  Gaussian Processes (GPs) are a popular approach to predict the output of a
parameterized experiment. They have many applications in the field of Computer
Experiments, in particular to perform sensitivity analysis, adaptive design of
experiments and global optimization. Nearly all of the applications of GPs
require the inversion of a covariance matrix that, in practice, is often
ill-conditioned. Regularization methodologies are then employed with
consequences on the GPs that need to be better understood.The two principal
methods to deal with ill-conditioned covariance matrices are i) pseudoinverse
and ii) adding a positive constant to the diagonal (the so-called nugget
regularization).The first part of this paper provides an algebraic comparison
of PI and nugget regularizations. Redundant points, responsible for covariance
matrix singularity, are defined. It is proven that pseudoinverse
regularization, contrarily to nugget regularization, averages the output values
and makes the variance zero at redundant points. However, pseudoinverse and
nugget regularizations become equivalent as the nugget value vanishes. A
measure for data-model discrepancy is proposed which serves for choosing a
regularization technique.In the second part of the paper, a distribution-wise
GP is introduced that interpolates Gaussian distributions instead of data
points. Distribution-wise GP can be seen as an improved regularization method
for GPs.
