358 lines
19 KiB
HTML
358 lines
19 KiB
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<title>Description of mr_hessian</title>
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<div><a href="../index.html">Home</a> > <a href="index.html">.</a> > mr_hessian.m</div>
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<!--<table width="100%"><tr><td align="left"><a href="../index.html"><img alt="<" border="0" src="../left.png"> Master index</a></td>
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<td align="right"><a href="index.html">Index for . <img alt=">" border="0" src="../right.png"></a></td></tr></table>-->
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<h1>mr_hessian
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</h1>
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<h2><a name="_name"></a>PURPOSE <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
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<div class="box"><strong>Copyright (C) 2004 Marco Ratto</strong></div>
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<h2><a name="_synopsis"></a>SYNOPSIS <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
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<div class="box"><strong>function [hessian_mat, gg, htol1, ihh, hh_mat0] = mr_hessian(func,x,hflag,htol0,varargin) </strong></div>
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<h2><a name="_description"></a>DESCRIPTION <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
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<div class="fragment"><pre class="comment"> Copyright (C) 2004 Marco Ratto
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adapted from Michel Juillard original rutine hessian.m
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[hessian_mat, gg, htol1, ihh, hh_mat0] = mr_hessian(func,x,hflag,htol0,varargin)
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numerical gradient and Hessian, with 'automatic' check of numerical
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error
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func = name of the function: func 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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x = parameter values
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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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hflag = 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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hflag = 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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varargin: other parameters of func</pre></div>
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<!-- crossreference -->
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<h2><a name="_cross"></a>CROSS-REFERENCE INFORMATION <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
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This function calls:
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<ul style="list-style-image:url(../matlabicon.gif)">
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</ul>
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This function is called by:
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<ul style="list-style-image:url(../matlabicon.gif)">
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<li><a href="newrat.html" class="code" title="function [xparam1, hh, gg, fval, igg] = newrat(func0, x, hh, gg, igg, ftol0, nit, flagg, varargin)">newrat</a> </li></ul>
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<!-- crossreference -->
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<h2><a name="_source"></a>SOURCE CODE <a href="#_top"><img alt="^" border="0" src="../up.png"></a></h2>
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<div class="fragment"><pre>0001 <span class="comment">% Copyright (C) 2004 Marco Ratto</span>
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0002 <span class="comment">% adapted from Michel Juillard original rutine hessian.m</span>
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0003 <span class="comment">%</span>
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0004 <span class="comment">% [hessian_mat, gg, htol1, ihh, hh_mat0] = mr_hessian(func,x,hflag,htol0,varargin)</span>
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0005 <span class="comment">%</span>
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0006 <span class="comment">% numerical gradient and Hessian, with 'automatic' check of numerical</span>
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0007 <span class="comment">% error</span>
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0008 <span class="comment">%</span>
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0009 <span class="comment">% func = name of the function: func must give two outputs:</span>
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0010 <span class="comment">% - the log-likelihood AND the single contributions at times t=1,...,T</span>
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0011 <span class="comment">% of the log-likelihood to compute outer product gradient</span>
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0012 <span class="comment">% x = parameter values</span>
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0013 <span class="comment">% hflag = 0, Hessian computed with outer product gradient, one point</span>
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0014 <span class="comment">% increments for partial derivatives in gradients</span>
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0015 <span class="comment">% hflag = 1, 'mixed' Hessian: diagonal elements computed with numerical second order derivatives</span>
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0016 <span class="comment">% with correlation structure as from outer product gradient;</span>
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0017 <span class="comment">% two point evaluation of derivatives for partial derivatives</span>
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0018 <span class="comment">% in gradients</span>
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0019 <span class="comment">% hflag = 2, full numerical Hessian, computes second order partial derivatives</span>
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0020 <span class="comment">% uses Abramowitz and Stegun (1965) formulas 25.3.24 and 25.3.27</span>
