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Solving the inverse problem for an ordinary differential equation using conjugation
An incremental approach to online dynamic mode decomposition for time-varying systems with applications to EEG data modeling
Electrical and Computer Engineering Department, Michigan State University, East Lansing, MI 48824, USA |
Dynamic Mode Decomposition (DMD) is a data-driven technique to identify a low dimensional linear time invariant dynamics underlying high-dimensional data. For systems in which such underlying low-dimensional dynamics is time-varying, a time-invariant approximation of such dynamics computed through standard DMD techniques may not be appropriate. We focus on DMD techniques for such time-varying systems and develop incremental algorithms for systems without and with exogenous control inputs. We build upon the work in [
References:
[1] |
M. Brand, Incremental singular value decomposition of uncertain data with missing values, in European Conference on Computer Vision, Springer, 2002,707–720. |
[2] |
M. Brand,
Fast low-rank modifications of the thin singular value decomposition, Linear Algebra Appl., 415 (2006), 20-30.
doi: 10.1016/j.laa.2005.07.021. |
[3] |
B. W. Brunton, L. A. Johnson, J. G. Ojemann and J. N. Kutz,
Extracting spatial–temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition, Journal of Neuroscience Methods, 258 (2016), 1-15.
|
[4] |
R. Chavarriaga and J. D. R. Millán,
Learning from EEG error-related potentials in noninvasive brain-computer interfaces, IEEE Transactions on Neural Systems and Rehabilitation Engineering, 18 (2010), 381-388.
doi: 10.1109/TNSRE.2010.2053387. |
[5] |
R. Chavarriaga, A. Sobolewski and J. D. R. Millán, Errare machinale est: The use of error-related potentials in brain-machine interfaces, Frontiers in Neuroscience, 8 (2014), 208.
doi: 10.3389/fnins.2014.00208. |
[6] |
X. Chen and K. S. Candan, LWI-SVD: Low-rank, windowed, incremental singular value decompositions on time-evolving data sets, in Proceedings of the 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, ACM, 2014,987–996.
doi: 10.1145/2623330.2623671. |
[7] |
A. C. Costa, T. Ahamed and G. J. Stephens,
Adaptive, locally linear models of complex dynamics, Proc. Natl. Acad. Sci. USA, 116 (2019), 1501-1510.
doi: 10.1073/pnas.1813476116. |
[8] |
A. Delorme and S. Makeig,
EEGLAB: An open source toolbox for analysis of single-trial eeg dynamics including independent component analysis, Journal of Neuroscience Methods, 134 (2004), 9-21.
doi: 10.1016/j.jneumeth.2003.10.009. |
[9] |
P. W. Ferrez and J. d. R. Millán,
Error-related EEG potentials generated during simulated brain–computer interaction, IEEE Transactions on Biomedical Engineering, 55 (2008), 923-929.
doi: 10.1109/TBME.2007.908083. |
[10] |
J. Grosek and J. N. Kutz, Dynamic mode decomposition for real-time background/foreground separation in video, arXiv preprint, arXiv: 1404.7592. |
[11] |
M. Gu and S. C. Eisenstat, A Stable and Fast Algorithm for Updating the Singular Value Decomposition, Technical Report YALEU/DCS/RR-966, Department of Computer Science, Yale University, New Haven, CT, 1993. |
[12] |
M. Hemati, E. Deem, M. Williams, C. W. Rowley and L. N. Cattafesta, Improving separation control with noise-robust variants of dynamic mode decomposition, in 54th AIAA Aerospace Sciences Meeting, (2016), 1103.
doi: 10.2514/6.2016-1103. |
[13] |
M. S. Hemati, M. O. Williams and C. W. Rowley, Dynamic mode decomposition for large and streaming datasets, Physics of Fluids, 26 (2014), 111701.
