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De-Biasing the Dynamic Mode Decomposition for Applied Koopman Spectral Analysis
Maziar S. Hemati, Clarence W. Rowley, Eric A. Deem, Louis N. Cattafesta
TL;DR
Noisy or imprecise snapshot measurements can bias standard DMD and reduce the reliability of Koopman spectral analysis and forecasts. The paper separates DMD into projection and operator-identification stages, then introduces total DMD (TDMD), which accounts for errors in all snapshots. TDMD removes the systematic bias, matches standard DMD for error-free data, and performs successfully on numerical and experimental fluid-flow examples.
Problem
Measurement noise affects all snapshots but standard DMD treats snapshots asymmetrically, leaving noise-induced bias insufficiently addressed.
Method
TDMD rewrites DMD as subspace projection followed by operator identification and uses an augmented snapshot matrix in a total-least-squares formulation.
Results
TDMD converges to exact spectra in a simple linear system and simulated cylinder flow and outperforms standard DMD on TR-PIV data from separated flow.
Takeaways & Limitations
The de-biasing procedure supports more reliable Koopman spectral analysis from noisy experimental data and generalizes to other DMD-like algorithms.
Takeaways & Limitations
TDMD can make solution procedures less stable because total-least-squares problems are less stable than least-squares problems, requiring more robust computational approaches.
Abstract
from arXiv · showhide
The Dynamic Mode Decomposition (DMD)---a popular method for performing data-driven Koopman spectral analysis---has gained increased adoption as a technique for extracting dynamically meaningful spatio-temporal 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 regularly fails 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.
1 Introduction
DMD provides data-driven dynamical descriptions and predictions for complex systems, but measurement noise can undermine its reliability. The paper identifies noise-induced bias in standard DMD and proposes a two-stage, noise-aware formulation to address it.
- 1 Introduction: DMD extracts dynamically relevant spatial structures and temporal characteristics from fluid-flow snapshots while approximating the Koopman operator.Its uses extend beyond fluid mechanics to fields including epidemiology, medicine, neuroscience, power systems, robotics, and video processing.
- 1 Introduction: Measurement errors can alter DMD-predicted growth and decay rates, complicating the identification of stable and unstable spatial modes.Existing mitigation approaches include rank reduction, ensemble averaging, cross-validation, and windowing.
- 1 Introduction: Standard DMD accounts for errors in only some snapshots, although measurement noise typically affects all snapshots, producing systematic bias.The paper addresses this issue directly rather than treating noise only through downstream mitigation methods.
- 1 Introduction: The proposed TDMD formulation rewrites DMD into subspace projection and operator identification stages, then modifies projection using an augmented snapshot matrix.This construction is designed to remove the source of bias introduced by the conventional asymmetric treatment of noisy snapshots.
- 1 Introduction: TDMD converges to exact spectra for a simple linear system and simulated cylinder flow, and outperforms standard DMD on TR-PIV data from separated flow.The paper focuses on measurement noise and data quality, while noting that process noise requires further investigation.
2 An Unbiased Formulation of DMD
The paper reformulates DMD as subspace projection followed by operator identification, revealing that asymmetric treatment of noisy snapshots causes bias. It removes this bias with total-least-squares projection using an augmented snapshot matrix, yielding TDMD, which agrees with standard DMD for noise-free data and can de-bias related methods.
- DMD and Koopman analysis: DMD approximates Koopman spectral properties from snapshot data by identifying a linear operator whose eigenvalues, modes, and eigenfunctions describe nonlinear dynamics.The approach uses paired observables evaluated at states and their one-step images.
- Source of bias: In overconstrained noisy settings, standard DMD treats snapshots in X and Y asymmetrically by correcting only Y, producing biased eigenvalues.Its least-squares interpretation corrects Y using a projection depending only on X.
- Noise-aware formulation: Under stated assumptions, total least squares converges to the exact noise-free solution as snapshot count increases, whereas least squares does not.This convergence requires an exact linear snapshot relationship in the noise-free case.
- Noise-aware formulation: The proposed TDMD formulation corrects both X and Y through a total-least-squares problem, accounting for measurement errors across all snapshots.It constructs an augmented snapshot matrix and projects onto its leading right-singular subspace before operator identification.
- Noise-free limit: When data are noise-free, standard least-squares DMD and TDMD are equivalent because their projected snapshot matrices equal the originals.The equivalence follows when Y = AX exactly.
