Generalization of Vandermonde determinants

Yang, Shangjun; Wu, Huazhang; Zhang, Quanbing

doi:10.1016/s0024-3795(01)00319-6

Cited by 14 publications

(8 citation statements)

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“…However, exact DMD has been shown to successfully extract modes from data that is not sequential. One potential starting point is the theory of exponential Vandermonde matrices [37,54]. It remains to be demonstrated that the modes extracted for non-sequential times data by exact DMD, or similar methods such as optimized DMD, are well-posed and unique.…”

Section: Background Subtraction For Video Surveillancementioning

confidence: 99%

Centering Data Improves the Dynamic Mode Decomposition

Hirsh¹,

Harris²,

Kutz³

et al. 2019

Preprint

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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.

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