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Koopman operators linearize nonlinear dynamical systems, making theirspectral information of crucial interest. Numerous algorithms have beendeveloped to approximate these spectral properties, and Dynamic ModeDecomposition (DMD) stands out as the poster child of projection-based methods.Although the Koopman operator itself is linear, the fact that it acts in aninfinite-dimensional space of observables poses challenges. These includespurious modes, essential spectra, and the verification of Koopman modedecompositions. While recent work has addressed these challenges fordeterministic systems, there remains a notable gap in verified DMD methods forstochastic systems, where the Koopman operator measures the expectation ofobservables. We show that it is necessary to go beyond expectations to addressthese issues. By incorporating variance into the Koopman framework, we addressthese challenges. Through an additional DMD-type matrix, we approximate the sumof a squared residual and a variance term, each of which can be approximatedindividually using batched snapshot data. This allows verified computation ofthe spectral properties of stochastic Koopman operators, controlling theprojection error. We also introduce the concept of variance-pseudospectra togauge statistical coherency. Finally, we present a suite of convergence resultsfor the spectral information of stochastic Koopman operators. Our studyconcludes with practical applications using both simulated and experimentaldata. In neural recordings from awake mice, we demonstrate howvariance-pseudospectra can reveal physiologically significant informationunavailable to standard expectation-based dynamical models.

Beyond expectations: Residual Dynamic Mode Decomposition and Variance for Stochastic Dynamical Systems