doi: 10.1063/1.4901016. |
[14] |
T. K. Huckle,
Efficient computation of sparse approximate inverses, Numer. Linear Algebra Appl., 5 (1998), 57-71.
doi: 10.1002/(SICI)1099-1506(199801/02)5:1<57::AID-NLA129>3.0.CO;2-C. |
[15] |
R. Isermann and M. Münchhof, Identification of Dynamic Systems: An Introduction with Applications, Springer, Heidelberg, 2011.
doi: 10.1007/978-3-540-78879-9. |
[16] |
S. M. Kay, Fundamentals of Statistical Signal Processing, Prentice Hall PTR, 1993. |
[17] |
B. O. Koopman,
Hamiltonian systems and transformation in Hilbert space, Proceedings of the National Academy of Sciences, 17 (1931), 315-318.
doi: 10.1073/pnas.17.5.315. |
[18] |
M. Korda and I. Mezić,
Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control, Automatica J. IFAC, 93 (2018), 149-160.
doi: 10.1016/j.automatica.2018.03.046. |
[19] |
J. N. Kutz, S. L. Brunton, B. W. Brunton and J. L. Proctor, Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems, vol. 149, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016.
doi: 10.1137/1.9781611974508. |
[20] |
S. Maćešić, N. Črnjarić-Žic and I. Mezić, Koopman operator family spectrum for nonautonomous systems-part 1, arXiv preprint, arXiv: 1703.07324. |
[21] |
D. Matsumoto and T. Indinger, On-the-fly algorithm for dynamic mode decomposition using incremental singular value decomposition and total least squares, arXiv preprint, arXiv: 1703.11004. |
[22] |
C. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, Philadelphia, PA, 2000.
doi: 10.1137/1.9780898719512. |
[23] |
I. Mezić,
Spectral properties of dynamical systems, model reduction and decompositions, Nonlinear Dynam., 41 (2005), 309-325.
doi: 10.1007/s11071-005-2824-x. |
[24] |
I. Mezić,
Analysis of fluid fows via spectral properties of the Koopman operator, Annual Review of Fluid Mechanics, 45 (2013), 357-378.
doi: 10.1146/annurev-fluid-011212-140652. |
[25] |
I. Mezić and A. Banaszuk,
Comparison of systems with complex behavior, Phys. D, 197 (2004), 101-133.
doi: 10.1016/j.physd.2004.06.015. |
[26] |
G. M. Oxberry, T. Kostova-Vassilevska, W. Arrighi and K. Chand,
Limited-memory adaptive snapshot selection for proper orthogonal decomposition, Internat. J. Numer. Methods Engrg., 109 (2017), 198-217.
doi: 10.1002/nme.5283. |
[27] |
J. L. Proctor, S. L. Brunton and J. N. Kutz,
Dynamic mode decomposition with control, SIAM J. Appl. Dyn. Syst., 15 (2016), 142-161.
doi: 10.1137/15M1013857. |
[28] |
J. L. Proctor and P. A. Eckhoff,
Discovering dynamic patterns from infectious disease data using dynamic mode decomposition, International Health, 7 (2015), 139-145.
doi: 10.1093/inthealth/ihv009. |
[29] |
C. W. Rowley, I. Mezić, S. Bagheri, P. Schlatter and D. S. Henningson,
Spectral analysis of nonlinear flows, J. Fluid Mech., 641 (2009), 115-127.
doi: 10.1017/S0022112009992059. |
[30] |
P. J. Schmid,
Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech., 656 (2010), 5-28.
doi: 10.1017/S0022112010001217. |
[31] |
J. Sherman and W. J. Morrison,
Adjustment of an inverse matrix corresponding to a change in one element of a given matrix, Ann. Math. Statistics, 21 (1950), 124-127.
doi: 10.1214/aoms/1177729893. |
[32] |
A. Surana, Koopman operator based nonlinear dynamic textures, in 2015 American Control Conference (ACC), IEEE, 2015, 1333–1338.
