% Illustration of linear least squares.
function main()
   
   % KSmrq's colors
   red    = [0.867 0.06 0.14];
   blue   = [0, 129, 205]/256;
   green  = [0, 200,  70]/256;
   yellow = [254, 194,   0]/256;
   white = 0.99*[1, 1, 1];
   gray = 0.2*white;
   
% Set up the grid and other parameters
   N = 100;
   A = -2.2; B = 2; 
   X = linspace(A, B, N);
   C=-4; D = 4;
   
   % Set up the figure
   lw = 4; % linewidth
   fs = 18; % font size
   figure(1); clf; hold on;
   set(gca, 'fontsize', fs);
   set(gca, 'linewidth', lw/2.5)
   hold on; grid on;

   % random numbers
   a = 1.1; b = 3.5; c = 1.4;
   M = 50;
   XX=linspace(A+0.3, B-0.3, M+1);
   Xr = 0*(1:M);
   Yr = Xr;
   for i=1:M
      r=rand(1);
      Xr(i) = XX(i)*r+XX(i+1)*(1-r);
      Yr(i) = a*Xr(i) + b + c*0.4*erfinv(2*rand(1) - 1);
   end

   myrad = 0.05;
   for i=1:length(Xr)
      ball(Xr(i), Yr(i), myrad, red);
   end


   % least squares fitting
   Yr = Yr';
   Mat = [Xr' (0*Xr+1)'];
   V=Mat'*Yr;
   V=(Mat'*Mat)\V;
   ae = V(1); be = V(2);   
   plot(X, ae*X+be, 'b', 'linewidth', lw);

   grid on;
   set(gca, 'GridLineStyle', '-', 'xcolor', gray);
   set(gca, 'GridLineStyle', '-', 'ycolor', gray);
   set(gca, 'XTick', [-2 -1 0 1 2]);


   axis equal;
   axis([-2, 2, 1, 6]);
   
   saveas(gcf, 'Linear_least_squares.eps', 'psc2'); % save as eps
   %plot2svg('Linear_least_squares.svg'); % save as svg

   
function ball(x, y, r, color)
   Theta=0:0.1:2*pi;
   X=r*cos(Theta)+x;
   Y=r*sin(Theta)+y;
   H=fill(X, Y, color);
   set(H, 'EdgeColor', 'none');