  Dynamic mode decomposition (DMD) is a data-driven method that models
high-dimensional time series as a sum of spatiotemporal modes, where the
temporal modes are constrained by linear dynamics. For nonlinear dynamical
systems exhibiting strongly coherent structures, DMD can be a useful
approximation to extract dominant, interpretable modes. In many domains with
large spatiotemporal data---including fluid dynamics, video processing, and
finance---the dynamics of interest are often perturbations about fixed points
or equilibria, which motivates the application of DMD to centered (i.e.
mean-subtracted) data. In this work, we show that DMD with centered data is
equivalent to incorporating an affine term in the dynamic model and is not
equivalent to computing a discrete Fourier transform. Importantly, DMD with
centering can always be used to compute eigenvalue spectra of the dynamics.
However, in many cases DMD without centering cannot model the corresponding
dynamics, most notably if the dynamics have full effective rank. Additionally,
we generalize the notion of centering to extracting arbitrary, but known, fixed
frequencies from the data. We corroborate these theoretical results numerically
on three nonlinear examples: the Lorenz system, a surveillance video, and brain
recordings. Since centering the data is simple and computationally efficient,
we recommend it as a preprocessing step before DMD; furthermore, we suggest
that it can be readily used in conjunction with many other popular
implementations of the DMD algorithm.