- Generalization: The projection step can serve as a de-biasing pre-processing stage for standard DMD and other DMD-like algorithms.Examples named by the paper include optimal mode decomposition, streaming DMD, sparsity-promoting DMD, and optimized DMD.
3 DMD on a linear system
TDMD is tested on a low-dimensional linear system with known dynamics and noisy high-dimensional observations. Across snapshot counts and repeated noise realizations, the spectra from standard DMD and TDMD are compared.
- The toy system evolves two state variables with eigenvalues 1.02e^0.1i and 1.04e^0.3i.
- The randomly chosen observable maps C2 into C250, while the subspace projection uses the known rank r = 2.
- Additive circularly symmetric complex Gaussian noise with (∆X, ∆Y ) ∼ CN(0, 0.05) corrupts the snapshot pairs.
- Standard and total DMD use m = {100, 200, 500} snapshot pairs assembled from multiple ensemble runs.
- Each method is evaluated over 200 independent noise realizations, and the resulting spectra are compared in Figure 1.
4 DMD on cylinder flow simulations
The cylinder-flow simulation evaluates TDMD under synthetic measurement noise using DNS vorticity data. The study examines whether TDMD outperforms standard DMD as snapshot availability changes.
- 4 DMD on cylinder flow simulations: TDMD outperforms standard DMD in the analysis of more complex systems, including flow past a cylinder.
- 4 DMD on cylinder flow simulations: Figure 1 compares true eigenvalues with mean and individual realizations from standard DMD and TDMD across 200 noise realizations.
- 4 DMD on cylinder flow simulations: The cylinder-flow demonstration uses vorticity data generated by direct numerical simulation and sampled at fs = 100Hz.
- 4 DMD on cylinder flow simulations: Synthetic measurement noise is used to model mild contamination while retaining control over the numerical flowfield data.
5 DMD on flow separation experiments
TDMD is evaluated on noisy experimental separated-flow data measured with time-resolved particle image velocimetry. Its dominant modes are compared with standard DMD using a rank reduction retaining over 99% of X’s energy.
- 5 DMD on flow separation experiments: The experiment measures velocity snapshots with TR-PIV at fs = 1600Hz, using n = 42 976 spatial measurements and m = 3 000 snapshots.
- 5 DMD on flow separation experiments: Both methods use rank-reduction level r = 25, retaining over 99% of the energy content based on an SVD of X.
- 5 DMD on flow separation experiments: TDMD extracts dominant oscillatory modes that are essentially non-decaying, unlike those identified by standard DMD.
- 5 DMD on flow separation experiments: TDMD reports one low-amplitude spurious eigenvalue with |α| = 10^-5, which becomes more damped and disappears at lower truncation levels.
- 5 DMD on flow separation experiments: Both methods identify similar frequencies except between 30Hz and 90Hz, where TDMD identifies 49Hz and standard DMD identifies 58Hz.
6 Concluding Discussion
The two-stage view of DMD identifies asymmetric snapshot treatment as a source of systematic noise-induced bias. TDMD removes this bias by using an augmented snapshot matrix, while improving the interpretation of separated-flow dynamics and supporting more representative forecasts from noisy data.
- Bias diagnosis: Standard DMD’s asymmetric treatment of snapshots systematically introduces noise-induced bias that averaging and cross-validation cannot remove.These approaches may reduce variance but do not eliminate systematic bias.
- Experimental spectra: TDMD predicts smaller decay rates than DMD for oscillatory modes extracted from separated-flow TR-PIV data.Figure 3 compares DMD eigenvalues as circles with TDMD eigenvalues as triangles.
- Experimental modes: Some TDMD spatial structures differ from standard DMD modes in separated-flow dynamics, although many modes remain qualitatively similar.Figure 4 orders oscillatory vorticity modes by decreasing frequency and uses r = 25.
- Debiasing formulation: TDMD uses an augmented snapshot matrix during subspace projection to account for errors in all available data and obtain an unbiased DMD formulation.The procedure is based on total-least-squares/error-in-variables reasoning.
- Practical implications: TDMD supports more valid Koopman spectral analysis and more representative forecasts when experimental snapshot measurements are noisy or imprecise.The framework is intended for practical real-world data-driven analysis with imperfect measurements.