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234 lines
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<div><a href="../index.html">Home</a> > <a href="index.html">.</a> > posterior_density_estimate.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>posterior_density_estimate
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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>%</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 [abscissa,f,h] = posterior_density_estimate(data,number_of_grid_points,bandwidth,kernel_function) </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">%
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% This function aims at estimating posterior univariate densities from realisations of a Metropolis-Hastings
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% algorithm. A kernel density estimator is used (see Silverman [1986]) and the main task of this function is
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% to obtain an optimal bandwidth parameter.
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%
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% * Silverman [1986], "Density estimation for statistics and data analysis".
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% * M. Skold and G.O. Roberts [2003], "Density estimation for the Metropolis-Hastings algorithm".
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%
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% The last section is adapted from Anders Holtsberg's matlab toolbox (stixbox).
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%
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% stephane.adjemian@cepremap.cnrs.fr [01/16/2004].</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="compDist.html" class="code" title="function compdist(xparam1, x2, pltopt, figurename)">compDist</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 <a name="_sub0" href="#_subfunctions" class="code">function [abscissa,f,h] = posterior_density_estimate(data,number_of_grid_points,bandwidth,kernel_function) </a>
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0002 <span class="comment">%%</span>
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0003 <span class="comment">%% This function aims at estimating posterior univariate densities from realisations of a Metropolis-Hastings</span>
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0004 <span class="comment">%% algorithm. A kernel density estimator is used (see Silverman [1986]) and the main task of this function is</span>
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0005 <span class="comment">%% to obtain an optimal bandwidth parameter.</span>
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0006 <span class="comment">%%</span>
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0007 <span class="comment">%% * Silverman [1986], "Density estimation for statistics and data analysis".</span>
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0008 <span class="comment">%% * M. Skold and G.O. Roberts [2003], "Density estimation for the Metropolis-Hastings algorithm".</span>
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0009 <span class="comment">%%</span>
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0010 <span class="comment">%% The last section is adapted from Anders Holtsberg's matlab toolbox (stixbox).</span>
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0011 <span class="comment">%%</span>
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0012 <span class="comment">%% stephane.adjemian@cepremap.cnrs.fr [01/16/2004].</span>
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0013
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0014 <span class="keyword">if</span> size(data,2) > 1 & size(data,1) == 1;
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0015 data = data';
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0016 <span class="keyword">elseif</span> size(data,2)>1 & size(data,1)>1;
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0017 error(<span class="string">'density_estimate: data must be a one dimensional array.'</span>);
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0018 <span class="keyword">end</span>;
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0019 test = log(number_of_grid_points)/log(2);
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0020 <span class="keyword">if</span> ( abs(test-round(test)) > 10^(-12));
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0021 error(<span class="string">'The number of grid points must be a power of 2.'</span>);
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0022 <span class="keyword">end</span>;
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0023
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0024 n = size(data,1);
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0025
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0026
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0027 <span class="comment">%% KERNEL SPECIFICATION...</span>
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0028
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0029 <span class="keyword">if</span> strcmp(kernel_function,<span class="string">'gaussian'</span>);
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0030 k = inline(<span class="string">'inv(sqrt(2*pi))*exp(-0.5*x.^2)'</span>);
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0031 k2 = inline(<span class="string">'inv(sqrt(2*pi))*(-exp(-0.5*x.^2)+(x.^2).*exp(-0.5*x.^2))'</span>); <span class="comment">% second derivate of the gaussian kernel</span>
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0032 k4 = inline(<span class="string">'inv(sqrt(2*pi))*(3*exp(-0.5*x.^2)-6*(x.^2).*exp(-0.5*x.^2)+(x.^4).*exp(-0.5*x.^2))'</span>); <span class="comment">% fourth derivate...</span>
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0033 k6 = inline(<span class="string">'inv(sqrt(2*pi))*(-15*exp(-0.5*x.^2)+45*(x.^2).*exp(-0.5*x.^2)-15*(x.^4).*exp(-0.5*x.^2)+(x.^6).*exp(-0.5*x.^2))'</span>); <span class="comment">% sixth derivate...</span>
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0034 mu02 = inv(2*sqrt(pi));
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0035 mu21 = 1;
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0036 <span class="keyword">elseif</span> strcmp(kernel_function,<span class="string">'uniform'</span>);
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0037 k = inline(<span class="string">'0.5*(abs(x) <= 1)'</span>);
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0038 mu02 = 0.5;
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0039 mu21 = 1/3;
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0040 <span class="keyword">elseif</span> strcmp(kernel_function,<span class="string">'triangle'</span>);
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0041 k = inline(<span class="string">'(1-abs(x)).*(abs(x) <= 1)'</span>);
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0042 mu02 = 2/3;
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0043 mu21 = 1/6;
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0044 <span class="keyword">elseif</span> strcmp(kernel_function,<span class="string">'epanechnikov'</span>);
