Computes the minimumvolume covering ellipoid that encloses N points in a Ddimensional space.
[A , c] = MinVolEllipse(P, tolerance)
Finds the minimum volume enclosing ellipsoid (MVEE) of a set of data points stored in matrix P. The following optimization problem is solved:
minimize log(det(A))
s.t. (P_i  c)'*A*(P_i  c)<= 1
in variables A and c, where P_i is the ith column of the matrix P.
The solver is based on Khachiyan Algorithm, and the final solution is different from the optimal value by the prespecified amount of 'tolerance'.

Outputs:
c : Ddimensional vector containing the center of the ellipsoid.
A : This matrix contains all the information regarding the shape of the ellipsoid. To get the radii and orientation of the ellipsoid take the Singular Value Decomposition ( svd function in matlab) of the output matrix A:
[U Q V] = svd(A);
the radii are given by:
r1 = 1/sqrt(Q(1,1));
r2 = 1/sqrt(Q(2,2));
...
rD = 1/sqrt(Q(D,D));
and matrix V is the rotation matrix that gives you the orientation of the ellipsoid.
For plotting in 2D or 3D, use MinVolEllipse_plot.m (see the link bellow)
1.2  A sample code is provided in the help section that shows a method to reduce the computation time drastically. 

To explain the outputs of the function. 
Inspired: Plot an ellipse in "center form", Approximate Lowner Ellipsoid, fakenmc/amvidc
Peter Lawrence (view profile)
I forgot to add to my original comment that, upon calculating [C, m] = MinVolEllipse(data,tol), you need to first invert the matrix. So please use
C=inv(C);
prior to the eigenvalue decomposition.
Peter Lawrence (view profile)
Works very well for low dimensional data.
@Ali, if you want to plot the ellipse obtained by [C, m] = MinVolEllipse(data,tol), use the following:
tq=linspace(pi,pi,M); % for display purposes, M is the number of points on the ellipse
[Ve,De]=eig(C);
De=sqrt(diag(De));
[l1,Ie] = max(De);
veig=Ve(:,Ie);
thu=atan2(veig(2),veig(1));
l2=De(setdiff([1 2],Ie));
U=[cos(thu) sin(thu);sin(thu) cos(thu)]*[l1*cos(tq);l2*sin(tq)];
plot(U(1,:)+m(1),U(2,:)+m(2))
Vincent Phan (view profile)
Ali Braytee (view profile)
Ali Braytee (view profile)
Hi Nima,
How to get the vertices or edge point of the ellipsoid??
`bg` (view profile)
Hi Nima,
There are a few optimization for your code which could make it much faster and more memory efficient.
Replace X = Q * diag(u) * Q';
by X = bsxfun(@times, Q, u') * Q';
Replace M = diag(Q' * inv(X) * Q);
by M = sum((X\Q).*Q,1);
And finally use:
c = P*u;
A = (1/d) * inv(bsxfun(@times, P, u')*P'  c*c');
Best,
`bg`
Uri Cohen (view profile)
Thanks for the great code!
I get the following warnings on numerical stability when calculating bounding ellipsoid of ~50 points in ~1000 dimensional space:
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND = 2.952356e23.
> In MinVolEllipse at 67
In calc_ellipsoid_properties at 33
Warning: Matrix is close to singular or badly scaled. Results may be inaccurate.
RCOND = 1.659718e21.
> In MinVolEllipse at 89
In calc_ellipsoid_properties at 33
Is the result valid? Is there a way to avoid this?
Nima (view profile)
The paper on MVEE can be downloaded here: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.116.7691
Muhammad Anoosh (view profile)
Thank you for your code. I have two questions:
Question 1 is related to your paper on MVEE and question 2 is regarding the link you mentioned in the comments and ratings of this code.
Q.1 You have mentioned in your paper that the volume of Ellipse is given by v0/(det(sqrt(E)). Can you please elaborate how have you obtained this volume (any reference)
Q.2 You have mentioned a link to the (unpublished) paper describing the algorithm and math behind the code:
https://sites.google.com/site/nimamoshtagh/software/MVEE.pdf?attredirects=0
The above link is not working can you please give the latest link.
Stefan (view profile)
Thank you for this code, helps a lot! I am also looking for a method of finding the minimum volume enclosing spheroid, i.e. an ellipsoid where two of the three semiaxes are identical. Do you know of any algorithms for this problem?
David (view profile)
Hmmm....
det(A) = pi^2 / area^2,
so it seems the optimization for minimum area is to maximize, rather than minimize, log(det(A)).
Also, for small d, it may be easier to obtain semi major/minor lengths (a, b) by finding the eigenvalues/vectors of A (eigenvalues are 1/a^2, 1/b^2, etc, rotation matrix columns are the eigenvectors).
I'm finding this algorithm gives me det(A) < 0 (a hyperbola) for some point sets. Also, A should be invariant under translation of all the input points (which should just result in a translation of c), but I find it is quite sensitive.
