Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition

AbstractDynamic 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: (1) a direct correction of the identified bias using known noise properties, (2) combining the results of performing DMD forwards and backwards in time, and (3) 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 the recent literature.

[1]  Xiuyuan Cheng,et al.  THE SPECTRUM OF RANDOM INNER-PRODUCT KERNEL MATRICES , 2012, 1202.3155.

[2]  I. Mezić,et al.  Analysis of Fluid Flows via Spectral Properties of the Koopman Operator , 2013 .

[3]  A. Peirce.  Computer Methods in Applied Mechanics and Engineering , 2010 .

[4]  Uri Shaham,et al.  Dynamic Mode Decomposition , 2013 .

[5]  T. Tao,et al.  Random covariance matrices: Universality of local statistics of eigenvalues , 2009, 0912.0966.

[6]  J. W. Silverstein,et al.  Spectral Analysis of Large Dimensional Random Matrices , 2009 .

[7]  Jan R. Wright,et al.  An eigensystem realization algorithm using data correlations (ERA/DC) for modal parameter identification , 1987 .

[8]  Lei Zheng,et al.  Image Noise Level Estimation by Principal Component Analysis , 2013, IEEE Transactions on Image Processing.

[9]  Clarence W. Rowley,et al.  A Kernel Approach to Data-Driven Koopman Spectral Analysis , 2014 .

[10]  J. Nathan Kutz,et al.  Dynamic Mode Decomposition for Real-Time Background/Foreground Separation in Video , 2014, ArXiv.

[11]  Gene H. Golub,et al.  Matrix computations , 1983 .

[12]  Amit Singer,et al.  Two-Dimensional Tomography from Noisy Projections Taken at Unknown Random Directions , 2013, SIAM J. Imaging Sci..

[13]  C. Williamson.  Vortex Dynamics in the Cylinder Wake , 1996 .

[14]  Clarence W. Rowley,et al.  Model Reduction for fluids, Using Balanced Proper Orthogonal Decomposition , 2005, Int. J. Bifurc. Chaos.

[15]  Ioannis G. Kevrekidis,et al.  Extending Dynamic Mode Decomposition: A Data--Driven Approximation of the Koopman Operator , 2014 .

[16]  J. Juang,et al.  Effects of Noise on Modal Parameters Identified by the Eigensystem Realization Algorithm , 1986 .

[17]  Tim Colonius,et al.  The immersed boundary method: A projection approach , 2007, J. Comput. Phys..

[18]  B. O. Koopman,et al.  Hamiltonian Systems and Transformation in Hilbert Space. , 1931, Proceedings of the National Academy of Sciences of the United States of America.

[19]  Steven L. Brunton,et al.  On dynamic mode decomposition: Theory and applications , 2013, 1312.0041.

[20]  M. Provansal,et al.  Bénard-von Kármán instability: transient and forced regimes , 1987, Journal of Fluid Mechanics.

[21]  C. Eckart,et al.  The approximation of one matrix by another of lower rank , 1936 .

[22]  Clarence W. Rowley,et al.  Variants of Dynamic Mode Decomposition: Boundary Condition, Koopman, and Fourier Analyses , 2012, J. Nonlinear Sci..

[23]  S. Bagheri.  Effects of weak noise on oscillating flows: Linking quality factor, Floquet modes, and Koopman spectrum , 2014 .

[24]  Β. L. HO,et al.  Editorial: Effective construction of linear state-variable models from input/output functions , 1966 .

[25]  P. Schmid,et al.  Dynamic mode decomposition of numerical and experimental data , 2008, Journal of Fluid Mechanics.

[26]  Damon Honnery,et al.  An error analysis of the dynamic mode decomposition , 2011, Experiments in Fluids.

[27]  Jinjun Wang,et al.  On the accuracy of dynamic mode decomposition in estimating instability of wave packet , 2015 .

[28]  Leonardo P. Chamorro,et al.  On the transient dynamics of the wake and trajectory of free falling cones with various apex angles , 2015 .

[29]  Clarence W. Rowley,et al.  Dynamic mode decomposition for large and streaming datasets , 2014, 1406.7187.

[30]  Brenden P. Epps,et al.  An error threshold criterion for singular value decomposition modes extracted from PIV data , 2010 .

[31]  P. Schmid,et al.  Applications of the dynamic mode decomposition , 2011 .

[32]  Michael Stewart,et al.  Perturbation of the SVD in the presence of small singular values , 2006 .

[33]  I. Mezić,et al.  Spectral analysis of nonlinear flows , 2009, Journal of Fluid Mechanics.

[34]  Clarence W. Rowley,et al.  Spectral analysis of fluid flows using sub-Nyquist-rate PIV data , 2014, Experiments in Fluids.

[35]  Jer-Nan Juang,et al.  An eigensystem realization algorithm for modal parameter identification and model reduction. [control systems design for large space structures] , 1985 .

[36]  Clarence W. Rowley,et al.  A parallelized model reduction library , 2013 .

[37]  Shervin Bagheri,et al.  Koopman-mode decomposition of the cylinder wake , 2013, Journal of Fluid Mechanics.

[38]  T. Colonius,et al.  A fast immersed boundary method using a nullspace approach and multi-domain far-field boundary conditions , 2008 .

[39]  Clarence W. Rowley,et al.  Algorithm 945 , 2014, ACM Trans. Math. Softw..

[40]  Matthew O. Williams,et al.  A Kernel-Based Approach to Data-Driven Koopman Spectral Analysis , 2014, 1411.2260.

[41]  Peter J. Schmid,et al.  Sparsity-promoting dynamic mode decomposition , 2012, 1309.4165.

[42]  Peter J. Schmid,et al.  Application of the dynamic mode decomposition to experimental data , 2011 .

[43]  Christian J. Kähler,et al.  Ultra-high-speed 3D astigmatic particle tracking velocimetry: application to particle-laden supersonic impinging jets , 2014 .

[44]  Paul J. Goulart,et al.  Optimal mode decomposition for high dimensional systems , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[45]  Yannick Bury,et al.  Lagrangian and spectral analysis of the forced flow past a circular cylinder using pulsed tangential jets , 2012, Journal of Fluid Mechanics.

[46]  Clarence W. Rowley,et al.  A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition , 2014, Journal of Nonlinear Science.

[47]  I. Mezić.  Spectral Properties of Dynamical Systems, Model Reduction and Decompositions , 2005 .

[48]  B. R. Noack,et al.  A hierarchy of low-dimensional models for the transient and post-transient cylinder wake , 2003, Journal of Fluid Mechanics.

[49]  Zhizhen Zhao,et al.  Fourier-Bessel rotational invariant eigenimages , 2012, Journal of the Optical Society of America. A, Optics, image science, and vision.

[50]  G. Stewart.  Perturbation theory for the singular value decomposition , 1990 .

[51]  P. Goulart,et al.  Optimal mode decomposition for unsteady flows , 2013, Journal of Fluid Mechanics.