I have some data that are clearly positively skewed and follow a log-normal distribution, lets assume the initial data is $Z = exp(Y)$, where $Y \sim N(\mu,\sigma^2)$.
A Gaussian process assumes that any subsets from the data is normally distributed. So I transformed the data $Z$ to $Y = ln(Z)$, to make it more Gaussian, and I performed a conventional Gaussian process regression. In the end I predict a future point $Y_*$ where $Y_*\sim N(\mu_*,\sigma_*^2)$.
1- How do I back-transform to the original log scale? i.e., how do I convert $Y_*$ back to $Z_*$? In log-normal kriging they predict the arithmetic mean of the $Z_*$, which is $$E(Z_*) = e^{(\mu_*+\sigma_*^2/2)}.$$ The variance of $Z_*$ will be $$Var(Z_*) = e^{(2\mu_*+\sigma_*^2)}*(e^{\sigma_*^2}-1).$$ Is this correct? or should I calculate the geometric mean of $Z_*$, $E(Z_*) = exp(\mu_*)$, instead?
2- How do I calculate the prediction interval as defined in this link? Normally for Gaussian process the prediction interval of 95% is calculated using the borders $\mu_* \pm 1.96\sigma^2$. How do I calculate the same for the back-transformed data (log-normal scale)? Note you cannot exponentiate the borders from the Gaussian scale to get the prediction interval (or in the reference's case the confidence interval) in the log-normal scale as shown in.
