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I am studying the Gaussian Process Regression for a personal project and knowledge. I know that Sigma is the noise standard deviation used by the algorithm. Anyway, looking in Matlab library, it set the initial value equal to std(y)/sqrt(2). Why did the software not use the std(y)? There is something in the literature about that?

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In a GP regression, the noise variance will be some fraction of the total sample variance. Assuming that you're using the initial value to find point estimates of the hyperparameters that maximize the marginal likelihood, the results can be very sensitive to the initial guesses. You wouldn't want to set the initial guess for sigma to std(y) because that's the maximum possible value. Setting the initial guess for the standard deviation to std(y) / sqrt(2) means you're setting the variance to half the var(y), which is halfway between it's minimum and maximum values, which I suppose is a reasonable starting point. In practice, because the optimization can be sensitive to the initialization, you'd want to try a few different initializations.

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