269 lines
11 KiB
Matlab
269 lines
11 KiB
Matlab
function [hessian_mat, gg, htol1, ihh, hh_mat0, hh1, hess_info] = mr_hessian(x,func,penalty,hflag,htol0,hess_info,bounds,prior_std,Save_files,varargin)
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% function [hessian_mat, gg, htol1, ihh, hh_mat0, hh1, hess_info] = mr_hessian(x,func,penalty,hflag,htol0,hess_info,bounds,prior_std,Save_files,varargin)
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% numerical gradient and Hessian, with 'automatic' check of numerical
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% error
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%
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% adapted from Michel Juillard original routine hessian.m
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%
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% Inputs:
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% - x parameter values
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% - func function handle. The function must give two outputs:
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% the log-likelihood AND the single contributions at times t=1,...,T
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% of the log-likelihood to compute outer product gradient
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% - penalty penalty due to error code
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% - hflag 0: Hessian computed with outer product gradient, one point
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% increments for partial derivatives in gradients
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% 1: 'mixed' Hessian: diagonal elements computed with numerical second order derivatives
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% with correlation structure as from outer product gradient;
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% two point evaluation of derivatives for partial derivatives
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% in gradients
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% 2: full numerical Hessian, computes second order partial derivatives
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% uses Abramowitz and Stegun (1965) formulas 25.3.24 and 25.3.27
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% p. 884.
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% - htol0 'precision' of increment of function values for numerical
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% derivatives
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% - hess_info structure storing the step sizes for
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% computation of Hessian
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% - bounds prior bounds of parameters
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% - prior_std prior standard devation of parameters (can be NaN)
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% - Save_files indicator whether files should be saved
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% - varargin other inputs
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% e.g. in dsge_likelihood
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% varargin{1} --> dataset_
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% varargin{2} --> dataset_info
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% varargin{3} --> options_
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% varargin{4} --> M_
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% varargin{5} --> estim_params_
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% varargin{6} --> bayestopt_
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% varargin{7} --> BoundsInfo
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% varargin{8} --> oo_
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%
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% Outputs
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% - hessian_mat hessian
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% - gg Jacobian
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% - htol1 updated 'precision' of increment of function values for numerical
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% derivatives
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% - ihh inverse outer product with modified std's
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% - hh_mat0 outer product hessian with modified std's
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% - hh1 updated hess_info.h1
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% - hess_info structure with updated step length
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% Copyright © 2004-2017 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 <https://www.gnu.org/licenses/>.
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n=size(x,1);
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[f0,~, ff0]=penalty_objective_function(x,func,penalty,varargin{:});
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h2=bounds(:,2)-bounds(:,1);
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hmax=bounds(:,2)-x;
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hmax=min(hmax,x-bounds(:,1));
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if isempty(ff0)
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outer_product_gradient=0;
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else
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outer_product_gradient=1;
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end
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hess_info.h1 = min(hess_info.h1,0.5.*hmax);
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if htol0<hess_info.htol
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hess_info.htol=htol0;
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end
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xh1=x;
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f1=zeros(size(f0,1),n);
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f_1=f1;
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if outer_product_gradient
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ff1=zeros(size(ff0));
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ff_1=ff1;
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ggh=zeros(size(ff0,1),n);
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end
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i=0;
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hhtol=hess_info.htol*ones(n,1);
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while i<n
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i=i+1;
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hess_info.htol=hhtol(i);
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h10=hess_info.h1(i);
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hcheck=0;
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xh1(i)=x(i)+hess_info.h1(i);
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[fx,~,ffx]=penalty_objective_function(xh1,func,penalty,varargin{:});
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it=1;
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dx=(fx-f0);
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ic=0;
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icount = 0;
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h0=hess_info.h1(i);
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while (abs(dx(it))<0.5*hess_info.htol || abs(dx(it))>(3*hess_info.htol)) && icount<10 && ic==0
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icount=icount+1;
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if abs(dx(it))<0.5*hess_info.htol
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if abs(dx(it)) ~= 0
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hess_info.h1(i)=min(max(1.e-10,0.3*abs(x(i))), 0.9*hess_info.htol/abs(dx(it))*hess_info.h1(i));
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else
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hess_info.h1(i)=2.1*hess_info.h1(i);
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end
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hess_info.h1(i) = min(hess_info.h1(i),0.5*hmax(i));
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hess_info.h1(i) = max(hess_info.h1(i),1.e-10);
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xh1(i)=x(i)+hess_info.h1(i);
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[fx,~,ffx]=penalty_objective_function(xh1,func,penalty,varargin{:});
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end
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if abs(dx(it))>(3*hess_info.htol)
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hess_info.h1(i)= hess_info.htol/abs(dx(it))*hess_info.h1(i);
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hess_info.h1(i) = max(hess_info.h1(i),1e-10);
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xh1(i)=x(i)+hess_info.h1(i);
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[fx,~,ffx]=penalty_objective_function(xh1,func,penalty,varargin{:});
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iter=0;
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while (fx-f0)==0 && iter<50
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hess_info.h1(i)= hess_info.h1(i)*2;
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xh1(i)=x(i)+hess_info.h1(i);
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[fx,~,ffx]=penalty_objective_function(xh1,func,penalty,varargin{:});
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ic=1;
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iter=iter+1;
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end
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end
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it=it+1;
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dx(it)=(fx-f0);
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h0(it)=hess_info.h1(i);
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if (hess_info.h1(i)<1.e-12*min(1,h2(i)) && hess_info.h1(i)<0.5*hmax(i))
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ic=1;
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hcheck=1;
