  In this paper, we consider modeling missing dynamics with a nonparametric
non-Markovian model, constructed using the theory of kernel embedding of
conditional distributions on appropriate Reproducing Kernel Hilbert Spaces
(RKHS), equipped with orthonormal basis functions. Depending on the choice of
the basis functions, the resulting closure model from this nonparametric
modeling formulation is in the form of parametric model. This suggests that the
success of various parametric modeling approaches that were proposed in various
domains of applications can be understood through the RKHS representations.
When the missing dynamical terms evolve faster than the relevant observable of
interest, the proposed approach is consistent with the effective dynamics
derived from the classical averaging theory. In the linear Gaussian case
without the time-scale gap, we will show that the proposed non-Markovian model
with a very long memory yields an accurate estimation of the nontrivial
autocovariance function for the relevant variable of the full dynamics.
Supporting numerical results on instructive nonlinear dynamics show that the
proposed approach is able to replicate high-dimensional missing dynamical terms
on problems with and without the separation of temporal scales.
