% RegressMultEx1.m
%
% Calculations for multiple regression example.

% Data pts x coordinates are generated as corners of a rotated square and
%   a center point.
% Then the square is offset by a constant vector added to all x pts.
% The y values are created as:
%  -1's at the corners and +4 at the center,
%  the y values are scaled by a constant,
%  the y values are shifted by a plane y = a1*x1+a2*x2*b.
% The plane in the last step is the multiple regression solution.

% Create the corners of the square and the center point at the origin.
X = [ 0,  0; 2,  1; -1,  2; -2, -1;  1, -2];

% Add offset to shift all x pts by vector.
X = X + ones(5,1)*[3,4];

% Create y pts as +4 at center and -1's at corners of square.
y = [4, -1, -1, -1, -1]';

% Scale y values by constant.
y = y * 3;

% Add offset and plane to y values.  ba = [b, a1, a2].
ba = [-2, -1, 5];

ones_X = [ones(5,1),X];

y = y + ones_X*ba';

% Print values.
data = [X,y]

% Now compute the multiple regression solution.
Xplus = pinv(ones_X);

% Find the hyperplane fit.
ba_soln = pinv(ones_X)*y

% Print out the pts for the fit.
fit = [X,ones_X*ba_soln]

hold off

% Make 3D plot.
plot3(data(:,1)',data(:,2)',data(:,3)','+k')
xlabel('x1'), ylabel('x2'), zlabel('y')
axis([0, 6, 0, 6, 0, 25])
grid on

hold on
for index = 1 : 5
  plot3([data(index,1);data(index,1)], ...
        [data(index,2);data(index,2)], ...
        [0,data(index,3)],'-k')
end

plot3(fit(:,1)',fit(:,2)',fit(:,3)','ok')

hold off