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Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition
Scott T. M. Dawson, Maziar S. Hemati, Matthew O. Williams, Clarence W. Rowley
TL;DR
Sensor noise can bias DMD and complicate identification of true dynamical features, especially when noise increases the apparent data rank. The paper derives this bias, develops three correction-oriented DMD variants, and evaluates them on synthetic, numerical, and experimental data. The modified algorithms improve identification of lower-amplitude eigenvalue dynamics, while their use depends on noise information, computational considerations, and the distinction between sensor and process noise.
Problem
Sensor noise can bias DMD and make true dynamical modes difficult to distinguish when noisy data have numerical rank exceeding the governing dynamics’ dimension.
Method
The paper derives the noise effect using DMD’s data-matrix formulation and develops noise-corrected, forward-backward, and total least-squares DMD algorithms.
Results
Modified DMD algorithms improve identification of lower-amplitude eigenvalues, particularly their real components, while regular DMD remains accurate for dominant dynamics.
Takeaways & Limitations
Researchers can choose among DMD variants according to data size, noise level, required accuracy, and computational resources.
Takeaways & Limitations
Separating sensor noise from process noise remains challenging, especially with limited statistical information and complex turbulent flows.
Abstract
from arXiv · showhide
Dynamic mode decomposition (DMD) provides a practical means of extracting insightful dynamical information from fluids datasets. Like any data processing technique, DMD's usefulness is limited by its ability to extract real and accurate dynamical features from noise-corrupted data. Here we show analytically that DMD is biased to sensor noise, and quantify how this bias depends on the size and noise level of the data. We present three modifications to DMD that can be used to remove this bias: (i) a direct correction of the identified bias using known noise properties, (ii) combining the results of performing DMD forwards and backwards in time, and (iii) a total least-squares-inspired algorithm. We discuss the relative merits of each algorithm, and demonstrate the performance of these modifications on a range of synthetic, numerical, and experimental datasets. We further compare our modified DMD algorithms with other variants proposed in recent literature.
1 Introduction
DMD extracts dynamical features directly from fluid-flow data, but sensor noise complicates the separation of true dynamics from noise. This motivates analyzing and correcting DMD’s noise sensitivity, particularly for experimental data.
- DMD identifies and analyzes dynamical features of time-evolving fluid flows from experimental or simulation data.
- Noise can raise the numerical rank far above the governing dynamics’ dimension, making true modes difficult to distinguish from noise-dominated modes.
- Common responses include projecting data onto a reduced basis, selecting dynamically important modes, or over-specifying before balanced truncation.
- DMD’s direct use of data without a system matrix makes its sensitivity to experimentally relevant noise especially important.
- The paper characterizes noise effects analytically and introduces noise-corrected, forward-backward, and total least-squares DMD modifications.
2 Characterizing noise in dynamic mode decomposition
This section characterizes how sensor noise biases DMD and develops three modifications designed to remove that bias: direct correction, forward-backward DMD, and total least-squares DMD.
- DMD formulation: DMD assembles time-separated snapshot pairs into X and Y, then identifies dynamics through a reduced propagation matrix in POD space.The data matrices contain m snapshot pairs of size n; projecting into POD space supports efficient computation and optional truncation.
- Sensor-noise model: Sensor noise is modeled as additive, independent, zero-mean Gaussian measurement noise in Xm = X + NX and Ym = Y + NY.The analysis is performed in the POD basis obtained from noisy measurements and neglects higher-order terms when noise is sufficiently small.
- Sensor-noise effects: DMD is biased to sensor noise, with the bias becoming dominant over random error when m^1/2SNR > n^1/2.The bias is the difference between the true and expected identified quantity, whereas random error fluctuates across noise realizations.
- Bias-correction methods: Three bias-removal strategies are proposed: direct noise correction, forward-backward DMD, and total least-squares DMD.The direct correction requires accurate noise covariance knowledge and assumes noise is smaller than the true data in retained POD modes; forward-backward DMD accounts for errors in both X and Y.
- Bias-correction methods: The direct correction can be computationally prohibitive because it requires inversion of large n × n matrices, although it could be more accurate without requiring close measured and true POD modes.The relevant matrix XX* is invertible only when m > n.
- DMD formulation: Standard DMD computes A = YX+ and thereby minimizes error in Y while implicitly treating X as error-free.This asymmetry explains why the derived sensor-noise bias depends on noise in X but not noise in Y.
3 Results with synthetic data
Synthetic-data experiments show that sensor noise biases regular DMD, while the proposed corrections reduce this bias and often improve eigenvalue or propagation-matrix estimates. Their benefits are strongest for small state sizes or larger noise, although no single algorithm is globally superior.
- 3.1 Example: A periodic linear system: Regular DMD exhibits a noise-induced bias whose error can saturate as the number of snapshots m increases.The saturation arises from a bias term whose magnitude is independent of m.
- 3.1 Example: A periodic linear system: The direct noise correction removes the bias, after which propagation-matrix error decreases proportional to m^-1/2.The theoretical noise covariance achieves almost the same reduction as the sampled covariance, and normalized ncDMD error curves collapse across noise levels.
