%
% Written by:
% -- 
% John L. Weatherwax                2008-02-20
% 
% email: wax@alum.mit.edu
% 
% Please send comments and especially bug reports to the
% above email address.
% 
%-----

clear all; 
close all; drawnow; 
clc; 

addpath( '../../Code/eda_toolbox' ); 

randn('seed',0); 

M=200; 
x = linspace( 0, +1, M ); 

% our dependent function "y": 
% 
% -- a function that is IN the space of functions we are able to approximate i.e. a polynomial
%y_truth = 4*x.^3 + 6*x.^2 - 1;    
% -- a function that is NOT in the space of functions we are able to approximate 
y_truth = 4*x.^5 + 6*x.^2 - 1 + (1./(x-1.1));    
%y_truth = 4*x.^5 + 6*x.^2 - 1 + sin( 2*pi*x );

% create some noise to add to "y":
sigma     = 1.5; 
noise_eps = sigma*randn(size(y_truth)); 

y       = y_truth + noise_eps; 

% plot the original and noised data: 
fd=figure; 
ht = plot( x, y_truth, '-g', 'LineWidth', 2 ); hold on; %grid on; 
ho = plot( x, y, 'xr' ); 

line_styles={'-',':','--'}; 
line_colors='brc'; 
data_plt_h = []; 

% try three polynomial models to "explain" this data: 
% 
for ii=1:3,

  % assume we *know* the dimension of the polynomial to fit: 
  [pcoef,s] = polyfit(x,y,ii); 
  
  % evaluate this polynomial model:
  yp = polyval(pcoef,x); 
  
  % plot this polynomial model:
  figure(fd); 
  tph=plot( x, yp, ['-',line_colors(ii)] ); 
  data_plt_h(end+1) = tph; 
    
  xlabel( 'predictor x' ); ylabel( 'residual' ); 
  title( ['residual dependence plot for a polynomial model of degree', num2str(ii)] ); 
  %axis( [0 1 -2.5 2.5] );
end

% finally construct a loess plot: 
alpha = 0.5; deg = 2; 
yl = loess( x, y, x, alpha, deg ); 

figure(fd); data_plt_h(end+1) = plot( x, yl, '-k', 'LineWidth', 1 ); 


figure(fd); 
legend([ht,data_plt_h], {'truth', 'poly deg 1','poly deg 2','poly deg 3','loess plot'}, 'location', 'northwest' ); 
saveas( gcf, ['../../WriteUp/Graphics/Chapter7/prob_7_1_M_',num2str(M)], 'epsc' );