  Low-rank approximation is a popular strategy to tackle the "big n problem"
associated with large-scale Gaussian process regressions. Basis functions for
developing low-rank structures are crucial and should be carefully specified.
Predictive processes simplify the problem by inducing basis functions with a
covariance function and a set of knots. The existing literature suggests
certain practical implementations of knot selection and covariance estimation;
however, theoretical foundations explaining the influence of these two factors
on predictive processes are lacking. In this paper, the asymptotic prediction
performance of the predictive process and Gaussian process predictions is
derived and the impacts of the selected knots and estimated covariance are
studied. We suggest the use of support points as knots, which best represent
data locations. Extensive simulation studies demonstrate the superiority of
support points and verify our theoretical results. Real data of precipitation
and ozone are used as examples, and the efficiency of our method over other
widely used low-rank approximation methods is verified.