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0045 k = inline(<span class="string">'0.75*(1-x.^2).*(abs(x) <= 1)'</span>);
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0046 mu02 = 3/5;
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0047 mu21 = 1/5;
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0048 <span class="keyword">elseif</span> strcmp(kernel_function,<span class="string">'quartic'</span>);
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0049 k = inline(<span class="string">'0.9375*((1-x.^2).^2).*(abs(x) <= 1)'</span>);
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0050 mu02 = 15/21;
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0051 mu21 = 1/7;
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0052 <span class="keyword">elseif</span> strcmp(kernel_function,<span class="string">'triweight'</span>);
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0053 k = inline(<span class="string">'1.09375*((1-x.^2).^3).*(abs(x) <= 1)'</span>);
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0054 k2 = inline(<span class="string">'(105/4*(1-x.^2).*x.^2-105/16*(1-x.^2).^2).*(abs(x) <= 1)'</span>);
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0055 k4 = inline(<span class="string">'(-1575/4*x.^2+315/4).*(abs(x) <= 1)'</span>);
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0056 k6 = inline(<span class="string">'(-1575/2).*(abs(x) <= 1)'</span>);
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0057 mu02 = 350/429;
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0058 mu21 = 1/9;
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0059 <span class="keyword">elseif</span> strcmp(kernel_function,<span class="string">'cosinus'</span>);
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0060 k = inline(<span class="string">'(pi/4)*cos((pi/2)*x).*(abs(x) <= 1)'</span>);
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0061 k2 = inline(<span class="string">'(-1/16*cos(pi*x/2)*pi^3).*(abs(x) <= 1)'</span>);
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0062 k4 = inline(<span class="string">'(1/64*cos(pi*x/2)*pi^5).*(abs(x) <= 1)'</span>);
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0063 k6 = inline(<span class="string">'(-1/256*cos(pi*x/2)*pi^7).*(abs(x) <= 1)'</span>);
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0064 mu02 = (pi^2)/16;
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0065 mu21 = (pi^2-8)/pi^2;
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0066 <span class="keyword">end</span>;
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0067
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0068
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0069 <span class="comment">%% OPTIMAL BANDWIDTH PARAMETER....</span>
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0070
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0071 <span class="keyword">if</span> bandwidth == 0; <span class="comment">% Rule of thumb bandwidth parameter (Silverman [1986] corrected by</span>
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0072 <span class="comment">% Skold and Roberts [2003] for Metropolis-Hastings).</span>
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0073 sigma = std(data);
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0074 h = 2*sigma*(sqrt(pi)*mu02/(12*(mu21^2)*n))^(1/5); <span class="comment">% Silverman's optimal bandwidth parameter.</span>
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0075 A = 0;
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0076 <span class="keyword">for</span> i=1:n;
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0077 j = i;
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0078 <span class="keyword">while</span> j<= n & data(j,1)==data(i,1);
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0079 j = j+1;
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0080 <span class="keyword">end</span>;
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0081 A = A + 2*(j-i) - 1;
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0082 <span class="keyword">end</span>;
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0083 A = A/n;
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0084 h = h*A^(1/5); <span class="comment">% correction</span>
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0085 <span class="keyword">elseif</span> bandwidth == -1; <span class="comment">% Adaptation of the Sheather and Jones [1991] plug-in estimation of the optimal bandwidth</span>
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0086 <span class="comment">% parameter for metropolis hastings algorithm.</span>
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0087 <span class="keyword">if</span> strcmp(kernel_function,<span class="string">'uniform'</span>) | <span class="keyword">...</span><span class="comment"> </span>
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0088 strcmp(kernel_function,<span class="string">'triangle'</span>) | <span class="keyword">...</span><span class="comment"> </span>
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0089 strcmp(kernel_function,<span class="string">'epanechnikov'</span>) | <span class="keyword">...</span><span class="comment"> </span>
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0090 strcmp(kernel_function,<span class="string">'quartic'</span>);
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0091 error(<span class="string">'I can''t compute the optimal bandwidth with this kernel... Try the gaussian, triweight or cosinus kernels.'</span>);
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0092 <span class="keyword">end</span>;
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0093 sigma = std(data);
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0094 A = 0;
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0095 <span class="keyword">for</span> i=1:n;
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0096 j = i;
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0097 <span class="keyword">while</span> j<= n & data(j,1)==data(i,1);
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0098 j = j+1;
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0099 <span class="keyword">end</span>;
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0100 A = A + 2*(j-i) - 1;
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0101 <span class="keyword">end</span>;
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0102 A = A/n;
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0103 Itilda4 = 8*7*6*5/(((2*sigma)^9)*sqrt(pi));
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0104 g3 = abs(2*A*k6(0)/(mu21*Itilda4*n))^(1/9);
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0105 Ihat3 = 0;
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0106 <span class="keyword">for</span> i=1:n;
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0107 Ihat3 = Ihat3 + sum(k6((data(i,1)-data)/g3));
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0108 <span class="keyword">end</span>;
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0109 Ihat3 = -Ihat3/((n^2)*g3^7);
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0110 g2 = abs(2*A*k4(0)/(mu21*Ihat3*n))^(1/7);
