Variable transformation for a multivariate normal distribution

[nilsgeiger]

TL;DR Summary

I want to transform a multivariate normal [$log (\delta_i + 1)$] distribution to a multivariate normal distribution of the $\delta_i$ .
Besides, i'm looking for a way to transform the random vectors with the components $log (\delta_i + 1)$ to vectors with components $\delta_i$.

Hello.

I would like to draw (sample) several random vectors x from a n-dimensional multivariate normal distribution.
For this purpose I want to use C++ and the GNU Scientific Library function gsl_ran_multivariate_gaussian .

https://www.gnu.org/software/gsl/manual/html_node/The-Multivariate-Gaussian-Distribution.html

The distribution has the usual density
$$p(x_1,\dots,x_k) dx_1 \dots dx_k = {1 \over \sqrt{(2 \pi)^k |\Sigma|}} \exp \left(-{1 \over 2} (x - \mu)^T \Sigma^{-1} (x - \mu)\right) dx_1 \dots dx_k$$
with $$\mu = 0$$ but with
$$ x = \begin{pmatrix} log (\delta_1 + 1) \\
log (\delta_2+1) \\
log (\delta_3 + 1) \\
log (\delta_4 + 1) \\
... \\
log (\delta_n + 1) \\
\end{pmatrix}$$

As stated the $log (\delta_i + 1)$ follow the multivariate normal distribution.

But I am actually only interested in the $\delta$ -vectors.
$$ \delta = \begin{pmatrix} \delta_1 \\
\delta_1 \\
\delta_2 \\
\delta_3 \\
... \\
\delta_n \\
\end{pmatrix}$$

1. How do you transform a x - vector to a $\delta$ - vector?
   With help of the covariances? But how exactly?
2. Alternatively, can you do a change of variables to the multivariate distribution of the $\delta_i$ und draw $\delta$ - vectors directly with the gsl_ran_multivariate_gaussian?
   Could you please tell me the formula to compute the appropriate new covariance matrix?
   Or is this not possible?
   I am aware that the multiariate log-normal distribution exists, but GSL can only sample the multivariate normal.

I'm so sorry, this are probably really stupid questions.
But I'm just a not particularly good bachelor physics student in his fourth semester who also started programming c++ for the very first time.
I'm really overwhelmed and began learning about multivariate statistics for the first time because of this task no more than a week ago.

It would really help me a lot if you could answer and explain my two questions in great detail and for idiots.

For literature references for general variable transformations for multivaraite distributions and multivariate normal distributions I would also be very very thankful.
Especially for multivariate normal distributions of $(log (x_i+1) )$ there must be formulas together with a detailed derivation, right?
Normally distributed logarithms have to occur and $+1$ just ensures that for $x_i$ greater zero the logarithm always remains positive, so they should also be quite common?

 

[andrewkirk]

From what you've written it sounds like you already know how to randomly generate x vectors from the required multivariate normal distribution. That's the difficult bit done. From there it's easy. Since for $i=1,2,...,n$ we have $x_i = \log(\delta_i+1)$, you can calculate a simulated $\mathbf \delta$ vector from its corresponding x vector by calculating $\delta_i = e^{x_i}-1$, where $\delta_i$ and $x_i$ are the $i$-th components of the $\mathbf \delta$ and x vectors respectively.