)=0.5*(sd0(j,1)+sdh(j,1));
|
|
end
|
|
ihh=ihh./(sd*sd').*(sd0*sd0'); %inverse outer product with modified std's
|
|
igg=inv(A)'*ihh*inv(A); % inverted rescaled outer product hessian with modified std's
|
|
hh_mat=inv(igg); % outer product rescaled hessian with modified std's
|
|
hh_mat0=inv(A)'*hh_mat*inv(A); % outer product hessian with modified std's
|
|
% sd0=sqrt(1./diag(hh0)); %rescaled 'standard errors' using second order derivatives
|
|
% sd=sqrt(diag(igg)); %rescaled 'standard errors' using outer product
|
|
% igg=igg./(sd*sd').*(sd0*sd0'); %rescaled inverse outer product with 'true' std's
|
|
% hh_mat=inv(igg); % rescaled outer product hessian with 'true' std's
|
|
% ihh=A'*igg*A; % inverted outer product hessian
|
|
% hh_mat0=inv(A)'*hh_mat*inv(A); % outer product hessian with 'true' std's
|
|
end
|
|
if hflag<2,
|
|
hessian_mat=hh_mat0(:);
|
|
end
|
|
|
|
if isnan(hessian_mat),
|
|
hh_mat0=eye(length(hh_mat0));
|
|
ihh=hh_mat0;
|
|
hessian_mat=hh_mat0(:);
|
|
end
|
|
hh1=h1;
|
|
htol1=htol;
|
|
save hess
|
|
% 11/25/03 SA Created from Hessian_sparse (removed sparse)
|
|
|
|
|