De-biasing the dynamic mode decomposition for applied Koopman spectral analysis of noisy datasets

Maziar S. Hemati, Clarence Worth Rowley, Eric A. Deem, Louis N. Cattafesta

Research output: Contribution to journalArticlepeer-review

206 Scopus citations


The dynamic mode decomposition (DMD)—a popular method for performing data-driven Koopman spectral analysis—has gained increased popularity for extracting dynamically meaningful spatiotemporal descriptions of fluid flows from snapshot measurements. Often times, DMD descriptions can be used for predictive purposes as well, which enables informed decision-making based on DMD model forecasts. Despite its widespread use and utility, DMD can fail to yield accurate dynamical descriptions when the measured snapshot data are imprecise due to, e.g., sensor noise. Here, we express DMD as a two-stage algorithm in order to isolate a source of systematic error. We show that DMD’s first stage, a subspace projection step, systematically introduces bias errors by processing snapshots asymmetrically. To remove this systematic error, we propose utilizing an augmented snapshot matrix in a subspace projection step, as in problems of total least-squares, in order to account for the error present in all snapshots. The resulting unbiased and noise-aware total DMD (TDMD) formulation reduces to standard DMD in the absence of snapshot errors, while the two-stage perspective generalizes the de-biasing framework to other related methods as well. TDMD’s performance is demonstrated in numerical and experimental fluids examples. In particular, in the analysis of time-resolved particle image velocimetry data for a separated flow, TDMD outperforms standard DMD by providing dynamical interpretations that are consistent with alternative analysis techniques. Further, TDMD extracts modes that reveal detailed spatial structures missed by standard DMD.

Original languageEnglish (US)
Pages (from-to)349-368
Number of pages20
JournalTheoretical and Computational Fluid Dynamics
Issue number4
StatePublished - Aug 1 2017

All Science Journal Classification (ASJC) codes

  • Computational Mechanics
  • Condensed Matter Physics
  • General Engineering
  • Fluid Flow and Transfer Processes


  • Data-driven dynamical systems
  • Experimental fluid mechanics
  • Koopman spectral analysis
  • Reduced-order model
  • Sensor noise
  • Total least-squares


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