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end
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end
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f1(:,i)=fx;
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if outer_product_gradient
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if any(isnan(ffx)) || isempty(ffx)
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ff1=ones(size(ff0)).*fx/length(ff0);
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else
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ff1=ffx;
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end
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end
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xh1(i)=x(i)-hess_info.h1(i);
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[fx,~,ffx]=penalty_objective_function(xh1,func,penalty,varargin{:});
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f_1(:,i)=fx;
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if outer_product_gradient
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if any(isnan(ffx)) || isempty(ffx)
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ff_1=ones(size(ff0)).*fx/length(ff0);
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else
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ff_1=ffx;
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end
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ggh(:,i)=(ff1-ff_1)./(2.*hess_info.h1(i));
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end
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xh1(i)=x(i);
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if hcheck && hess_info.htol<1
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hess_info.htol=min(1,max(min(abs(dx))*2,hess_info.htol*10));
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hess_info.h1(i)=h10;
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hhtol(i) = hess_info.htol;
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i=i-1;
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end
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end
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h_1=hess_info.h1;
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xh1=x;
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xh_1=xh1;
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gg=(f1'-f_1')./(2.*hess_info.h1);
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if outer_product_gradient
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if hflag==2
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% full numerical Hessian
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hessian_mat = zeros(size(f0,1),n*n);
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for i=1:n
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if i > 1
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k=[i:n:n*(i-1)];
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hessian_mat(:,(i-1)*n+1:(i-1)*n+i-1)=hessian_mat(:,k);
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end
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hessian_mat(:,(i-1)*n+i)=(f1(:,i)+f_1(:,i)-2*f0)./(hess_info.h1(i)*h_1(i));
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temp=f1+f_1-f0*ones(1,n);
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for j=i+1:n
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xh1(i)=x(i)+hess_info.h1(i);
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xh1(j)=x(j)+h_1(j);
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xh_1(i)=x(i)-hess_info.h1(i);
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xh_1(j)=x(j)-h_1(j);
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temp1 = penalty_objective_function(xh1,func,penalty,varargin{:});
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temp2 = penalty_objective_function(xh_1,func,penalty,varargin{:});
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hessian_mat(:,(i-1)*n+j)=-(-temp1 -temp2+temp(:,i)+temp(:,j))./(2*hess_info.h1(i)*h_1(j));
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xh1(i)=x(i);
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xh1(j)=x(j);
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xh_1(i)=x(i);
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xh_1(j)=x(j);
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end
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end
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elseif hflag==1
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% full numerical 2nd order derivs only in diagonal
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hessian_mat = zeros(size(f0,1),n*n);
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for i=1:n
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dum = (f1(:,i)+f_1(:,i)-2*f0)./(hess_info.h1(i)*h_1(i));
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hessian_mat(:,(i-1)*n+i)=dum;
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if any(dum<=eps)
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hessian_mat(dum<=eps,(i-1)*n+i)=max(eps, gg(i)^2);
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end
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end
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end
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gga=ggh.*kron(ones(size(ff1)),2.*hess_info.h1'); % re-scaled gradient
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hh_mat=gga'*gga; % rescaled outer product hessian
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hh_mat0=ggh'*ggh; % outer product hessian
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A=diag(2.*hess_info.h1); % rescaling matrix
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% igg=inv(hh_mat); % inverted rescaled outer product hessian
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ihh=A'*(hh_mat\A); % inverted outer product hessian (based on rescaling)
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if hflag>0 && min(eig(reshape(hessian_mat,n,n)))>0
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hh0 = A*reshape(hessian_mat,n,n)*A'; %rescaled second order derivatives
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hh = reshape(hessian_mat,n,n); %second order derivatives
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sd0=sqrt(diag(hh0)); %rescaled 'standard errors' using second order derivatives
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sd=sqrt(diag(hh_mat)); %rescaled 'standard errors' using outer product
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hh_mat=hh_mat./(sd*sd').*(sd0*sd0'); %rescaled inverse outer product with 'true' std's
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ihh=A'*(hh_mat\A); % update inverted outer product hessian with 'true' std's
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sd=sqrt(diag(ihh)); %standard errors
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sdh=sqrt(1./diag(hh)); %diagonal standard errors
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for j=1:length(sd)
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% some heuristic normalizations of the standard errors that
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% avoid numerical issues in outer product
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sd0(j,1)=min(prior_std(j), sd(j)); %prior std
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sd0(j,1)=10^(0.5*(log10(sd0(j,1))+log10(sdh(j,1))));
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end
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inv_A=inv(A);
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ihh=ihh./(sd*sd').*(sd0*sd0'); %inverse outer product with modified std's
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igg=inv_A'*ihh*inv_A; % inverted rescaled outer product hessian with modified std's
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% hh_mat=inv(igg); % outer product rescaled hessian with modified std's
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hh_mat0=inv_A'/igg*inv_A; % outer product hessian with modified std's
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% sd0=sqrt(1./diag(hh0)); %rescaled 'standard errors' using second order derivatives
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% sd=sqrt(diag(igg)); %rescaled 'standard errors' using outer product
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% igg=igg./(sd*sd').*(sd0*sd0'); %rescaled inverse outer product with 'true' std's
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% hh_mat=inv(igg); % rescaled outer product hessian with 'true' std's
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% ihh=A'*igg*A; % inverted outer product hessian
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% hh_mat0=inv(A)'*hh_mat*inv(A); % outer product hessian with 'true' std's
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end
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if hflag<2
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hessian_mat=hh_mat0(:);
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end
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if any(isnan(hessian_mat))
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hh_mat0=eye(length(hh_mat0));
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ihh=hh_mat0;
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hessian_mat=hh_mat0(:);
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end
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hh1=hess_info.h1;
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if Save_files
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save('hess.mat','hessian_mat')
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end
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else
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hessian_mat=[];
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ihh=[];
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hh_mat0 = [];
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hh1 = [];
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end
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htol1=hhtol;
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