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%% Generate and Exponentiate Normal Random Variables % % Copyright 2015 The MathWorks, Inc. %% % 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') %% % 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); %% % 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 $\rho$ 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.