In a previous post, I asked the question if the method described here can be used with non-Gaussian distributions.
Let us explore that by considering two independent zero mean, unit variance distributions that are not Gaussian. Let us sample \(x_1\) from a triangular distribution, and \(x_2\) from a uniform distribution.
We consider a triangular distribution with zero mean and unit variance, which is symmetric about zero (spans -sqrt(6) to +sqrt(6)). Similarly, we consider a symmetric uniform distribution, which spans -sqrt(3) to +sqrt(3).
Samples from these independent random variables are shown below.
Let us explore that by considering two independent zero mean, unit variance distributions that are not Gaussian. Let us sample \(x_1\) from a triangular distribution, and \(x_2\) from a uniform distribution.
We consider a triangular distribution with zero mean and unit variance, which is symmetric about zero (spans -sqrt(6) to +sqrt(6)). Similarly, we consider a symmetric uniform distribution, which spans -sqrt(3) to +sqrt(3).
Samples from these independent random variables are shown below.
When we use a correlation coefficient of 0.2, and use the previous recipe, we get correlated random variables with zero mean and the same covariance matrix, but ...
... the marginals are not exactly the same!
This is evident when we increase the correlation coefficient to say 0.5.
The sharp edges of the uniform distribution get smoothened out.
Did the method fail?
Not really. If you paid attention, the method is designed to preserve the mean and the covariance matrix (which is does). It doesn't really guarantee the preservation of the marginal distributions.
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