Clarifying the effect of mean subtraction on Dynamic Mode Decomposition

TL;DR

The paper analyzes how subtracting the temporal mean in Dynamic Mode Decomposition affects the resulting spectral estimates when the observables admit only a finite number of Koopman modes. It establishes a geometric necessary-and-sufficient condition for muDMD to be equivalent to a temporal DFT and shows that linear consistency plus over-sampling prevents this equivalence, while under- or just-sampling can preserve it. Delay embedding is proposed as a practical tool to attain a Koopman-invariant dictionary, thereby severing muDMD–DFT equivalence in data-rich regimes. The work combines rigorous theory with numerical experiments on LTIs, the Van der Pol oscillator, and lid-driven cavity flow, and introduces the Relative distance to DFT as a numerically robust indicator of equivalence and data sufficiency. Collectively, the results provide a principled way to diagnose data adequacy for DMD and to design preprocessing or embedding strategies that avoid misleading DMD-DFT identifications.

Abstract

Any autonomous nonlinear dynamical system can be viewed as a superposition of infinitely many linear processes, through the so-called Koopman mode decomposition. Its data-driven approximation- Dynamic Mode Decomposition (DMD)- has been extensively developed and deployed across a plethora of fields. In this work, we study the effect of subtracting the temporal mean on the DMD approximation, for observables possessing only a finite number of Koopman modes. Pre-processing time-sequential training data by removing the temporal mean has been a point of contention in the Companion matrix formulation of DMD. This stems from the potential of said pre-processing to render DMD equivalent to a temporal Discrete Fourier Transform (DFT). We prove that this equivalence is impossible when the training data is linearly consistent and the order of the DMD model exceeds the number of Koopman modes. Since model order and training set size are synonymous in this variant of DMD, the parity of DMD and DFT can, therefore, be indicative of inadequate training data.

Figures (6)

• Figure 1: For the linear time-invariant (LTI) systems described by \ref{['eq:LTI1ab_LTI3']}, box plots of $\texttt{Relative distance to DFT}$ reveal its dependence on the model order $(\theta)$ and the number of time delays (Colour-coded). The magenta line indicates the system order $r$, which is $7$ for all three examples. When a minimum of 6 time delays are taken and the model order is at least 7 (8) for $\textrm{LTI}_{1a} (\textrm{LTI}_{1b})$, Relative distance to DFT is larger than at lower model orders, as predicted by \ref{['cor:DMD_DFT_Numerical']}. The pertinent prediction for $\textrm{LTI}_{3}$ - large values of Relative distance to DFT for $\theta \geq 8$ and $d \geq 6$ - also appears to hold for $d = 6$ i.e., when only the minimal required time delays are taken. However, increasing the delay embedding dimension to $24$ significantly diminishes the spike in Relative distance to DFT at $\theta = 8$, to the extent that it could be overlooked.
• Figure 1: Variation of $\sigma_{\rm Tail}$ corresponding to the studies visualized in \ref{['fig:cms_transit_knownss']}. Given a choice of $(\theta,d)$, a low value of $\sigma_{\rm Tail}$ means we have accurately computed Relative distance to DFT while a large value indicates that we have only calculated an upper bound. Hence, for all three systems, when we only take $3$ or $6$ time delays, Relative distance to DFT is accurately computed for all model orders. In contrast, when $24$ time delays are taken, $\sigma_{\rm Tail}$ spikes at $\theta=14,16~\textrm{and}~21$ for $\textrm{LTI}_{1a},\textrm{LTI}_{1b}~\textrm{and}~\textrm{LTI}_{3}$ respectively, and remains high thereafter. Therefore, the concomitant computations in \ref{['fig:cms_transit_knownss']} for $d=24$ are only accurate until the aforementioned values of $\theta$. Beyond this, what is visualized in only an upper bound on Relative distance to DFT.
• Figure 2: The system order, $r$, is unknown for the \ref{['eq:Dictionary__Van_Der_Pol']} to study the Van der Pol oscillator \ref{['eq:ODE__Van_Der_Pol']}. Nonetheless, sufficient time delays lead to a step-like trend in $\texttt{Relative distance to DFT}$. By \ref{['cor:DMD_DFT_Numerical']}, the location of the jump might be indicative of $\boldsymbol{\mathbf{\psi}}$ possessing no more than 11 Koopman modes.
• Figure 2: Variation in $\sigma_{\rm Tail}$ for the computations with the Van der Pol oscillator in \ref{['fig:DistanceToDFT__VanDerPol']}. Unless the model order is more than $20$ and $d=25$, $\sigma_{\rm Tail}$ remains low. Hence, the trends seen in \ref{['fig:DistanceToDFT__VanDerPol']}, particularly the spike at $\theta=11$, are dynamically relevant and not an artifact of the low-rank approximation used in computing Relative distance to DFT.
• Figure 3: As $Re$ increases, the lid-driven cavity flow grows in complexity. This correlation is reflected in the above box plots of (\ref{['s:DisrepancyChecks']}) $\texttt{Relative distance to DFT}$. Panels (a)-(c) display jumps that occur at larger values of $\theta$. In contrast, the final plot (d) is conspicuous in its lack of a discontinuity. By \ref{['cor:DMD_DFT_Numerical']}, we can infer that $\boldsymbol{\mathbf{\psi}}$ possesses at-least 26 Koopman modes. This observation agrees with the fact that the Koopman operator does not possess any eigenfunctions when the underlying dynamics is chaotic.

Theorems & Definitions (61)

• Theorem 1.1
• Definition 3.1: Koopman operator koopman1931hamiltonian
• Definition 3.2: Koopman eigen-functions (KEFs)
• Definition 3.3: Koopman Mode Decomposition (KMD) mezic2004comparisonmezic2005spectral
• Theorem 3.4
• Definition 3.5: Order of a DMD model
• Definition 3.6: Forecasting with $\boldsymbol{\mathbf{\mu}}\textrm{DMD}$
• Theorem 3.7: $\boldsymbol{\mathbf{\mu}}\textrm{DMD}$ perfectly forecasts constant KEFs
• Proof 1
• Lemma 3.8