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Phys Rev E. 2017 Sep;96(3-1):033310. doi: 10.1103/PhysRevE.96.033310. Epub 2017 Sep 18.

Subspace dynamic mode decomposition for stochastic Koopman analysis.

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Department of Aeronautics and Astronautics, The University of Tokyo, Bunkyo, Tokyo 113-8656, Japan.
The Institute of Scientific and Industrial Research, Osaka University, Ibaraki, Osaka 567-0047, Japan.
RIKEN Center for Advanced Intelligence Project, Chuo, Tokyo 103-0027, Japan.


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 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.


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