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

arXiv: https://arxiv.org/abs/2308.10697 · HTML: https://mlsanity.com/paper/2308.10697/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 91 references that are in the map.
- First posted on arXiv 21 Aug 2023.

## Builds on (most cited first, 14 of 14)

- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Dynamic mode decomposition with control](https://mlsanity.com/paper/1409.6358/) (2014)
- [On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator](https://mlsanity.com/paper/1703.04680/) (2017)
- [Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2205.14027/) (2022)
- [Data-driven discovery of Koopman eigenfunctions for control](https://mlsanity.com/paper/1707.01146/) (2017)
- [Extracting spatial-temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition](https://mlsanity.com/paper/1409.5496/) (2014)
- [Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems](https://mlsanity.com/paper/2111.14889/) (2021)
- [Linear dynamical neural population models through nonlinear embeddings](https://mlsanity.com/paper/1605.08454/) (2016)
- [Residual Dynamic Mode Decomposition: Robust and verified Koopmanism](https://mlsanity.com/paper/2205.09779/) (2022)
- [Finite-data error bounds for Koopman-based prediction and control](https://mlsanity.com/paper/2108.07102/) (2021)
- [Modeling and Control of Soft Robots Using the Koopman Operator and Model Predictive Control](https://mlsanity.com/paper/1902.02827/) (2019)
- [The mpEDMD Algorithm for Data-Driven Computations of Measure-Preserving Dynamical Systems](https://mlsanity.com/paper/2209.02244/) (2022)
- [On Robust Computation of Koopman Operator and Prediction in Random Dynamical Systems](https://mlsanity.com/paper/1803.08562/) (2018)
- [Kernel Autocovariance Operators of Stationary Processes: Estimation and Convergence](https://mlsanity.com/paper/2004.00891/) (2020)

## Built on by (most cited first, 11 of 11)

- [Koopman-Assisted Reinforcement Learning](https://mlsanity.com/paper/2403.02290/) (2024)
- [Modeling Unknown Stochastic Dynamical System Subject to External Excitation](https://mlsanity.com/paper/2406.15747/) (2024)
- [Trustworthy Koopman Operator Learning: Invariance Diagnostics and Error Bounds](https://mlsanity.com/paper/2603.15091/) (2026)
- [On the Convergence of Hermitian Dynamic Mode Decomposition](https://mlsanity.com/paper/2401.03192/) (2024)
- [Weighted Birkhoff Averages Accelerate Data-Driven Methods](https://mlsanity.com/paper/2511.17772/) (2025)
- [Residual-Guided Dictionary Learning for Spectrally Accurate Koopman Approximation](https://mlsanity.com/paper/2606.29083/) (2026)
- [Bayesian Transfer Operators in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2509.22482/) (2025)
- [Automatic feature identification in least-squares policy iteration using the Koopman operator framework](https://mlsanity.com/paper/2603.26464/) (2026)
- [Finding Koopman Invariant Subspaces via Personalized PageRank](https://mlsanity.com/paper/2605.24666/) (2026)
- [Koopman early warning signals for bifurcation and rate-induced tipping](https://mlsanity.com/paper/2608.14716/) (2026)
- [Using Composition Operators to Linearize LLM Semantic Transformations](https://mlsanity.com/paper/2609.22143/) (2026)

## Abstract (arXiv)

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.