  Originally introduced in the fluid mechanics community, dynamic mode
decomposition (DMD) has emerged as a powerful tool for analyzing the dynamics
of nonlinear systems. However, existing DMD theory deals primarily with
sequential time series for which the measurement dimension is much larger than
the number of measurements taken. We present a theoretical framework in which
we define DMD as the eigendecomposition of an approximating linear operator.
This generalizes DMD to a larger class of datasets, including nonsequential
time series. We demonstrate the utility of this approach by presenting novel
sampling strategies that increase computational efficiency and mitigate the
effects of noise, respectively. We also introduce the concept of linear
consistency, which helps explain the potential pitfalls of applying DMD to
rank-deficient datasets, illustrating with examples. Such computations are not
considered in the existing literature, but can be understood using our more
general framework. In addition, we show that our theory strengthens the
connections between DMD and Koopman operator theory. It also establishes
connections between DMD and other techniques, including the eigensystem
realization algorithm (ERA), a system identification method, and linear inverse
modeling (LIM), a method from climate science. We show that under certain
conditions, DMD is equivalent to LIM.
