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<h1>fig6_1
</h1>

<h2><a name="_name"></a>PURPOSE <a href="#_top"><img alt="^" border="0" src="../../../up.png"></a></h2>
<div class="box"><strong>Local Regression and Likelihood, Figure 6.1.</strong></div>

<h2><a name="_synopsis"></a>SYNOPSIS <a href="#_top"><img alt="^" border="0" src="../../../up.png"></a></h2>
<div class="box"><strong>This is a script file. </strong></div>

<h2><a name="_description"></a>DESCRIPTION <a href="#_top"><img alt="^" border="0" src="../../../up.png"></a></h2>
<div class="fragment"><pre class="comment"> Local Regression and Likelihood, Figure 6.1.

 Derivative (local slope) estimation, for the Old Faithful Geyser Data.
 The 'deriv' argument specifies derivative estimation,
   'deriv',1        First-order derivative.
   'deriv',[1 1]    Second-order derivative.
   'deriv',2        For bivariate fits, partial deriv. wrt second variable.
   'deriv',[1 2]    Mixed second-order derivative.

 Density estimation is done on the log-scale. That is, the estimate
 is of g(x) = log(f(x)), where f(x) is the density.

 The relation between derivatives is therefore
     f'(x) = f(x)g'(x) = g'(x)exp(g(x)).
 To estimate f'(x), we must estimate g(x) and g'(x) (fit1 and fit2 below),
 evaluate on a grid of points (p1 and p2), and apply the back-transformation.

 Disclaimer: I don't consider derivative estimation from noisy data
 to be a well-defined problem. Use at your own risk.

 Author: Catherine Loader

 NEED: m argument passed to lfmarg().</pre></div>

<!-- crossreference -->
<h2><a name="_cross"></a>CROSS-REFERENCE INFORMATION <a href="#_top"><img alt="^" border="0" src="../../../up.png"></a></h2>
This function calls:
<ul style="list-style-image:url(../../../matlabicon.gif)">
<li><a href="../../../chronux_2_10/locfit/m/lfmarg.html" class="code" title="function xfit = lfmarg(fit)">lfmarg</a>	computes grid margins from a locfit object, used for plotting.</li><li><a href="../../../chronux_2_10/locfit/m/locfit.html" class="code" title="function fit=locfit(varargin)">locfit</a>	Smoothing noisy data using Local Regression and Likelihood.</li><li><a href="../../../chronux_2_10/locfit/m/predict.html" class="code" title="function [y, se] = predict(varargin)">predict</a>	Interpolate a fit produced by locfit().</li></ul>
This function is called by:
<ul style="list-style-image:url(../../../matlabicon.gif)">
<li><a href="runbook.html" class="code" title="">runbook</a>	</li></ul>
<!-- crossreference -->


<h2><a name="_source"></a>SOURCE CODE <a href="#_top"><img alt="^" border="0" src="../../../up.png"></a></h2>
<div class="fragment"><pre>0001 <span class="comment">% Local Regression and Likelihood, Figure 6.1.</span>
0002 <span class="comment">%</span>
0003 <span class="comment">% Derivative (local slope) estimation, for the Old Faithful Geyser Data.</span>
0004 <span class="comment">% The 'deriv' argument specifies derivative estimation,</span>
0005 <span class="comment">%   'deriv',1        First-order derivative.</span>
0006 <span class="comment">%   'deriv',[1 1]    Second-order derivative.</span>
0007 <span class="comment">%   'deriv',2        For bivariate fits, partial deriv. wrt second variable.</span>
0008 <span class="comment">%   'deriv',[1 2]    Mixed second-order derivative.</span>
0009 <span class="comment">%</span>
0010 <span class="comment">% Density estimation is done on the log-scale. That is, the estimate</span>
0011 <span class="comment">% is of g(x) = log(f(x)), where f(x) is the density.</span>
0012 <span class="comment">%</span>
0013 <span class="comment">% The relation between derivatives is therefore</span>
0014 <span class="comment">%     f'(x) = f(x)g'(x) = g'(x)exp(g(x)).</span>
0015 <span class="comment">% To estimate f'(x), we must estimate g(x) and g'(x) (fit1 and fit2 below),</span>
0016 <span class="comment">% evaluate on a grid of points (p1 and p2), and apply the back-transformation.</span>
0017 <span class="comment">%</span>
0018 <span class="comment">% Disclaimer: I don't consider derivative estimation from noisy data</span>
0019 <span class="comment">% to be a well-defined problem. Use at your own risk.</span>
0020 <span class="comment">%</span>
0021 <span class="comment">% Author: Catherine Loader</span>
0022 <span class="comment">%</span>
0023 <span class="comment">% NEED: m argument passed to lfmarg().</span>
0024 
0025 load geyser;
0026 fit1 = <a href="../../../chronux_2_10/locfit/m/locfit.html" class="code" title="function fit=locfit(varargin)">locfit</a>(geyser,<span class="string">'alpha'</span>,[0.1 0.6],<span class="string">'ll'</span>,1,<span class="string">'ur'</span>,6);
0027 fit2 = <a href="../../../chronux_2_10/locfit/m/locfit.html" class="code" title="function fit=locfit(varargin)">locfit</a>(geyser,<span class="string">'alpha'</span>,[0.1 0.6],<span class="string">'ll'</span>,1,<span class="string">'ur'</span>,6,<span class="string">'deriv'</span>,1);
0028 z = <a href="../../../chronux_2_10/locfit/m/lfmarg.html" class="code" title="function xfit = lfmarg(fit)">lfmarg</a>(fit1);
0029 p1 = <a href="../../../chronux_2_10/locfit/m/predict.html" class="code" title="function [y, se] = predict(varargin)">predict</a>(fit1,z);
0030 p2 = <a href="../../../chronux_2_10/locfit/m/predict.html" class="code" title="function [y, se] = predict(varargin)">predict</a>(fit2,z);
0031 figure(<span class="string">'Name'</span>,<span class="string">'fig6_1: slope estimation: Old faithful data'</span> );
0032 plot(z{1},p2.*exp(p1));
0033 xlabel(<span class="string">'Eruption Duration (Minutes)'</span>);
0034 ylabel(<span class="string">'Density Derivative'</span>);</pre></div>
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