  We address the issue of knots selection for Gaussian predictive process
methodology. Predictive process approximation provides an effective solution to
the cubic order computational complexity of Gaussian process models. This
approximation crucially depends on a set of points, called knots, at which the
original process is retained, while the rest is approximated via a
deterministic extrapolation. Knots should be few in number to keep the
computational complexity low, but provide a good coverage of the process domain
to limit approximation error. We present theoretical calculations to show that
coverage must be judged by the canonical metric of the Gaussian process. This
necessitates having in place a knots selection algorithm that automatically
adapts to the changes in the canonical metric affected by changes in the
parameter values controlling the Gaussian process covariance function. We
present an algorithm toward this by employing an incomplete Cholesky
factorization with pivoting and dynamic stopping. Although these concepts
already exist in the literature, our contribution lies in unifying them into a
fast algorithm and in using computable error bounds to finesse implementation
of the predictive process approximation. The resulting adaptive predictive
process offers a substantial automatization of Guassian process model fitting,
especially for Bayesian applications where thousands of values of the
covariance parameters are to be explored.
