%% Annotated Matlab transcript -- 5/1
%% CS 323 - Doug DeCarlo

% least squares fitting

% data from the book
x = 1:0.2:5;
y = [-1.945 -1.253 -1.140 -1.087 -0.760 -0.682 -0.424 -0.012 -0.190 0.452 0.337 0.764 0.532 1.073 1.286 1.502 1.582 1.993 2.473 2.503 2.322];
plot(x,y,'.');
% here is the data with the line that results from a least squares fit
plot(x,y,'.',x,1.06338*x-2.74605);

% solving the system...
A = [size(x,2) sum(x); sum(x), sum(x.^2)]
f = [sum(y); sum(x.*y)]
z = A \ f
plot(x,y,'.',x,1.06338*x-2.74605);

% the fit locations
yf = 1.06338*x-2.74605;
% measuring the error between these locations and the data
sqrt(1/size(x,2)*sum((y - yf).^2))

% we notice this error increases when we tweak the line equation
% (as the above solution minimizes this error)
yf = 1.07338*x-2.74605;
sqrt(1/size(x,2)*sum((y - yf).^2))
yf = 2.07338*x-5.74605;
sqrt(1/size(x,2)*sum((y - yf).^2))
% and here we see a plot of one of these tweaked lines...
plot(x,y,'.',x,2.07338*x-5.74605);