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.

翻译：我们运用随机矩阵理论研究测量不确定度对动态模态分解的影响。具体而言，当测量值服从正态概率密度函数时，我们展示了该密度函数的矩如何通过动态模态分解进行传播。虽然本文重点分析一阶和二阶矩，但所推导的解析表达式具有普适性，可扩展至高阶矩。此外，提出的不确定度传播数值方法不依赖于特定动态模态分解形式。特别值得注意的是，估计的二阶矩可为可信度指标提供置信区间——即有限维线性算子表征底层动力系统的可信程度。我们在两个典型系统上开展数值实验，通过将矩估计结果与蒙特卡洛模拟结果对比，验证了所估计置信水平的可靠性。