I'm doing a Gaussian Process regression using a SquaredExponential kernel. Due to very different scales in my multidimensional inputs -some components lie in [0, 0.2] while others in [1000, 1000000]- I decided to standardize the inputs so that all their components lie in [0,1]. I standardized the outputs as well so that they have 0 mean and they lie in [-0.5, 0.5].
Testing it on a known function I can reconstruct it from few points and I am able to rescale it to its original size. However, I don't know what to do to scale back the variance of the predicted points, since I assume it is smaller due to the standardization. How can I scale it back to the size it's supposed to be?