  Many physical systems are described by nonlinear differential equations that
are too complicated to solve in full. A natural way to proceed is to divide the
variables into those that are of direct interest and those that are not,
formulate solvable approximate equations for the variables of greater interest,
and use data and statistical methods to account for the impact of the other
variables. In the present paper the problem is considered in a fully
discrete-time setting, which simplifies both the analysis of the data and the
numerical algorithms. The resulting time series are identified by a NARMAX
(nonlinear autoregression moving average with exogenous input) representation
familiar from engineering practice. The connections with the Mori-Zwanzig
formalism of statistical physics are discussed, as well as an application to
the Lorenz 96 system.
