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. Furthermore, the proposed numerical method for propagating 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 with those obtained from Monte Carlo simulations.

翻译：本文应用随机矩阵理论研究测量不确定性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各阶矩如何通过动态模态分解进行传递。虽然我们主要关注一阶矩和二阶矩，但推导出的解析表达式具有普适性，可扩展至高阶矩情形。此外，所提出的不确定性传递数值方法不依赖于特定的动态模态分解公式。特别重要的是，估计得到的二阶矩可提供置信边界，该边界可作为可信度的度量标准，即评估有限维线性算子表征底层动力系统的可靠程度。我们在两个经典系统上进行了数值实验，并通过将矩估计结果与蒙特卡洛模拟结果进行比较，验证了估计置信水平的有效性。