  Model reduction methods aim to describe complex dynamic phenomena using only
relevant dynamical variables, decreasing computational cost, and potentially
highlighting key dynamical mechanisms. In the absence of special dynamical
features such as scale separation or symmetries, the time evolution of these
variables typically exhibits memory effects. Recent work has found a variety of
data-driven model reduction methods to be effective for representing such
non-Markovian dynamics, but their scope and dynamical underpinning remain
incompletely understood. Here, we study data-driven model reduction from a
dynamical systems perspective. For both chaotic and randomly-forced systems, we
show the problem can be naturally formulated within the framework of Koopman
operators and the Mori-Zwanzig projection operator formalism. We give a
heuristic derivation of a NARMAX (Nonlinear Auto-Regressive Moving Average with
eXogenous input) model from an underlying dynamical model. The derivation is
based on a simple construction we call Wiener projection, which links
Mori-Zwanzig theory to both NARMAX and to classical Wiener filtering. We apply
these ideas to the Kuramoto-Sivashinsky model of spatiotemporal chaos and a
viscous Burgers equation with stochastic forcing.