Anton Semechko (view profile)
sweet, thanks!
Pengju (view profile)
Javier (view profile)
Hi,
thanks for sharing the code it's really good. One question, please could you explain me which it's the differente between this method and least squares?
Thank you very much
banjo (view profile)
thanks. This is good.
Sepehr Farhand (view profile)
Glib (view profile)
Nima Moshtagh (view profile)
Here is a link to the (unpublished) paper describing the algorithm and math behind the code:
https://sites.google.com/site/nimamoshtagh/software/MVEE.pdf?attredirects=0
Reza Jafari (view profile)
I would like to thank you the author for the useful code which he developed. It solved many of my problems. The only problem which I faced was due to the singularity of Matrix X in line 76 and singularity of Matrix P * U * P'  (P * u)*(P*u)' in line 96. Do you have any suggestion to avoid singularity for these two matrices?
Appreciated.
Regards,
Reza
Rashed (view profile)
Hi,
I have a problem in 3D. Say, I have an 3D array of P(5,5,5) and how can I make this to work? Please suggest. Or, do I need to produce a matrix P containing the coordinates of the region?
Rashed (view profile)
Thanks a lot. I was looking for something like this.
Nima Moshtagh (view profile)
Here is a paper describing the algorithm and math behind the code: http://www.seas.upenn.edu/~nima/papers/Mim_vol_ellipse.pdf
Avinash Uppuluri (view profile)
Can you also provide a good journal reference for this algorithm. Thanks!
Avinash Uppuluri (view profile)
Hello Nima Moshtagh, The code posted was very useful. Is there a way to add a weight to the points, giving the importance of each point to be included within the ellipse. (Instead of using the tolerance which excludes points through the periphery). With the idea that points that are of less significance need not be included in the ellipse. Where can I include such a weighing parameter. I can probably do it in the preprocessing stage but is there a way to include it within your code. Thanks!
THank you
Very good program. It has helped me a lot to do with the matlab. Thanks to Nima Moshtagh for submitting the code.
Dear Ondrej Danko,
Thank you for bringing this point to my attention. The outcome that a number of points fall outside of the ellipsoid is expected, because we are truncating the algorithm before reaching its optimal value. The amount of error that we will get by doing this, is of the order of parameter 'tolerance', which represents the amount of error you can tolerate in the final result. For example for t=0.001 the average distance of an outside point is of the order of 10^3.
"Computes the minimumvolume covering ellipoid that encloses N points in a Ddimensional space"...
I would expect that resulting ellipsoid COVERS all points, I mean no points are outside of the resulting ellipsoid...
I had ran a small test if it is so... and surprisingly not all points are necessarily inside.
P = rand(3,N);
[A , C] = MinVolEllipse(P, t);
for j = 1:N
if ((P(:,j)C)'*A*(P(:,j)C) > 1)
disp('Point is outside...');
end
end
I am missing something, or it is expected? If expected, any suggestions to modify the algorithm to output ellipsoid strictly covering all points?
I find that the run time can be radically improved by recognizing that the only points that determine the bounding ellipsoid are on the convex hull, as used by John D'Errico in his minboundsphere. Given points P, the following code finds the reduced set of points lying on the hull.
K=convhulln(P.');
K=unique(K(:));
Q=P(:,K);
for P=randn(3,500)
my computer runs MinVolEllipse(P, .001) in 12.54 seconds
and finds Q followed by MinVolEllipse(Q, .001) in 0.08 seconds
This is a very good program for doing minimum volume ellipsoids. it has been very helpful and the author has as well.
Excellent stuff! This was exactly what I was looking for and works quite well. Although the math behind it all was a bit beyond me, the author was more than helpful in helping me make the most of what the code has to offer, thanks again Nima!
For tolerance=0.01 in 2D here is how long it takes (approximatly) to find the covering ellipse on a 2.4GHz computer:
#point time(sec)
 
100 0.03
500 1.24
1000 5.20
1500 13.26
2000 25.27
This could be useful IF you have small sets of data.
I tested this for several problems, in 1, 2 and 5 dimensions. All seemed to work. The only downside is data of even moderate size. The author gives an example with 100 points, solved fairly quickly. But expand that to only 500 points (in 2d) and it takes 140 seconds to run on my computer.