The mean is a linear combination of regression functions stored in f (x i ) with coefficient vector β . The (i, j)th element k(x i , x j ; φ , σ 2 , τ 2 ) of the covariance matrix K(x 1 , . . . x n ; φ , σ 2 , τ 2 ) is defined through the correlation kernel κ(x i , x j ; φ ), variance parameter σ 2 > 0 and a regularization parameter, or nugget, τ 2 ≥ 0. For non-deterministic data, the nugget measures random variation around the mean response. For deterministic systems, addition of a nugget is still common and beneficial, improving the conditioning of the prior covariance matrix and providing some robustness of the assumed form of the correlation kernel. Hence