Here’s a Matlab function I wrote to draw a set of random numbers from a uniform distribution within an n-dimensional ellipsoid, where the size and orientation of the ellipsoid is defined by a covariance matrix.

function pnts = draw_from_ellipsoid( covmat, cent, npts )
% function pnts = draw_from_ellipsoid( covmat, cent, npts )
%
% This function draws points uniformly from an n-dimensional ellipsoid
% with edges and orientation defined by the the covariance matrix covmat.
% The number of points produced in the n-dimensional space is given by
% npts, with an output array of npts by ndims. The centre of each dimension
% is given in the vector cent.
% get number of dimensions of the covariance matrix
ndims = length(covmat);
% calculate eigenvalues and vectors of the covariance matrix
[v, e] = eig(covmat);
% check size of cent and transpose if necessary
sc = size(cent);
if sc(1) > 1
cent = cent';
end
% generate radii of hyperspheres
rs = rand(npts,1);
% generate points
pt = randn(npts,ndims);
% get scalings for each point onto the surface of a unit hypersphere
fac = sum(pt(:,:)'.^2);
% calculate scaling for each point to be within the unit hypersphere
% with radii rs
fac = (rs.^(1/ndims)) ./ sqrt(fac');
pnts = zeros(npts,ndims);
% scale points to the ellipsoid using the eigenvalues and rotate with
% the eigenvectors and add centroid
d = sqrt(diag(e));
for i=1:npts
% scale points to a uniform distribution within unit hypersphere
pnts(i,:) = fac(i)*pt(i,:);
% scale and rotate to ellipsoid
pnts(i,:) = (pnts(i,:) .* d' * v') + cent;
end
end

*Related*

Thank you so much for the article Matthew. Really helped me a lot 🙂

Thanks – I needed that and translated it into R. Will send you code if you want.

Thanks very much for this. In case anyone wants the code for Python, here you go:

Hi, thanks a lot! Can you point me to the underlying theory?

I get the intuition:

You sample points from a normal distribution and normalize them to be on the unit sphere. That way, you get points with a uniform distribution in their orientation. Then you combine the uniform orientations with uniform radii (is that results still uniform?). Where is the exponent in r^*(1/ndim) coming from?

Finally, you rotate and rescale.

Looking forward to some hint,

Felix

It seems similar to what is done here, even though I was hoping for something even simpler:

https://ieeexplore.ieee.org/document/758215

Someone needs this in c++, here you go..

Eigen::MatrixXf generate_samples_from_ellipsoid(Eigen::MatrixXf covmat, Eigen::VectorXf cent, int npts){

int ndims = covmat.rows();

Eigen::EigenSolver eigensolver;

eigensolver.compute(covmat);

Eigen::VectorXf eigen_values = eigensolver.eigenvalues().real();

Eigen::MatrixXf eigen_vectors = eigensolver.eigenvectors().real();

std::vector<std::tuple> eigen_vectors_and_values;

for(int i=0; i<eigen_values.size(); i++){

std::tuple vec_and_val(eigen_values[i], eigen_vectors.row(i));

eigen_vectors_and_values.push_back(vec_and_val);

}

std::sort(eigen_vectors_and_values.begin(), eigen_vectors_and_values.end(),

[&](const std::tuple& a, const std::tuple& b) -> bool{

return std::get(a) <= std::get(b);

});

int index = 0;

for(auto const vect : eigen_vectors_and_values){

eigen_values(index) = std::get(vect);

eigen_vectors.row(index) = std::get(vect);

index++;

}

Eigen::MatrixXf eigen_values_as_matrix = eigen_values.asDiagonal();

std::random_device rd{};

std::mt19937 gen{rd()};

std::uniform_real_distribution dis(0, 1);

std::normal_distribution normal_dis{0.0f, 1.0f};

Eigen::MatrixXf pt = Eigen::MatrixXf::Zero(npts, ndims).unaryExpr([&](float dummy){return (float)normal_dis(gen);});

Eigen::VectorXf rs = Eigen::VectorXf::Zero(npts).unaryExpr([&](float dummy){return dis(gen);});

Eigen::VectorXf fac = pt.array().pow(2).rowwise().sum();

Eigen::VectorXf fac_sqrt = fac.array().sqrt();

Eigen::VectorXf rs_pow = rs.array().pow(1.0/ndims);

fac = rs_pow.array()/fac_sqrt.array();

Eigen::MatrixXf points = Eigen::MatrixXf::Zero(npts, ndims);

Eigen::VectorXf d = eigen_values_as_matrix.diagonal().array().sqrt();

for(auto i(0); i<npts; i++){

points.row(i) = fac(i)*pt.row(i).array();

Eigen::MatrixXf fff = (points.row(i).array()*d.transpose().array());

Eigen::VectorXf bn = eigen_vectors*fff.transpose();

points.row(i) = bn.array() + cent.array();

}

std::cout << "points: " << points << std::endl;

return points;

}