# Centering Data Improves the Dynamic Mode Decomposition

## TL;DR

Centering data before applying Dynamic Mode Decomposition (DMD) enhances its ability to model and analyze the dynamics of nonlinear systems, especially when data are perturbations around fixed points, by enabling eigenvalue spectrum computation and improving model fidelity.

## Contribution

This work demonstrates that centering data in DMD is equivalent to adding an affine term, clarifies its relation to Fourier transforms, and extends the concept to extracting known fixed frequencies, with practical benefits shown through numerical examples.

## Key findings

- Centered DMD can always compute eigenvalue spectra.
- Without centering, DMD may fail to model full-rank dynamics.
- Centering improves DMD performance on nonlinear systems.

## 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.