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Koopman Spectral Analysis from Noisy Measurements based on Bayesian Learning and Kalman Smoothing

Zhexuan Zeng, Jun Zhou, Yasen Wang, Zuowei Ping

TL;DR

The paper addresses the problem of extracting Koopman spectral properties from noisy measurements. It introduces the Koopman Bayesian Kalman smoothing (KBK) framework that jointly identifies a finite-dimensional Koopman operator and estimates the lifted states using time-delay observables, Bayesian inference, and Kalman smoothing via an EM algorithm. The spectrum is obtained from the eigenvalues of the lifted operator, with continuous-time interpretation via $\log(\lambda)/(MT_s)$, and the method demonstrates superior accuracy and state reconstruction compared to DMD-based approaches across several nonlinear systems with real, imaginary, and complex eigenvalues. This approach decouples projection error from measurement noise, providing a robust, data-efficient tool for data-driven spectral analysis in nonlinear dynamics, especially when noise characteristics are uncertain.

Abstract

Koopman spectral analysis plays a crucial role in understanding and modeling nonlinear dynamical systems as it reveals key system behaviors and long-term dynamics. However, the presence of measurement noise poses a significant challenge to accurately extracting spectral properties. In this work, we propose a robust method for identifying the Koopman operator and extracting its spectral characteristics in noisy environments. To address the impact of noise, our approach tackles an identification problem that accounts for both systematic errors from finite-dimensional approximations and measurement noise in the data. By incorporating Bayesian learning and Kalman smoothing, the method simultaneously identifies the Koopman operator and estimates system states, effectively decoupling these two error sources. The method's efficiency and robustness are demonstrated through extensive experiments, showcasing its accuracy across varying noise levels.

Koopman Spectral Analysis from Noisy Measurements based on Bayesian Learning and Kalman Smoothing

TL;DR

The paper addresses the problem of extracting Koopman spectral properties from noisy measurements. It introduces the Koopman Bayesian Kalman smoothing (KBK) framework that jointly identifies a finite-dimensional Koopman operator and estimates the lifted states using time-delay observables, Bayesian inference, and Kalman smoothing via an EM algorithm. The spectrum is obtained from the eigenvalues of the lifted operator, with continuous-time interpretation via , and the method demonstrates superior accuracy and state reconstruction compared to DMD-based approaches across several nonlinear systems with real, imaginary, and complex eigenvalues. This approach decouples projection error from measurement noise, providing a robust, data-efficient tool for data-driven spectral analysis in nonlinear dynamics, especially when noise characteristics are uncertain.

Abstract

Koopman spectral analysis plays a crucial role in understanding and modeling nonlinear dynamical systems as it reveals key system behaviors and long-term dynamics. However, the presence of measurement noise poses a significant challenge to accurately extracting spectral properties. In this work, we propose a robust method for identifying the Koopman operator and extracting its spectral characteristics in noisy environments. To address the impact of noise, our approach tackles an identification problem that accounts for both systematic errors from finite-dimensional approximations and measurement noise in the data. By incorporating Bayesian learning and Kalman smoothing, the method simultaneously identifies the Koopman operator and estimates system states, effectively decoupling these two error sources. The method's efficiency and robustness are demonstrated through extensive experiments, showcasing its accuracy across varying noise levels.
Paper Structure (14 sections, 20 equations, 3 figures)

This paper contains 14 sections, 20 equations, 3 figures.

Table of Contents

  1. INTRODUCTION
  2. Notation
  3. Preliminary and Problem formulation
  4. Koopman operator theory
  5. Problem formulation
  6. Description of proposed method: KBK approach
  7. Koopman operator-based reconstruction framework
  8. Matrix identification and sample estimation
  9. Spectral approximation and state estimation
  10. Numerical experiments
  11. Nonlinear system with real Koopman eigenvalues
  12. Nonlinear system with pure imaginary eigenvalues
  13. Nonlinear system with complex eigenvalues
  14. Conclusion

Figures (3)

  • Figure 1: The spectral analysis performance of the nonlinear system \ref{['eq:system3']} using DMD, TDMD, fbDMD, KFDMD and KBK methods. (a) the spectral approximation when $\sigma^2 = 10^{-2}$, (b) the spectral approximation errors across varying noise variances $\sigma^2$, (c) the state reconstruction errors across varying noise variances $\sigma^2$.
  • Figure 2: The spectral analysis performance of the nonlinear system \ref{['eq:system2']} using DMD, TDMD, fbDMD, KFDMD and KBK methods. (a) the spectral approximation when $\sigma^2 = 10^{-2}$, (b) the spectral approximation errors across varying noise variances $\sigma^2$, (c) the state reconstruction errors across varying noise variances $\sigma^2$.
  • Figure 3: The spectral analysis performance of the nonlinear system \ref{['eq:system1']} using DMD, TDMD, fbDMD, KFDMD and KBK methods. (a) the spectral approximation when $\sigma^2 = 10^{-2}$, (b) the spectral approximation errors across varying noise variances $\sigma^2$, (c) the state reconstruction errors across varying noise variances $\sigma^2$.

Theorems & Definitions (3)

  • Definition 1: Koopman eigenvalue and eigenfunctions
  • Remark 1
  • Remark 2: Convergence of optimization