- 3.2 A periodic linear system with a high-dimensional state of observables: In high-dimensional observables, Algorithms 2–4 produce mean eigenvalues closer to the true value than regular DMD.For small state sizes, Algorithms 3 and 4 also yield smaller confidence ellipses; as state size increases, bias becomes smaller relative to random error and ellipse size decreases proportional to n^-1/2.
- 3.3 Comparison to other modified DMD algorithms: Forward-backward DMD and total least-squares DMD give the best estimate of the true eigenvalue in the reported comparison with OMD and spDMD.spDMD occasionally produced erroneous outliers, whereas the proposed algorithms are given in closed form and do not require optimization parameters or tolerances.
- 3.3 Comparison to other modified DMD algorithms: Results from one dataset do not establish global superiority for any algorithm.The authors note that different datasets or metrics could favor different methods.
- 3.4 Identifying hidden dynamics: The proposed corrections are most useful when dynamics are small or quickly decaying and can be hidden among measurement noise.For a system combining a growing traveling wave with a decaying signal, sensor-noise correction accurately corrects the eigenvalue shift with and without process noise.
4 Results with numerical and experimental data
Numerical and experimental cylinder-wake data show that standard DMD misidentifies growth and decay rates under noise, especially for lower-energy, higher-frequency modes. fbDMD and tlsDMD substantially correct these errors while preserving the identified modes, and tlsDMD improves experimental reduced-order models without requiring noise-characteristic estimates.
- DNS cylinder wake: At Re = 100, standard DMD produces significant growth-rate errors for the highest-frequency eigenvalues in noisy DNS data.The resulting model predicts decay for dominant low-frequency POD modes.
- DNS cylinder wake: The DNS comparison evaluates eigenvalues and POD coefficients for DMD, ncDMD, fbDMD, and tlsDMD under Gaussian noise with σ = 0.2U/D.
- DNS cylinder wake: fbDMD and tlsDMD almost completely remove the erroneous decay of high-frequency modes, while ncDMD improves performance marginally.
- DNS cylinder wake: The modified algorithms are also validated against clean-data DMD modes using visual comparisons and normalized inner products.
- Water-channel experiment: In experimental data, tlsDMD yields more accurate low-dimensional models than DMD, whose left-shifted eigenvalues cause erroneous decay of less energetic, rapidly oscillating POD coefficients.This improvement requires no explicit knowledge of process or sensor noise characteristics.
- Water-channel experiment: The experimental comparison uses eigenvalues and predicted POD coefficients from DMD and tlsDMD after projection onto the 15 most energetic POD modes.
5 Discussion and conclusions
The authors conclude that sensor noise biases standard DMD, particularly for low-amplitude modes’ growth and decay rates, and propose algorithm choices based on noise, data size, accuracy, and computational resources. They identify limitations involving noise assumptions, projection choices, data availability, and preprocessing.
- Conclusions: Direct bias correction nearly eliminates DMD’s bias but requires an accurate noise covariance and can be unsuitable with untruncated small singular values.
- Conclusions: fbDMD and tlsDMD correct the bias without requiring noise characteristics and reduce random error across many noise conditions.
- Practical guidance: Modified DMD algorithms most improve eigenvalues of lower-amplitude modes, especially their real components, whereas standard DMD can accurately identify dominant dynamics.
- Mechanism: tlsDMD addresses sensor-noise bias by accounting for noise in both data matrices, unlike least-squares formulations that treat only one direction as noisy.
- Comparisons and scope: Compared with sparsity-promoting DMD and optimized DMD, the proposed algorithms performed better in the reported comparisons, although POD projection can degrade results when important modes differ from dominant POD modes.
- Limitations: Real datasets usually lack the many trials used for synthetic statistical testing, limiting confidence in algorithm selection and reported results.
- Limitations: The study focuses mainly on sensor noise; process-noise characterization and removal remain a separate problem with different frequency-dependent effects.
- Future work: The effects of experimental averaging and smoothing preprocessing on subsequent DMD analysis remain to be investigated.
Appendix 1: Quantifying the size of the bias in DMD
Under uniform, independent sensor-noise assumptions, the appendix quantifies DMD’s bias and compares it with random error. The bias reduces computed dynamics, especially for low-energy modes, and can dominate even as more snapshots are added.
- Assumptions: The bias estimate assumes uniform, spatially and temporally independent noise and sufficiently stable noise-covariance estimates.The analysis also assumes POD energy fractions remain constant when the data dimensions vary.
- Bias magnitude: The bias magnitude is represented by mσ_N^2Σ^-2, making its diagonal effect larger for POD modes with smaller energy.The ith diagonal bias entry scales inversely with the corresponding mode energy.
- Dynamical consequence: The bias predicts overly rapid decay and continuous-time eigenvalues shifted farther into the left half plane, especially for lower-energy modes.The appendix identifies growth rates as particularly vulnerable to this noise-induced distortion.
- Random error: Noise-induced cross terms form uncorrelated sums over m random terms, with each term’s variance determined by the POD energy fraction and noise variance.This scaling is used to estimate the random error in the identified DMD operator.
- Bias versus random error: The bias is independent of m, so it can become dominant for many low-dimensional snapshots and cannot always be reduced by collecting more data.The appendix compares this bias with the random error component to determine when the bias dominates.