Koopman operators linearize nonlinear dynamical systems, making their spectral information of crucial interest. Numerous algorithms have been developed to approximate these spectral properties, and Dynamic Mode Decomposition (DMD) stands out as the poster child of projection-based methods. Although the Koopman operator itself is linear, the fact that it acts in an infinite-dimensional space of observables poses challenges. These include spurious modes, essential spectra, and the verification of Koopman mode decompositions. While recent work has addressed these challenges for deterministic systems, there remains a notable gap in verified DMD methods for stochastic systems, where the Koopman operator measures the expectation of observables. We show that it is necessary to go beyond expectations to address these issues. By incorporating variance into the Koopman framework, we address these challenges. Through an additional DMD-type matrix, we approximate the sum of a squared residual and a variance term, each of which can be approximated individually using batched snapshot data. This allows verified computation of the spectral properties of stochastic Koopman operators, controlling the projection error. We also introduce the concept of variance-pseudospectra to gauge statistical coherency. Finally, we present a suite of convergence results for the spectral information of stochastic Koopman operators. Our study concludes with practical applications using both simulated and experimental data. In neural recordings from awake mice, we demonstrate how variance-pseudospectra can reveal physiologically significant information unavailable to standard expectation-based dynamical models.

翻译：库普曼算子将非线性动力系统线性化，使其谱信息具有关键意义。为近似这些谱特性，研究者已发展出众多算法，其中动态模态分解（DMD）作为基于投影方法的典范脱颖而出。尽管库普曼算子本身是线性的，但其作用在无限维观测函数空间这一事实带来了诸多挑战，包括伪模态、本质谱以及库普曼模态分解的验证问题。尽管近期研究已针对确定性系统解决了这些挑战，但在随机系统中仍存在已验证DMD方法的显著空白——此类系统中库普曼算子度量的是观测函数的期望值。我们研究表明，要解决这些问题必须超越期望的范畴。通过将方差纳入库普曼框架，我们成功应对了这些挑战。借助额外的DMD型矩阵，我们近似了平方残差项与方差项之和，而这两项均可通过分批快照数据分别近似。这使得随机库普曼算子谱特性的可验证计算成为可能，并实现了对投影误差的控制。我们还引入方差伪谱概念以量化统计一致性。最后，我们给出随机库普曼算子谱信息的一系列收敛性结论。本研究通过模拟数据与实验数据展示了实际应用。在对清醒小鼠的神经记录中，我们演示了方差伪谱如何揭示标准基于期望的动态模型无法获得的生理显著信息。