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0021 <span class="comment">% p. 884.</span>
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0022 <span class="comment">% htol0 = 'precision' of increment of function values for numerical</span>
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0023 <span class="comment">% derivatives</span>
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0024 <span class="comment">%</span>
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0025 <span class="comment">% varargin: other parameters of func</span>
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0026 <span class="comment">%</span>
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0027
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0028 <a name="_sub0" href="#_subfunctions" class="code">function [hessian_mat, gg, htol1, ihh, hh_mat0] = mr_hessian(func,x,hflag,htol0,varargin)</a>
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0029 <span class="keyword">global</span> options_ bayestopt_
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0030 <span class="keyword">persistent</span> h1 htol
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0031
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0032 gstep_=options_.gstep;
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0033 <span class="keyword">if</span> isempty(htol), htol = 1.e-4; <span class="keyword">end</span>
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0034 func = str2func(func);
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0035 [f0, ff0]=feval(func,x,varargin{:});
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0036 n=size(x,1);
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0037 h2=bayestopt_.ub-bayestopt_.lb;
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0038 <span class="comment">%h1=max(abs(x),gstep_*ones(n,1))*eps^(1/3);</span>
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0039 <span class="comment">%h1=max(abs(x),sqrt(gstep_)*ones(n,1))*eps^(1/6);</span>
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0040 <span class="keyword">if</span> isempty(h1),
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0041 h1=max(abs(x),sqrt(gstep_)*ones(n,1))*eps^(1/4);
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0042 <span class="keyword">end</span>
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0043
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0044 <span class="keyword">if</span> htol0<htol,
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0045 htol=htol0;
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0046 <span class="keyword">end</span>
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0047 xh1=x;
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0048 f1=zeros(size(f0,1),n);
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0049 f_1=f1;
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0050 ff1=zeros(size(ff0));
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0051 ff_1=ff1;
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0052
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0053 <span class="comment">%for i=1:n,</span>
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0054 i=0;
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0055 <span class="keyword">while</span> i<n,
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0056 i=i+1;
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0057 h10=h1(i);
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0058 hcheck=0;
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0059 dx=[];
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0060 xh1(i)=x(i)+h1(i);
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0061 [fx, ffx]=feval(func,xh1,varargin{:});
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0062 it=1;
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0063 dx=(fx-f0);
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0064 ic=0;
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0065 <span class="comment">% if abs(dx)>(2*htol),</span>
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0066 <span class="comment">% c=mr_nlincon(xh1,varargin{:});</span>
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0067 <span class="comment">% while c</span>
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0068 <span class="comment">% h1(i)=h1(i)*0.9;</span>
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0069 <span class="comment">% xh1(i)=x(i)+h1(i);</span>
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0070 <span class="comment">% c=mr_nlincon(xh1,varargin{:});</span>
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0071 <span class="comment">% ic=1;</span>
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0072 <span class="comment">% end</span>
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0073 <span class="comment">% if ic,</span>
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0074 <span class="comment">% [fx, ffx]=feval(func,xh1,varargin{:});</span>
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0075 <span class="comment">% dx=(fx-f0);</span>
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0076 <span class="comment">% end</span>
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0077 <span class="comment">% end</span>
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0078
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0079 icount = 0;
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0080 h0=h1(i);
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0081 <span class="keyword">while</span> (abs(dx(it))<0.5*htol | abs(dx(it))>(2*htol)) & icount<10 & ic==0,
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0082 <span class="comment">%while abs(dx(it))<0.5*htol & icount< 10 & ic==0,</span>
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0083 icount=icount+1;
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0084 <span class="comment">%if abs(dx(it)) ~= 0,</span>
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0085 <span class="keyword">if</span> abs(dx(it))<0.5*htol
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0086 <span class="keyword">if</span> abs(dx(it)) ~= 0,
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0087 h1(i)=min(0.3*abs(x(i)), 0.9*htol/abs(dx(it))*h1(i));
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0088 <span class="keyword">else</span>
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0089 h1(i)=2.1*h1(i);
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0090 <span class="keyword">end</span>
