MATLAB: Generating correlated random variables

correlated random numbersrandom number generatorStatistics and Machine Learning Toolbox

Dear all,
I have already been reading a little bit about this issue, and I see, that if the variables are not normally distributed, the problem is not at all trivial.
My specific problem is: I need three variables; first and second has lognormal distribution (mu1, sigma1, mu2, sigma2 specified). The third variable has uniform distribution on a given interval. Even the full (3×3) correlation matrix is specified.
For the first two variables I can use MvLogNRand on File Exchange, but can't cope with the third one. I don't know if copula method works here. Code from stat gurus would be appreciated.
Thanks in advance!

Best Answer

The Higher-Order Copulas section of the documentation on Simulating Dependent Random Variables Using Copulas has a very good explanation of the general approach.
The first example in that section shows how to generate three correlated distributions. I've adapted that to your case, using two lognormals and one uniform distribution. Note that it is crucial that MATLAB has the ability to generate the inverses of all those distributions, because that is key to the copula method.
mu1    = 0;
sigma1 = 0.5;
mu2    = 0;
sigma2 = 0.5;
a3     = 0;
b3     = 1;
figure
% subplot(1,1,1);
n = 5000;
Rho = [1.0  0.4  0.2;
       0.4  1.0 -0.8;
       0.2 -0.8 1.0];
Z = mvnrnd([0 0 0], Rho, n);
U = normcdf(Z,0,1);
X = [logninv(U(:,1),mu1,sigma1) logninv(U(:,2),mu2,sigma2) unifinv(U(:,3),a3,b3)];
plot3(X(:,1),X(:,2),X(:,3),'.');
grid on;
view([-55, 15]);
xlabel('U1');
ylabel('U2');
zlabel('U3');
figure
subplot(3,1,1); histogram(X(:,1)); xlabel('U1')
subplot(3,1,2); histogram(X(:,2)); xlabel('U2')
subplot(3,1,3); histogram(X(:,3)); xlabel('U3')
See the resulting distributions below. They distributions also have the expected correlation structure, as can be shown using
corr(X)
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