  In statistical modeling with Gaussian Process regression, it has been shown
that combining (few) high-fidelity data with (many) low-fidelity data can
enhance prediction accuracy, compared to prediction based on the few
high-fidelity data only. Such information fusion techniques for multifidelity
data commonly approach the high-fidelity model $f_h(t)$ as a function of two
variables $(t,y)$, and then using $f_l(t)$ as the $y$ data. More generally, the
high-fidelity model can be written as a function of several variables
$(t,y_1,y_2....)$; the low-fidelity model $f_l$ and, say, some of its
derivatives, can then be substituted for these variables. In this paper, we
will explore mathematical algorithms for multifidelity information fusion that
use such an approach towards improving the representation of the high-fidelity
function with only a few training data points. Given that $f_h$ may not be a
simple function -- and sometimes not even a function -- of $f_l$, we
demonstrate that using additional functions of $t$, such as derivatives or
shifts of $f_l$, can drastically improve the approximation of $f_h$ through
Gaussian Processes. We also point out a connection with "embedology" techniques
from topology and dynamical systems.