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0091 xh1(i)=x(i)+h1(i);
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0092 <span class="comment">% c=mr_nlincon(xh1,varargin{:});</span>
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0093 <span class="comment">% while c</span>
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0094 <span class="comment">% h1(i)=h1(i)*0.9;</span>
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0095 <span class="comment">% xh1(i)=x(i)+h1(i);</span>
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0096 <span class="comment">% c=mr_nlincon(xh1,varargin{:});</span>
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0097 <span class="comment">% ic=1;</span>
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0098 <span class="comment">% end</span>
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0099 [fx, ffx]=feval(func,xh1,varargin{:});
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0100 <span class="keyword">end</span>
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0101 <span class="keyword">if</span> abs(dx(it))>(2*htol),
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0102 h1(i)= htol/abs(dx(it))*h1(i);
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0103 xh1(i)=x(i)+h1(i);
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0104 [fx, ffx]=feval(func,xh1,varargin{:});
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0105 <span class="keyword">while</span> (fx-f0)==0,
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0106 h1(i)= h1(i)*2;
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0107 xh1(i)=x(i)+h1(i);
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0108 [fx, ffx]=feval(func,xh1,varargin{:});
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0109 ic=1;
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0110 <span class="keyword">end</span>
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0111 <span class="keyword">end</span>
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0112 it=it+1;
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0113 dx(it)=(fx-f0);
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0114 h0(it)=h1(i);
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0115 <span class="keyword">if</span> h1(i)<1.e-12*min(1,h2(i)),
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0116 ic=1;
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0117 hcheck=1;
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0118 <span class="keyword">end</span>
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0119 <span class="comment">%else</span>
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0120 <span class="comment">% h1(i)=1;</span>
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0121 <span class="comment">% ic=1;</span>
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0122 <span class="comment">%end</span>
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0123 <span class="keyword">end</span>
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0124 <span class="comment">% if (it>2 & dx(1)<10^(log10(htol)/2)) ,</span>
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0125 <span class="comment">% [dum, is]=sort(h0);</span>
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0126 <span class="comment">% if find(diff(sign(diff(dx(is)))));</span>
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0127 <span class="comment">% hcheck=1;</span>
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0128 <span class="comment">% end</span>
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0129 <span class="comment">% elseif (it>3 & dx(1)>10^(log10(htol)/2)) ,</span>
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0130 <span class="comment">% [dum, is]=sort(h0);</span>
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0131 <span class="comment">% if find(diff(sign(diff(dx(is(1:end-1))))));</span>
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0132 <span class="comment">% hcheck=1;</span>
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0133 <span class="comment">% end</span>
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0134 <span class="comment">% end</span>
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0135 f1(:,i)=fx;
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0136 ff1=ffx;
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0137 <span class="keyword">if</span> hflag, <span class="comment">% two point based derivatives</span>
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0138 xh1(i)=x(i)-h1(i);
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0139 <span class="comment">% c=mr_nlincon(xh1,varargin{:});</span>
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0140 <span class="comment">% ic=0;</span>
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0141 <span class="comment">% while c</span>
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0142 <span class="comment">% h1(i)=h1(i)*0.9;</span>
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0143 <span class="comment">% xh1(i)=x(i)-h1(i);</span>
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0144 <span class="comment">% c=mr_nlincon(xh1,varargin{:});</span>
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0145 <span class="comment">% ic = 1;</span>
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0146 <span class="comment">% end</span>
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0147 [fx, ffx]=feval(func,xh1,varargin{:});
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0148 f_1(:,i)=fx;
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0149 ff_1=ffx;
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0150 <span class="comment">% if ic,</span>
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0151 <span class="comment">% xh1(i)=x(i)+h1(i);</span>
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0152 <span class="comment">% [f1(:,i), ff1]=feval(func,xh1,varargin{:});</span>
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0153 <span class="comment">% end</span>
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0154 ggh(:,i)=(ff1-ff_1)./(2.*h1(i));
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0155 <span class="keyword">else</span>
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0156 ggh(:,i)=(ff1-ff0)./h1(i);
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0157 <span class="keyword">end</span>
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0158 xh1(i)=x(i);
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0159 <span class="keyword">if</span> hcheck & htol<1,
