  We develop a computational procedure to estimate the covariance
hyperparameters for semiparametric Gaussian process regression models with
additive noise. Namely, the presented method can be used to efficiently
estimate the variance of the correlated error, and the variance of the noise
based on maximizing a marginal likelihood function. Our method involves
suitably reducing the dimensionality of the hyperparameter space to simplify
the estimation procedure to a univariate root-finding problem. Moreover, we
derive bounds and asymptotes of the marginal likelihood function and its
derivatives, which are useful to narrowing the initial range of the
hyperparameter search. Using numerical examples, we demonstrate the
computational advantages and robustness of the presented approach compared to
traditional parameter optimization.