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0111 Ihat2 = 0;
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0112 <span class="keyword">for</span> i=1:n;
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0113 Ihat2 = Ihat2 + sum(k4((data(i)-data)/g2));
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0114 <span class="keyword">end</span>;
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0115 Ihat2 = Ihat2/((n^2)*g2^5);
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0116 h = (A*mu02/(n*Ihat2*mu21^2))^(1/5); <span class="comment">% equation (22) in Skold and Roberts [2003] --> h_{MH}</span>
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0117 <span class="keyword">elseif</span> bandwidth == -2; <span class="comment">% Bump killing... We construct local bandwith parameters in order to remove</span>
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0118 <span class="comment">% spurious bumps introduced by long rejecting periods.</span>
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0119 <span class="keyword">if</span> strcmp(kernel_function,<span class="string">'uniform'</span>) | <span class="keyword">...</span><span class="comment"> </span>
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0120 strcmp(kernel_function,<span class="string">'triangle'</span>) | <span class="keyword">...</span><span class="comment"> </span>
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0121 strcmp(kernel_function,<span class="string">'epanechnikov'</span>) | <span class="keyword">...</span><span class="comment"> </span>
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0122 strcmp(kernel_function,<span class="string">'quartic'</span>);
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0123 error(<span class="string">'I can''t compute the optimal bandwidth with this kernel... Try the gaussian, triweight or cosinus kernels.'</span>);
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0124 <span class="keyword">end</span>;
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0125 sigma = std(data);
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0126 A = 0;
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0127 T = zeros(n,1);
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0128 <span class="keyword">for</span> i=1:n;
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0129 j = i;
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0130 <span class="keyword">while</span> j<= n & data(j,1)==data(i,1);
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0131 j = j+1;
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0132 <span class="keyword">end</span>;
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0133 T(i) = (j-i);
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0134 A = A + 2*T(i) - 1;
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0135 <span class="keyword">end</span>;
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0136 A = A/n;
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0137 Itilda4 = 8*7*6*5/(((2*sigma)^9)*sqrt(pi));
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0138 g3 = abs(2*A*k6(0)/(mu21*Itilda4*n))^(1/9);
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0139 Ihat3 = 0;
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0140 <span class="keyword">for</span> i=1:n;
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0141 Ihat3 = Ihat3 + sum(k6((data(i,1)-data)/g3));
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0142 <span class="keyword">end</span>;
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0143 Ihat3 = -Ihat3/((n^2)*g3^7);
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0144 g2 = abs(2*A*k4(0)/(mu21*Ihat3*n))^(1/7);
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0145 Ihat2 = 0;
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0146 <span class="keyword">for</span> i=1:n;
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0147 Ihat2 = Ihat2 + sum(k4((data(i)-data)/g2));
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0148 <span class="keyword">end</span>;
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0149 Ihat2 = Ihat2/((n^2)*g2^5);
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0150 h = ((2*T-1)*mu02/(n*Ihat2*mu21^2)).^(1/5); <span class="comment">% Note that h is a column vector (local banwidth parameters).</span>
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0151 <span class="keyword">elseif</span> bandwidth > 0;
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0152 h = bandwidth;
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0153 <span class="keyword">else</span>;
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0154 error(<span class="string">'density_estimate: bandwidth must be positive or equal to 0,-1 or -2.'</span>);
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0155 <span class="keyword">end</span>;
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0156
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0157 <span class="comment">%% COMPUTE DENSITY ESTIMATE, using the optimal bandwidth parameter.</span>
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0158 <span class="comment">%%</span>
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0159 <span class="comment">%% This section is adapted from Anders Holtsberg's matlab toolbox</span>
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0160 <span class="comment">%% (stixbox --> plotdens.m).</span>
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0161
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0162
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0163 a = min(data) - (max(data)-min(data))/3;
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0164 b = max(data) + (max(data)-min(data))/3;
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0165 abscissa = linspace(a,b,number_of_grid_points)';
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0166 d = abscissa(2)-abscissa(1);
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0167 xi = zeros(number_of_grid_points,1);
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0168 xa = (data-a)/(b-a)*number_of_grid_points;
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0169 <span class="keyword">for</span> i = 1:n;
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0170 indx = floor(xa(i));
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0171 temp = xa(i)-indx;
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0172 xi(indx+[1 2]) = xi(indx+[1 2]) + [1-temp,temp]';
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0173 <span class="keyword">end</span>;
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0174 xk = [-number_of_grid_points:number_of_grid_points-1]'*d;
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0175 kk = k(xk/h);
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0176 kk = kk / (sum(kk)*d*n);
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0177 f = ifft(fft(fftshift(kk)).*fft([xi ;zeros(size(xi))]));
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0178 f = real(f(1:number_of_grid_points));</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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