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0160 htol=min(1,max(min(abs(dx))*2,htol*10));
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0161 h1(i)=h10;
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0162 i=0;
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0163 <span class="keyword">end</span>
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0164 save hess
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0165 <span class="keyword">end</span>
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0166
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0167 h_1=h1;
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0168 xh1=x;
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0169 xh_1=xh1;
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0170
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0171 <span class="keyword">if</span> hflag,
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0172 gg=(f1'-f_1')./(2.*h1);
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0173 <span class="keyword">else</span>
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0174 gg=(f1'-f0)./h1;
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0175 <span class="keyword">end</span>
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0176
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0177 <span class="keyword">if</span> hflag==2,
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0178 gg=(f1'-f_1')./(2.*h1);
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0179 hessian_mat = zeros(size(f0,1),n*n);
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0180 <span class="keyword">for</span> i=1:n
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0181 <span class="keyword">if</span> i > 1
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0182 k=[i:n:n*(i-1)];
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0183 hessian_mat(:,(i-1)*n+1:(i-1)*n+i-1)=hessian_mat(:,k);
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0184 <span class="keyword">end</span>
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0185 hessian_mat(:,(i-1)*n+i)=(f1(:,i)+f_1(:,i)-2*f0)./(h1(i)*h_1(i));
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0186 temp=f1+f_1-f0*ones(1,n);
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0187 <span class="keyword">for</span> j=i+1:n
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0188 xh1(i)=x(i)+h1(i);
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0189 xh1(j)=x(j)+h_1(j);
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0190 xh_1(i)=x(i)-h1(i);
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0191 xh_1(j)=x(j)-h_1(j);
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0192 <span class="comment">%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));</span>
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0193 <span class="comment">%temp1 = feval(func,xh1,varargin{:});</span>
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0194 <span class="comment">% c=mr_nlincon(xh1,varargin{:});</span>
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0195 <span class="comment">% lam=1;</span>
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0196 <span class="comment">% while c,</span>
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0197 <span class="comment">% lam=lam*0.9;</span>
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0198 <span class="comment">% xh1(i)=x(i)+h1(i)*lam;</span>
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0199 <span class="comment">% xh1(j)=x(j)+h_1(j)*lam;</span>
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0200 <span class="comment">% %disp( ['hessian warning cross ', num2str(c) ]),</span>
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0201 <span class="comment">% c=mr_nlincon(xh1,varargin{:});</span>
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0202 <span class="comment">% end</span>
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0203 <span class="comment">% temp1 = f0+(feval(func,xh1,varargin{:})-f0)/lam;</span>
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0204 temp1 = feval(func,xh1,varargin{:});
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0205
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0206 <span class="comment">% c=mr_nlincon(xh_1,varargin{:});</span>
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0207 <span class="comment">% while c,</span>
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0208 <span class="comment">% lam=lam*0.9;</span>
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0209 <span class="comment">% xh_1(i)=x(i)-h1(i)*lam;</span>
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0210 <span class="comment">% xh_1(j)=x(j)-h_1(j)*lam;</span>
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0211 <span class="comment">% %disp( ['hessian warning cross ', num2str(c) ]),</span>
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0212 <span class="comment">% c=mr_nlincon(xh_1,varargin{:});</span>
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0213 <span class="comment">% end</span>
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0214 <span class="comment">% temp2 = f0+(feval(func,xh_1,varargin{:})-f0)/lam;</span>
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0215 temp2 = feval(func,xh_1,varargin{:});
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0216
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0217 hessian_mat(:,(i-1)*n+j)=-(-temp1 -temp2+temp(:,i)+temp(:,j))./(2*h1(i)*h_1(j));
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0218 xh1(i)=x(i);
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0219 xh1(j)=x(j);
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0220 xh_1(i)=x(i);
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0221 xh_1(j)=x(j);
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0222 j=j+1;
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0223 save hess
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0224 <span class="keyword">end</span>
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0225 i=i+1;
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0226 <span class="keyword">end</span>
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0227
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0228 <span class="keyword">elseif</span> hflag==1,
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0229 hessian_mat = zeros(size(f0,1),n*n);
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0230 <span class="keyword">for</span> i=1:n,
