how to fit a surface to 3d data points

Hi, I have a set of 3D data points (x,y,z) that I want to fit using the equation

A*x^2+B*y^2+C*x*y = [ z (1-z) ]^2

A, B, C being the parameters I have to estimate.

can anyone help me?

Answer 1

[Star Strider]

If your data points are each vectors, try this:

x = rand(1, 10);                                            % Create Data
y = rand(1, 10);                                            % Create Data
z = rand(1, 10);                                            % Create Data

DM = [x(:).^2, y(:).^2, x(:).*y(:)];                        % Design Matrix
P = DM \ (z(:).*(1-z(:))).^2;                               % Estimate Parameters

A = P(1);
B = P(2);
C = P(3)

Experiment to get the result you want.

[Alessandraro]

Thanks, the fact is now I changed the equation in

A*x^2+B*y^2+C*x*y = [ (z-D)/(1-D) (1-z) ]^2

A, B, C, D being the parameters I have to estimate.

Is it still possible to use your solution?

[Star Strider]

As always, my pleasure.

‘Is it still possible to use your solution?’

Yes. The ‘P’ calculation then becomes:

P = DM \ [(z(:)-D(:))./(1-D(:)) (1-z(:))].^2;               % Estimate Parameters

and ‘P’ is now a (3x2) matrix, so the parameters are now:

A = P(1,:);
B = P(2,:);
C = P(3,:);

[Alessandraro]

Thank you for your answer. Can I, also, ask you how to evaluate the goodness of the fit? R^2?

[Star Strider]

As always, my pleasure.

I’m not certain which model you’re referring ot, so I calculated it for both:

x = rand(1, 10);                                            % Create Data
y = rand(1, 10);                                            % Create Data
z = rand(1, 10);                                            % Create Data
D = rand(1, 10);                                            % Create Data

DM = [x(:).^2, y(:).^2, x(:).*y(:)];                        % Design Matrix
P = DM \ (z(:).*(1-z(:))).^2;                               % Estimate Parameters
SStot = sum(((z(:).*(1-z(:))).^2 - mean(z(:))).^2);
SSres = sum(((z(:).*(1-z(:))).^2 - DM*P).^2);
Rsq = 1 - SSres/SStot

P = DM \ [(z(:)-D(:))./(1-D(:)) (1-z(:))].^2;               % Estimate Parameters
SStot = sum(([(z(:)-D(:))./(1-D(:)) (1-z(:))].^2 - mean(z(:))).^2);
SSres = sum(([(z(:)-D(:))./(1-D(:)) (1-z(:))].^2 - DM*P).^2);
Rsq = 1 - SSres/SStot

[Alessandraro]

Thank you. I am sorry, can I ask you why you calculate SStot

SStot = sum(([(z(:)-D(:))./(1-D(:)) (1-z(:))].^2 - mean(z(:))).^2);

calculating mean(z(:)) instead of mean([(z(:)-D(:))./(1-D(:)) (1-z(:))].^2)?

Thank you.

[Star Strider]

Because that is how ‘SStot’ is calculated, at least as I interpret it in the context of yoiur model. See the Wikipedia article on: Coefficient of determination (link).