Best fit plane for 3D data

Question

Ok, so the goal is to do a planar fit to your data. We presume that there MAY be NaN elements in at some of the arrays. So lets do an errors in variables fit to the data.

% First, remove NaNs. All the NaN (if any) will be in the same places,
% so we need do only one test. Testing each of X,Y,Z leads to confusion
% and leads to the expectation that the NaNs are NOT in the same places
% with some potential.
k = isfinite(X);
if all(k(:))
  % in this case there were no nans, so reshape the arrays into vectors
  % While X(k) would convert an array into a column vector anyway in
  % this case, it seems far more sane to do the reshape explicitly,
  % rather than let X(k) do it implicitly. Again, a source of ambiguity
  % removed.
  X = X(:);
  Y = Y(:);
  Z = Z(:);
else
  X = X(k);
  Y = Y(k);
  Z = Z(k);
end

% Combine X,Y,Z into one array
XYZ = [X,Y,Z];

% column means. This will allow us to do the fit properly with no
% constant term needed. It is necessary to do it this way, as
% otherwise the fit would not be properly scale independent
cm = mean(XYZ,1);

% subtract off the column means
XYZ0 = bsxfun(@minus,XYZ,cm);

% The "regression" as a planar fit is now accomplished by SVD.
% This presumes errors in all three variables. In fact, it makes
% presumptions that the noise variance is the same for all the
% variables. Be very careful, as this fact is built into the model.
% If your goal is merely to fit z(x,y), where x and y were known
% and only z had errors in it, then this is the wrong way
% to do the fit.
[U,S,V] = svd(XYZ0,0);

% The singular values are ordered in decreasing order for svd.
% The vector corresponding to the zero singular value tells us
% the direction of the normal vector to the plane. Note that if
% your data actually fell on a straight line, this will be a problem
% as then there are two vectors normal to your data, so no plane fit.
% LOOK at the values on the diagonal of S. If the last one is NOT
% essentially small compared to the others, then you have a problem
% here. (If it is numerically zero, then the points fell exactly in
% a plane, with no noise.)
diag(S)

% Assuming that S(3,3) was small compared to S(1,1), AND that S(2,2)
% is significantly larger than S(3,3), then we are ok to proceed.
% See that if the second and third singular values are roughly equal,
% this would indicate a points on a line, not a plane.
% You do need to check these numbers, as they will be indicative of a
% potential problem.
% Finally, the magnitude of S(3,3) would be a measure of the noise
% in your regression. It is a measure of the deviations from your
% fitted plane.

% The normal vector is given by the third singular vector, so the
% third (well, last in general) column of V. I'll call the normal
% vector P to be consistent with the question notation.
P = V(:,3);

% The equation of the plane for ANY point on the plane [x,y,z]
% is given by
%
%    dot(P,[x,y,z] - cm) == 0
%
% Essentially this means we subtract off the column mean from our
% point, and then take the dot product with the normal vector. That
% must yield zero for a point on the plane. We can also think of it
% in a different way, that if a point were to lie OFF the plane,
% then this dot product would see some projection along the normal
% vector.
%
% So if your goal is now to predict Z, as a function of X and Y,
% we simply expand that dot product.
% 
%   dot(P,[x,y,z]) - dot(P,cm) = 0
%
%   P(1)*X + P(2)*Y + P(3)*Z - dot(P,cm) = 0
%
% or simply (assuming P(3), the coefficient of Z) is not zero...

Zhat = (dot(P,cm) - P(1)*X) - P(2)*Y)/P(3);

Be careful, in that if the coefficient of Z was small, then the resulting prediction will be very sensitive to small perturbations in that coefficient. Essentially it would means that the plane was a vertical plane, or nearly so.

Again, all of this assumes that your goal was truly an errors in variables computation, where we had noise in all three variables X,Y,Z. If the noise was only in Z, then the proper solution is far simpler.