# effects of Box-Cox transformation on covariance

I'm trying to synthesize data for a Monte Carlo simulation. I have a stationary random process $x$ and can readily estimate its covariance matrix $S$. I know that if the increments of the process are normally distributed, the Cholesky Decomposition

$LL^T = S$

can be used to generate processes $y$ which posses the same covariance as the original process using

$Lv = y$,

where $v$ is a zero mean, unit variance vector with normally distributed elements.

Unfortunately the processes I am considering are not always normally distributed, but can be transformed fairly nicely using the Box-Cox transform. My question is: is it feasible to apply the Box-Cox transform $x_\lambda = f_\lambda(x)$ and use this process to estimate a new covariance matrix $S_\lambda$ which I could then use to generate the process $y_\lambda$. Finally I could use the inverse Box-Cox transform to get the process back into the original domain (which I need), i.e., $f^{-1}(y_\lambda)$?

Is such an approach sound? I'm concerned that something may be lost during the transformations or a bias introduced.

• Could you explain what you mean by "apply the transform $f_\lambda(x)$ obtain a covariance matrix"? Does this mean you would estimate $S_\lambda$ from the transformed process? If so, then what exactly is the relevance of the $S$ in the first place? – whuber Jan 13 '15 at 16:38
• Sorry for the confusion, I meant that I use the transformed process to estimate a new covariance matrix. The original $S$ can be discarded (AFAIK there is no simple transformation between the two) – ws6079 Jan 13 '15 at 16:45
• Just to be clear, then, it sounds like you would like to simulate your process $x$ by means of a nonlinear transformation of some Gaussian process $y$ and you propose to estimate the transformation from the data and then estimate the parameters of $y$ (its mean and covariance) from the transformed data. If that is so, then how is one to assess alternative approaches? What are they supposed to achieve? How are they to be compared to your proposed approach? – whuber Jan 13 '15 at 16:50
• Essentially I need many realizations of the process $x$ for the MC simulation. In the end, ideally the obtained process $y$ should have the same variance and covariance and distribution as the original process. However I'm wondering if something is lost during the transformations or if there a bias is introduced. (I don't need alternatives, if the outlined approach is sound). – ws6079 Jan 13 '15 at 16:58