  We apply random matrix theory to study the impact of measurement uncertainty
on dynamic mode decomposition. Specifically, when the measurements follow a
normal probability density function, we show how the moments of that density
propagate through the dynamic mode decomposition. While we focus on the first
and second moments, the analytical expressions we derive are general and can be
extended to higher-order moments. Further, the proposed numerical method to
propagate uncertainty is agnostic of specific dynamic mode decomposition
formulations. Of particular relevance, the estimated second moments provide
confidence bounds that may be used as a metric of trustworthiness, that is, how
much one can rely on a finite-dimensional linear operator to represent an
underlying dynamical system. We perform numerical experiments on two canonical
systems and verify the estimated confidence levels by comparing the moments to
those obtained from Monte Carlo simulations.
