Paper Info
Title
Centering Data Improves the Dynamic Mode Decomposition
Abstract
Dynamic mode decomposition (DMD) is a data-driven method that models high-dimensional time series as a sum of spatiotemporal modes, where the temporal modes are constrained by linear dynamics. For nonlinear dynamical systems exhibiting strongly coherent structures, DMD can be a useful approximation to extract dominant, interpretable modes. In many domains with large spatiotemporal data-including fluid dynamics, video processing, and finance-the dynamics of interest are often perturbations about fixed points or equilibria, which motivates the application of DMD to centered (i.e., mean-subtracted) data. In this work, we show that DMD with centered data is equivalent to incorporating an affine term in the dynamic model and is not equivalent to computing a discrete Fourier transform. Importantly, DMD with centering can always be used to compute eigenvalue spectra of the dynamics. However, in many cases DMD without centering cannot model the corresponding dynamics, most notably if the dynamics have full effective rank. Additionally, we generalize the notion of centering to extracting arbitrary, but known, fixed frequencies from the data. We corroborate these theoretical results numerically on three nonlinear examples: the Lorenz system, a surveillance video, and brain recordings. Since centering the data is simple and computationally efficient, we recommend it as a preprocessing step before DMD; furthermore, we suggest that it can be readily used in conjunction with many other popular implementations of the DMD algorithm.
Year
DOI
Venue
2020
10.1137/19M1289881
SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS
Keywords
DocType
Volume
dynamic mode decomposition,spatiotemporal decomposition,centering,equilibrium
Journal
19
Issue
ISSN
Citations 
3
1536-0040
0
PageRank 
References 
Authors
0.34
0
4
Name
Order
Citations
PageRank
Hirsh Seth M.100.34
Harris Kameron Decker200.34
J. Nathan Kutz322547.13
Bingni W Brunton431.10