  In this study we investigate a data-driven stochastic methodology to
parameterize small-scale features in a prototype multiscale dynamical system,
the Lorenz '96 (L96) model. We propose to model the small-scale features using
a vector autoregressive process with exogenous variable (VARX), estimated from
given sample data. To reduce the number of parameters of the VARX we impose a
diagonal structure on its coefficient matrices. We apply the VARX to two
different configurations of the 2-layer L96 model, one with common parameter
choices giving unimodal invariant probability distributions for the L96 model
variables, and one with non-standard parameters giving trimodal distributions.
We show through various statistical criteria that the proposed VARX performs
very well for the unimodal configuration, while keeping the number of
parameters linear in the number of model variables. We also show that the
parameterization performs accurately for the very challenging trimodal L96
configuration by allowing for a dense (non-diagonal) VARX covariance matrix.
