Generate and Exponentiate Normal Random Variables
The lognrnd function simulates independent lognormal random variables. In the following example, the mvnrnd function generates n pairs of independent normal random variables, and then exponentiates them. Notice that the covariance matrix used here is diagonal.
n = 1000; sigma = .5; SigmaInd = sigma.^2 .* [1 0; 0 1] rng('default'); % For reproducibility ZInd = mvnrnd([0 0],SigmaInd,n); XInd = exp(ZInd); plot(XInd(:,1),XInd(:,2),'.') axis([0 5 0 5]) axis equal xlabel('X1') ylabel('X2')
SigmaInd = 0.2500 0 0 0.2500
Dependent bivariate lognormal random variables are also easy to generate using a covariance matrix with nonzero off-diagonal terms.
rho = .7; SigmaDep = sigma.^2 .* [1 rho; rho 1] ZDep = mvnrnd([0 0],SigmaDep,n); XDep = exp(ZDep);
SigmaDep = 0.2500 0.1750 0.1750 0.2500
A second scatter plot demonstrates the difference between these two bivariate distributions.
plot(XDep(:,1),XDep(:,2),'.') axis([0 5 0 5]) axis equal xlabel('X1') ylabel('X2')
It is clear that there is a tendency in the second data set for large values of X1 to be associated with large values of X2, and similarly for small values. The correlation parameter of the underlying bivariate normal determines this dependence. The conclusions drawn from the simulation could well depend on whether you generate X1 and X2 with dependence. The bivariate lognormal distribution is a simple solution in this case; it easily generalizes to higher dimensions in cases where the marginal distributions are different lognormals.
Other multivariate distributions also exist. For example, the multivariate t and the Dirichlet distributions simulate dependent t and beta random variables, respectively. But the list of simple multivariate distributions is not long, and they only apply in cases where the marginals are all in the same family (or even the exact same distributions). This can be a serious limitation in many situations.