Skip to main content
SpringerLink
Log in

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

  • Original Article
  • Published:
Theoretical and Computational Fluid Dynamics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Akaike, H.: A new look at the statistical model identification. IEEE Trans. Autom. Control 19(6), 716–723 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bagheri, S.: Effects of weak noise on oscillating flows: linking quality factor, floquet modes, and Koopman spectrum. Phys. Fluids 26, 094104 (2014)

    Article  Google Scholar 

  3. Bendat, J.S., Piersol, A.G.: Random Data: Analysis and Measurement Procedures, vol. 729. Wiley, New York (2011)

    MATH  Google Scholar 

  4. Berger, E., Sastuba, M., Vogt, D., Jung, B., Ben Amor, H.: Estimation of perturbations in robotic behavior using dynamic mode decomposition. Adv. Robot. 29(5), 331–343 (2015)

    Article  Google Scholar 

  5. Bourantas, G.C., Ghommem, M., Kagadis, G.C., Katsanos, K., Loukopoulos, V.C., Burganos, V.N., Nikiforidis, G.C.: Real-time tumor ablation simulation based on dynamic mode decomposition method. Med. Phys. 41, 053301 (2014)

    Article  Google Scholar 

  6. Box, G.E., Jenkins, G.M., Reinsel, G.C.: Time Series Analysis, 3rd edn. Prentice-Hall, Englewood Cliffs, NJ (1994)

    MATH  Google Scholar 

  7. Brunton, B.W., Johnson, L.A., Ojemann, J.G., Kutz, J.N.: Extracting spatial–temporal coherent patterns in large-scale neural recording using dynamic mode decomposition. J. Neurosci. Methods. 258, 1–15 (2016). doi:10.1016/j.jneumeth.2015.10.010

  8. Budis̆ić, M., Mohr, R., Mezić, I.: Applied Koopmanism. Chaos 22, 047510 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, K., Tu, J., Rowley, C.: Variants of dynamic mode decomposition: boundary condition, Koopman, and Fourier analysis. J. Nonlinear Sci. 22(6), 887–915 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Davison, E.J.: A method for simplifying linear dynamic systems. IEEE Trans. Autom. Control 11(1), 93–101 (1966)

    Article  Google Scholar 

  11. Dawson, S.T.M., Hemati, M.S., Williams, M.O., Rowley, C.W.: Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition. Exp. Fluids 57(42), (2016)

  12. Duke, D., Honnery, D., Soria, J.: Experimental investigation of nonlinear instabilities in annular liquid sheets. J. Fluid Mech. 691, 594–604 (2012)

    Article  MATH  Google Scholar 

  13. Duke, D., Soria, J., Honnery, D.: An error analysis of the dynamic mode decomposition. Exp. Fluids 52(2), 529–542 (2012)

    Article  MATH  Google Scholar 

  14. Fierro, R.D., Bunch, J.R.: Orthogonal projection and total least squares. Numer. Linear Algebra Appl. 2(2), 135–153 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fierro, R.D., Bunch, J.R.: Perturbation theory and orthogonal projection methods with applications to least squares and total least squares. Linear Algebra Its Appl. 234, 71–96 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fierro, R.D., Golub, G.H., Hansen, P.C., O’Leary, P.: Regularization by truncated total least squares. SIAM J. Sci. Comput. 18(4), 1223–1241 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gleser, L.J.: Estimation in a multivariate “error-in-variables” regression model: large sample results. Ann. Stat. 9(1), 24–44 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  18. Golub, G.H., Heath, M., Wahba, G.: Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics 21, 215–223 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  19. Golub, G.H., Van Loan, C.F.: An analysis of the total least squares problem. SIAM J. Numer. Anal. 17(6), 883–893 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  20. Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore, MD (1996)

    MATH  Google Scholar 

  21. Goulart, P., Wynn, A., Pearson, D.: Optimal mode decomposition for high dimensional systems. In: 51st IEEE Conference on Descision and Control (2012)

  22. Grosek, J., Kutz, J.N.: Dynamic mode decomposition for real-time background/foreground separation in video. arXiv:1404.7592v1 (2014)

