  Dynamic mode decomposition (DMD) is a leading tool for equation-free analysis
of high-dimensional dynamical systems from observations. In this work, we focus
on a combination of delay-coordinates embedding and DMD, i.e.,
delay-coordinates DMD, which accommodates the analysis of a broad family of
observations. An important utility of DMD is the compact and reduced-order
spectral representation of observations in terms of the DMD eigenvalues and
modes, where the temporal information is separated from the spatial
information. From a spatiotemporal viewpoint, we show that when DMD is applied
to delay-coordinates embedding, temporal information is intertwined with
spatial information, inducing a particular spectral structure on the DMD
components. We formulate and analyze this structure, which we term the
spatiotemporal coupling in delay-coordinates DMD. Based on this spatiotemporal
coupling, we propose a new method for DMD components selection. When using
delay-coordinates DMD that comprises redundant modes, this selection is an
essential step for obtaining a compact and reduced-order representation of the
observations. We demonstrate our method on noisy simulated signals and various
dynamical systems and show superior component selection compared to a
commonly-used method that relies on the amplitudes of the modes.
