Subspace dynamic mode decomposition for stochastic Koopman analysis
This work addresses the problem of inaccurate spectral estimation in Koopman analysis due to observation noise, which is relevant for practitioners analyzing nonlinear stochastic systems.
The paper proposes subspace DMD to handle observation noise in random dynamical systems for Koopman analysis, showing convergence to the true stochastic Koopman operator spectra under standard assumptions and improved empirical performance.
The analysis of nonlinear dynamical systems based on the Koopman operator is attracting attention in various applications. Dynamic mode decomposition (DMD) is a data-driven algorithm for Koopman spectral analysis, and several variants with a wide range of applications have been proposed. However, popular implementations of DMD suffer from observation noise on random dynamical systems and generate an inaccurate estimation of the spectra of the stochastic Koopman operator. In this paper, we propose subspace DMD as an algorithm for the Koopman analysis of random dynamical systems with observation noise. Subspace DMD first computes the orthogonal projection of future snapshots to the space of past snapshots and then estimates the spectra of a linear model, and its output converges to the spectra of the stochastic Koopman operator under standard assumptions. We investigate the empirical performance of subspace DMD with several dynamical systems and show its utility for the Koopman analysis of random dynamical systems.