  We propose to adopt statistical regression as the projection operator to
enable data-driven learning of the operators in the Mori--Zwanzig formalism. We
present a principled method to extract the Markov and memory operators for any
regression models. We show that the choice of linear regression results in a
recently proposed data-driven learning algorithm based on Mori's projection
operator, which is a higher-order approximate Koopman learning method. We show
that more expressive nonlinear regression models naturally fill in the gap
between the highly idealized and computationally efficient Mori's projection
operator and the most optimal yet computationally infeasible Zwanzig's
projection operator. We performed numerical experiments and extracted the
operators for an array of regression-based projections, including linear,
polynomial, spline, and neural-network-based regressions, showing a progressive
improvement as the complexity of the regression model increased. Our
proposition provides a general framework to extract memory-dependent
corrections and can be readily applied to an array of data-driven learning
methods for stationary dynamical systems in the literature.
