How to fit a line on the plane?

How to fit a line onto dots on the plane? polyfit is not the solution, because it is good for functions and not for points on the plane. ax+b function fit fails for vertical lines for example, and noise model also differs

Answer 1

[David Goodmanson]

Hi M,

Here is a way to fit a line in 2d with the equation

a1*x1 + a2*x2 = 1

where (x1,x2) are (x,y). This representation gets rid of infinite slope problems.

The same code works in general in m dimensions to fit an m-1 dimensional plane

a1*x1 + a2*x2 + ... am*xm  =  1.

For n points, let X be an nxm matrix where each row in X represents a point in m dimensions.

X = rand(20,2)                % for example

Xcm = mean(X);                % coordinates of center of mass
[~,s,v] = svd(X-Xcm,'econ');
n = v(:,end);    % unit vector for the smallest singular value (they are sorted) 
L = Xcm*n;       % displacement of the best fit plane from the origin
if L < 0         % turn displacement into distance
  L = -L;
  n = -n;
end
a = n/L;                      % fitted plane, a1*x1 + a2*x2 + ... am*xm  =  1.
drms = rms(X*n -L);
% L = 1/norm(a), so drms also equals (1/norm(a))*rms(X*a-1)

drms is the rms distance of the set of points to the plane.

For older versions of Matlab, in the svd line you would have to use

X-repmat(Xcm,size(X,1),1)    in place of    X-Xcm