Generating 2D different shapes with same mean and covariance matrix

Question

Im trying to reproduce the synthetic data used in this article.

The authors claim the data was generated from densities with the same mean and covariance matrix.

Is there any principled way I to generate such data in matlab? I know the functions to generate random values but I don't know how to make "good guesses" on the covariance matrix to generate this kind of dataset. It's important for me that I keep the same mean and covariance matrix across the examples.

Answer 1

If you can generate data roughly in the shape you want, you can impose your own mean and covariance to match the ones you have in the first sample. This should be similar in any software. Here's an example using MATLAB code to generate data in the shape of an 8 and to give it the desired mean and covariance.

First generate two noisy circles, one on top of the other, not worrying about their means or covariances

% Generate Figure 8
theta = linspace(0,2*pi)';
x1 = [sin(theta),cos(theta)] + randn(length(theta),2)/20;
x2 = [sin(theta),cos(theta)] + randn(length(theta),2)/20;
x1(:,2) = x1(:,2)+1;     % top of figure 8
x2(:,2) = -x2(:,2)-1;    % bottom of figure 8
x = [x1;x2];
subplot(1,2,1)
plot(x(:,1),x(:,2),'b.')

Now standardize. First subtract the column means from X using bsxfun to get Y. Next remove the covariance. Suppose the covariance of Y is $C=cov(Y)$ and it has cholesky factor A, $C=A'A$. Then the covariance of $Z=YA^{-1}$ (written Y/A) is $cov(Z)=A'^{-1}(A'A)A^{-1}=I$, the identity.

% Remove existing mean and covariance
y = bsxfun(@minus,x,mean(x));
z = y / chol(cov(y));

So if I want mean 0 and covariance I, I'm done. But suppose I generated the first set of points and found they have some particular mean M and covariance C, such as

% Mean and covariance I want
M = [9 12];
C = [20 5;5 10];

I can modify my generated points to have the same mean and covariance

% Impose these
y = z*chol(C)            % if A=chol(C), cov(y)=A'*A=C
x = bsxfun(@plus,M,y);   % add column means M to columns of y
mean(x)
cov(x)
subplot(1,2,2)
plot(x(:,1),x(:,2),'r.')

The resulting figure has points roughly in the same of a figure 8, but slanted somewhat to match the desired covariance.