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0231 dum = (f1(:,i)+f_1(:,i)-2*f0)./(h1(i)*h_1(i));
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0232 <span class="keyword">if</span> dum>eps,
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0233 hessian_mat(:,(i-1)*n+i)=dum;
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0234 <span class="keyword">else</span>
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0235 hessian_mat(:,(i-1)*n+i)=max(eps, gg(i)^2);
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0236 <span class="keyword">end</span>
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0237 <span class="keyword">end</span>
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0238 <span class="comment">%hessian_mat2=hh_mat(:)';</span>
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0239 <span class="keyword">end</span>
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0240
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0241 gga=ggh.*kron(ones(size(ff1)),2.*h1'); <span class="comment">% re-scaled gradient</span>
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0242 hh_mat=gga'*gga; <span class="comment">% rescaled outer product hessian</span>
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0243 hh_mat0=ggh'*ggh; <span class="comment">% outer product hessian</span>
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0244 A=diag(2.*h1); <span class="comment">% rescaling matrix</span>
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0245 igg=inv(hh_mat); <span class="comment">% inverted rescaled outer product hessian</span>
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0246 ihh=A'*igg*A; <span class="comment">% inverted outer product hessian</span>
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0247 <span class="keyword">if</span> hflag>0 & min(eig(reshape(hessian_mat,n,n)))>0,
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0248 hh0 = A*reshape(hessian_mat,n,n)*A'; <span class="comment">%rescaled second order derivatives</span>
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0249 hh = reshape(hessian_mat,n,n); <span class="comment">%rescaled second order derivatives</span>
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0250 sd0=sqrt(diag(hh0)); <span class="comment">%rescaled 'standard errors' using second order derivatives</span>
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0251 sd=sqrt(diag(hh_mat)); <span class="comment">%rescaled 'standard errors' using outer product</span>
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0252 hh_mat=hh_mat./(sd*sd').*(sd0*sd0'); <span class="comment">%rescaled inverse outer product with 'true' std's</span>
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0253 igg=inv(hh_mat); <span class="comment">% rescaled outer product hessian with 'true' std's</span>
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0254 ihh=A'*igg*A; <span class="comment">% inverted outer product hessian</span>
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0255 hh_mat0=inv(A)'*hh_mat*inv(A); <span class="comment">% outer product hessian with 'true' std's</span>
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0256 sd=sqrt(diag(ihh)); <span class="comment">%standard errors</span>
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0257 sdh=sqrt(1./diag(hh)); <span class="comment">%diagonal standard errors</span>
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0258 <span class="keyword">for</span> j=1:length(sd),
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0259 sd0(j,1)=min(bayestopt_.pstdev(j), sd(j)); <span class="comment">%prior std</span>
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0260 sd0(j,1)=10^(0.5*(log10(sd0(j,1))+log10(sdh(j,1))));
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0261 <span class="comment">%sd0(j,1)=0.5*(sd0(j,1)+sdh(j,1));</span>
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0262 <span class="keyword">end</span>
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0263 ihh=ihh./(sd*sd').*(sd0*sd0'); <span class="comment">%inverse outer product with modified std's</span>
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0264 igg=inv(A)'*ihh*inv(A); <span class="comment">% inverted rescaled outer product hessian with modified std's</span>
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0265 hh_mat=inv(igg); <span class="comment">% outer product rescaled hessian with modified std's</span>
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|
0266 hh_mat0=inv(A)'*hh_mat*inv(A); <span class="comment">% outer product hessian with modified std's</span>
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|
0267 <span class="comment">% sd0=sqrt(1./diag(hh0)); %rescaled 'standard errors' using second order derivatives</span>
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|
0268 <span class="comment">% sd=sqrt(diag(igg)); %rescaled 'standard errors' using outer product</span>
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|
0269 <span class="comment">% igg=igg./(sd*sd').*(sd0*sd0'); %rescaled inverse outer product with 'true' std's</span>
|
|
0270 <span class="comment">% hh_mat=inv(igg); % rescaled outer product hessian with 'true' std's</span>
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|
0271 <span class="comment">% ihh=A'*igg*A; % inverted outer product hessian</span>
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|
0272 <span class="comment">% hh_mat0=inv(A)'*hh_mat*inv(A); % outer product hessian with 'true' std's</span>
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0273 <span class="keyword">end</span>
|
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0274 <span class="keyword">if</span> hflag<2,
|
|
0275 hessian_mat=hh_mat0(:);
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0276 <span class="keyword">end</span>
|
|
0277
|
|
0278 <span class="keyword">if</span> isnan(hessian_mat),
|
|
0279 hh_mat0=eye(length(hh_mat0));
|
|
0280 ihh=hh_mat0;
|
|
0281 hessian_mat=hh_mat0(:);
|
|
0282 <span class="keyword">end</span>
|
|
0283 hh1=h1;
|
|
0284 htol1=htol;
|
|
0285 save hess
|
|
0286 <span class="comment">% 11/25/03 SA Created from Hessian_sparse (removed sparse)</span>
|
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0287
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0288</pre></div>
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<hr><address>Generated on Fri 16-Jun-2006 09:09:06 by <strong><a href="http://www.artefact.tk/software/matlab/m2html/">m2html</a></strong> © 2003</address>
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