fig6_1

 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().

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