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A simple method to generate correlated lognormal variables $X_i$ that obey a covariance matrix $C_{\mathrm{ln}}$ with elements $c_{\mathrm{ln}}^{ij}$ is to first compute the covariance matrix $C_{\mathrm{g}}$ for the associated Gaussian variables $\ln(X_i)$, where the elements $c_{\mathrm{g}}^{ij}$ of $C_{\mathrm{g}}$ are given by:

$c_{\mathrm{g}}^{ij} = \ln \left(\frac{c_{\mathrm{ln}}^{ij}}{\langle X_i\rangle\langle X_j\rangle}+1\right)$,

where $\langle X_i \rangle$ is the mean of $X_i$. Then, we use $C_{\mathrm{g}}$ to generate correlated Gaussian variables that are later exponentiated to return $X_i$.

My question is: why some positive-definite $C_{\mathrm{ln}}$ result in non-positive-definite $C_{\mathrm{g}}$? For a $2\times2$ covariance matrix the answer is here, but even when such restrictions are satisfied the resulting $C_{\mathrm{g}}$ might still be non-positive-definite for $N\times N$ matrices with $N>2$.

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hsxavier
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    can you provide a reference for this property about the covariance matrix? and explicit the symbol $$? – Xi'an Jul 30 '15 at 17:04
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    @Xi'an See https://en.wikipedia.org/wiki/Log-normal_distribution#Multivariate_log-normal. The angle brackets are a notation used by physicists to denote expectations. – whuber Jul 30 '15 at 20:47
  • @Xi'an I change the text, now it explains $\langle X_i \rangle$. One reference for the equation relating $c_{\mathrm{g}}^{ij}$ and $c_{\mathrm{ln}}^{ij}$ is Appendix B of [Chiang et al. 2013](http://arxiv.org/abs/1306.4157v2) – hsxavier Aug 02 '15 at 08:44

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Not all positive-definite $C_{ln}$ give valid $C_g$ because some covariance matrices are impossible to reach in this setup, because of the non-linearity between the original random variable and the variable of interest.

For example, if you considered the following problem: $X_1$ and $X_2$ and gaussian random variables of mean 0, possibly correlated, and $Y_1 = X_1$ and $Y_2 = (X_2)^2$ Then $Y_1$ and $Y_2$ would always be uncorrelated, no matter the original correlation between $X_1$ and $X_2$

A similar phenomenon is taking place for the log-normal tuning-curves. Because you go through the exponential non-linearity, some correlation values become impossible to reach

Guillaume Dehaene
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