De-Biasing the Dynamic Mode Decomposition for Applied Koopman Spectral Analysis

Maziar S. Hemati, Clarence W. Rowley

Published 2015-02-12Version 1

The Dynamic Mode Decomposition (DMD)---a popular method for performing Koopman spectral analysis in numerous application areas---processes snapshot measurements sampled from a time-evolving system to extract dynamically meaningful spatio-temporal descriptions of the underlying process. Often times, DMD descriptions can be used for predictive purposes as well, which enables informed decision-making based on DMD model-forecasts. Despite its widespread use and utility, DMD regularly fails to yield accurate dynamical descriptions when the measured snapshot data are even slightly imprecise due to, e.g., sensor noise. Here, we express DMD as a two-stage algorithm in order to isolate a source of systematic error. We show that DMD's first stage, a subspace projection step, systematically introduces bias errors by processing snapshots asymmetrically. In order to remove this systematic error, we propose utilizing an augmented snapshot matrix in a subspace projection step, as in problems of total least-squares, in order to account for the error present in all snapshots. The resulting unbiased and noise-robust total DMD (TDMD) formulation reduces to standard DMD in the absence of snapshot errors, while the two-stage perspective generalizes the de-biasing framework to other related methods as well. Several examples of TDMD's superior performance over standard DMD are presented, including in numerical and experimental studies of fluid flow over a cylinder.