# Subspace dynamic mode decomposition for stochastic Koopman analysis

## TL;DR

This paper introduces subspace DMD, a novel algorithm for Koopman spectral analysis of stochastic systems that effectively handles observation noise, providing accurate spectral estimates for random dynamical systems.

## Contribution

The paper proposes subspace DMD, a new method that improves Koopman analysis of noisy stochastic systems by ensuring convergence to the true spectra under standard assumptions.

## Key findings

- Subspace DMD accurately estimates spectra in noisy stochastic systems.
- The method converges to the true Koopman spectra under standard conditions.
- Empirical tests demonstrate its utility across various dynamical systems.

## Abstract

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.