  23. Guéniat, F., Mathelin, L., Pastur, L.R.: A dynamic mode decomposition approach for large and arbitrarily sampled systems. Phys. Fluids 27, 025113 (2015)

    Article  Google Scholar 

  24. Hansen, P.C.: Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev. 32, 561–580 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hasselmann, K.: PIPs and POPs: the reduction of complex dynamical systems using principal interaction and oscillation patterns. J. Geophys. Res. 93(D9), 11015–11021 (1988)

    Article  Google Scholar 

  26. Hemati, M.S., Williams, M.O., Rowley, C.W.: Dynamic mode decomposition for large and streaming datasets. Phys. Fluids 26, 111701 (2014)

    Article  Google Scholar 

  27. Ho, B.L., Kalman, R.E.: Effective construction of linear, state-variable models from input/output functions. Regelungstechnik 14(12), 545–548 (1966)

    MATH  Google Scholar 

  28. Holmes, P., Lumley, J.L., Berkooz, G., Rowley, C.W.: Turbulence, Coherent Structures, Dynamical Systems and Symmetry, 2nd edn. Cambridge University Press, Cambridge (2012)

    Book  MATH  Google Scholar 

  29. Jovanović, M.R., Schmid, P.J., Nichols, J.W.: Sparsity promoting dynamic mode decomposition. Phys. Fluids 26, 024103 (2014)

    Article  Google Scholar 

  30. Juang, J.N., Pappa, R.S.: An eigensystem realization algorithm for modal parameter-identification and model-reduction. J. Guid. Control Dyn. 8(5), 620–627 (1985)

    Article  MATH  Google Scholar 

  31. Koopman, B.O.: Hamiltonian systems and transformation in Hilbert space. Proc. Natl. Acad. Sci. 17(5), 315–318 (1931)

    Article  MATH  Google Scholar 

  32. Koopman, B.O., von Neumann, J.: Dynamical systems of continuous spectra. Proc. Natl. Acad. Sci. 18(3), 255–263 (1932)

    Article  MATH  Google Scholar 

  33. Kung, S.Y.: A new identification and model reduction algorithm via singular value decomposition. In: Proceedings of the 12th Asilomar Conference on Circuits, Systems, and Computers, pp. 705–714

  34. Lanczos, C.: Applied Analysis. Dover Publications, New York (1988)

    MATH  Google Scholar 

  35. Markovsky, I., Van Huffel, S.: Overview of total least squares methods. Signal Process. 87(10), 2283–2302 (2007)

    Article  MATH  Google Scholar 

  36. Martinsson, P.G., Rokhlin, V., Tygert, M.: A randomized algorithm for the decomposition of matrices. Appl. Comput. Harmonic Anal. 30(1), 47–68 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. Mezić, I.: Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dyn. 41(1–3), 309–325 (2005)

    MathSciNet  MATH  Google Scholar 

  38. Mezić, I.: Analysis of fluid flows via spectral properties of the Koopman operator. Ann. Rev. Fluid Mech. 45, 357–378 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  39. Morozov, V.A.: Methods for Solving Incorrectly Posed Problems. Springer, New York (1984)

    Book  Google Scholar 

  40. Noack, B.R., Afanasiev, K., Morzynski, M., Tadmor, G., Thiele, F.: A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335–363 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  41. Proctor, J.L., Eckhoff, P.A.: Discovering dynamic patterns from infectious disease data using dynamic mode decomposition. Int. Health 7(2), 139–145 (2015)

    Article  Google Scholar 

  42. Rowley, C.W., Mezić, I., Bagheri, S., Schlatter, P., Henningson, D.: Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115–127 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  43. Schmid, P.: Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  44. Schmid, P.: Application of the dynamic mode decomposition to experimental data. Exp. Fluids 50, 1123–1130 (2011)

    Article  Google Scholar 

  45. Schmid, P., Li, L., Juniper, M., Pust, O.: Applications of the dynamic mode decomposition. Theor. Comput. Fluid Dyn. 25, 249–259 (2011)

