How to fit largest ellipsoid for 3d data points such that it covers all points?

Hi, I have search a lot for ellipsoid fit to 3d data and come up with some answer but I want some improvement in my method such that it covers all the data points. My code is here.

Y=data;  % data in non-principle coordinate 
X = data;
oldmu =mean(X);
X= bsxfun(@minus,X,oldmu); % mean-centered data
[pc val] = eig(X'*X  ); 

nC = X*pc;   % data in principle coordinate 
a2=(max(nC(:,1))-min(nC(:,1)))/2;  %range of data in X 
b2=(max(nC(:,2))-min(nC(:,2)))/2;  %range of data in Y
c2=(max(nC(:,3))-min(nC(:,3)))/2;  %range of data in Z
[x, y, z] = ellipsoid(0,0,0,a2,b2,c2,40); 
% fit ellipsoid in principle coordinate 

   tt=[x(:) y(:) z(:)]*inv(pc);%ellipsoid in non-principle coordinate
nx=reshape(tt(:,1),size(x,1),size(x,2));
ny=reshape(tt(:,2),size(x,1),size(x,2));
nz=reshape(tt(:,3),size(x,1),size(x,2));
% plot ellipsoid in non-principle coordinate 
hSurface=surf(nx+oldmu(1), ny+oldmu(2), nz+oldmu(3),    'FaceColor','r','EdgeColor','none','FaceAlpha',0.6);

   hold on 
   %plot data in non-principle coordinate 
   plot3(Y(:,1),Y(:,2),Y(:,3),'k.-')

end

The data file is here.

[Baptiste Ottino]

What do you mean by "covers all the data points". Do you want an ellipsoid that encompasses all the data points? Or is all your data located on the surface of an ellipsoid for sure?

Answer 1

[Image Analyst]

I think it's pretty difficult. John did the same thing in 2-D here http://www.mathworks.com/matlabcentral/fileexchange/34767-a-suite-of-minimal-bounding-objects If it were easy he'd probably have done it already.

[ankit agrawal]

Thank you for answer. Someone told me about Khachiyan Algorithm and it is working very fine. https://github.com/minillinim/ellipsoid/blob/master/ellipsoid.py