.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. Preprocessing 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 preprocessing 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.KeywordsKoopman operatordynamic mode decompositioncompanion matrixmean subtractiondiscrete Fourier transformMSC codes37M1037N1047N20
Seenivasaharagavan et al. (Tue,) studied this question.