    Article  MATH  Google Scholar 

  46. Schmid, P., Sesterhenn, J.: Dynamic mode decomposition of numerical and experimental data. In: 61st Annual Meeting of the APS Division of Fluid Dynamics (2008)

  47. Semeraro, O., Bellani, G., Lundell, F.: Analysis of time-resolved PIV masurements of a confined turbulent jet using POD and Koopman modes. Exp. Fluids 53, 1203–1220 (2012)

    Article  Google Scholar 

  48. Sima, D.M., Van Huffel, S.: Level choice in truncated total least squares. Comput. Stat. Data Anal. 52(2), 1103–1118 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  49. Timmins, B.H., Wilson, B.W., Smith, B.L., Vlachos, P.P.: A method for automatic estimation of instantaneous local uncertainty in particle image velocimetry measurements. Exp. Fluids 53(4), 1133–1147 (2012)

    Article  Google Scholar 

  50. Tu, J.H., Rowley, C.W., Luchtenburg, D.M., Brunton, S.L., Kutz, J.N.: On dynamic mode decomposition: theory and applications. J. Comput. Dyn. 1(2), 391–421 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  51. Van Huffel, S., Vandewalle, J.: On the accuracy of total least squares and least squares techniques in the presence of errors on all data. Automatica 25(5), 765–769 (1989)

    Article  MATH  Google Scholar 

  52. Van Huffel, S., Vandewalle, J.: The total least squares problem: computational aspects and analysis. In: Frontiers in Applied Mathematics, vol. 9. SIAM, Philadelphia, PA (1991). doi:10.1137/1.9781611971002

  53. von Neumann, J.: Proof of the quasi-ergodic hypothesis. Proc. Natl. Acad. Sci. 18, 70–82 (1932)

    Article  MATH  Google Scholar 

  54. Wieneke, B.: PIV uncertainty quantification from correlation statistics. Meas. Sci. Technol. 26, 074002 (2015)

    Article  Google Scholar 

  55. Williams, M.O., Kevrekidis, I.G., Rowley, C.W.: A data-driven approximation of the Koopman operator: extending dynamic mode decomposition. J. Nonlinear. Sci. 25(6), 1307–1346 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  56. Williams, M.O., Rowley, C.W., Kevrekidis, I.G.: A kernel-based method for data-driven Koopman spectral analysis. J. Comput. Dyn. 2(2), 247–265 (2015)

    MathSciNet  MATH  Google Scholar 

  57. Wilson, B.M., Smith, B.L.: Uncertainty on piv mean and fluctuating velocity due to bias and random errors. Meas. Sci. Technol. 24(3), 035,302 (2013)

    Article  Google Scholar 

  58. Wynn, A., Pearson, D., Ganapathisubramani, B., Goulart, P.: Optimal mode decomposition for unsteady flows. J. Fluid Mech. 733, 473–503 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  59. Zoltowski, M.D.: Generalized minimum norm and constrained total least squares with applications to array signal processing. Proc. SPIE 975, 78–85 (1988)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN, 55455, USA

    Maziar S. Hemati

  2. Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ, 08544, USA

    Clarence W. Rowley

  3. Florida Center for Advanced Aero-Propulsion, Florida State University, Tallahassee, FL, 32310, USA

    Eric A. Deem & Louis N. Cattafesta

Authors
  1. Maziar S. Hemati
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Clarence W. Rowley
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Eric A. Deem
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Louis N. Cattafesta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maziar S. Hemati.

Additional information

Communicated by Oleg V. Vasilyev.

This work was supported by the Air Force Office of Scientific Research under Grant FA9550-14-1-0289 and the Office of Naval Research under MURI Grant 00014-14-1-0533. M.S.H. gratefully acknowledges support from the Department of Aerospace Engineering and Mechanics at the University of Minnesota.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (m 0 KB)

Supplementary material 2 (m 3 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hemati, M.S., Rowley, C.W., Deem, E.A. et al. De-biasing the dynamic mode decomposition for applied Koopman spectral analysis of noisy datasets . Theor. Comput. Fluid Dyn. 31, 349–368 (2017). https://doi.org/10.1007/s00162-017-0432-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00162-017-0432-2

Keywords

Search

Navigation