% Example 6.2: Robust regression using the Huber penalty
% Section 6.1.2, Figure 6.5
% Boyd & Vandenberghe "Convex Optimization"
% Original by Lieven Vandenberghe
% Adapted for CVX by Joelle Skaf - 09/07/05
%
% Compares the solution of regular Least-squares:
%           minimize    sum(y_i - alpha - beta*t_i)^2
% to the solution of the following:
%           minimize    sum( phi_h (y_i - alpha - beta*t_i)^2 )
% where phi_h is the Huber penalty function, (t_i,y_i) are data points in a
% plane.

% Input data
randn('seed',1);
rand('seed',1);

m=40;  n=2;    A = randn(m,n);
xex = [5;1];
pts = -10+20*rand(m,1);
A = [ones(m,1) pts];
b = A*xex + .5*randn(m,1);
outliers = [-9.5; 9];  outvals = [20; -15];
A = [A; ones(length(outliers),1), outliers];
b = [b; outvals];
m = size(A,1);
pts = [pts;outliers];

% Least Squares
fprintf(1,'Computing the solution of the least-squares problem...');

xls =  A\b;

fprintf(1,'Done! \n');

% Huber
fprintf(1,'Computing the solution of the huber-penalized problem...');

cvx_begin quiet
    variable xhub(n)
    minimize(sum(huber(A*xhub-b)))
cvx_end

fprintf(1,'Done! \n');

% Plots
figure(1);  hold off
plot(pts,b,'o', [-11; 11], [1 -11; 1 11]*xhub, '-', ...
     [-11; 11], [1 -11; 1 11]*xls, '--');
axis([-11 11 -20 25])
title('Least-square fit vs robust least-squares fit (Huber-penalized)');
xlabel('x');
ylabel('y');
legend('Data points','Huber penalty','Regular LS','Location','Best');
%print -deps robustls.eps