doi: 10.1109/ACC.2015.7170918. |
[33] |
J. H. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. Brunton and J. N. Kutz,
On dynamic mode decomposition: Theory and applications, J. Comput. Dyn., 1 (2014), 391-421.
doi: 10.3934/jcd.2014.1.391. |
[34] |
M. O. Williams, I. G. Kevrekidis and C. W. Rowley,
A data–driven approximation of the Koopman operator: Extending dynamic mode decomposition, J. Nonlinear Sci., 25 (2015), 1307-1346.
doi: 10.1007/s00332-015-9258-5. |
[35] |
H. Zhang, C. W. Rowley, E. A. Deem and L. N. Cattafesta,
Online dynamic mode decomposition for time-varying systems, SIAM J. Appl. Dyn. Syst., 18 (2019), 1586-1609.
doi: 10.1137/18M1192329. |
show all references
References:
[1] |
M. Brand, Incremental singular value decomposition of uncertain data with missing values, in European Conference on Computer Vision, Springer, 2002,707–720. |
[2] |
M. Brand,
Fast low-rank modifications of the thin singular value decomposition, Linear Algebra Appl., 415 (2006), 20-30.
doi: 10.1016/j.laa.2005.07.021. |
[3] |
B. W. Brunton, L. A. Johnson, J. G. Ojemann and J. N. Kutz,
Extracting spatial–temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition, Journal of Neuroscience Methods, 258 (2016), 1-15.
|
[4] |
R. Chavarriaga and J. D. R. Millán,
Learning from EEG error-related potentials in noninvasive brain-computer interfaces, IEEE Transactions on Neural Systems and Rehabilitation Engineering, 18 (2010), 381-388.
doi: 10.1109/TNSRE.2010.2053387. |
[5] |
R. Chavarriaga, A. Sobolewski and J. D. R. Millán, Errare machinale est: The use of error-related potentials in brain-machine interfaces, Frontiers in Neuroscience, 8 (2014), 208.
doi: 10.3389/fnins.2014.00208. |
[6] |
X. Chen and K. S. Candan, LWI-SVD: Low-rank, windowed, incremental singular value decompositions on time-evolving data sets, in Proceedings of the 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, ACM, 2014,987–996.
doi: 10.1145/2623330.2623671. |
[7] |
A. C. Costa, T. Ahamed and G. J. Stephens,
Adaptive, locally linear models of complex dynamics, Proc. Natl. Acad. Sci. USA, 116 (2019), 1501-1510.
doi: 10.1073/pnas.1813476116. |
[8] |
A. Delorme and S. Makeig,
EEGLAB: An open source toolbox for analysis of single-trial eeg dynamics including independent component analysis, Journal of Neuroscience Methods, 134 (2004), 9-21.
doi: 10.1016/j.jneumeth.2003.10.009. |
[9] |
P. W. Ferrez and J. d. R. Millán,
Error-related EEG potentials generated during simulated brain–computer interaction, IEEE Transactions on Biomedical Engineering, 55 (2008), 923-929.
doi: 10.1109/TBME.2007.908083. |
[10] |
J. Grosek and J. N. Kutz, Dynamic mode decomposition for real-time background/foreground separation in video, arXiv preprint, arXiv: 1404.7592. |
[11] |
M. Gu and S. C. Eisenstat, A Stable and Fast Algorithm for Updating the Singular Value Decomposition, Technical Report YALEU/DCS/RR-966, Department of Computer Science, Yale University, New Haven, CT, 1993. |
[12] |
M. Hemati, E. Deem, M. Williams, C. W. Rowley and L. N. Cattafesta, Improving separation control with noise-robust variants of dynamic mode decomposition, in 54th AIAA Aerospace Sciences Meeting, (2016), 1103.
doi: 10.2514/6.2016-1103. |
[13] |
M. S. Hemati, M. O. Williams and C. W. Rowley, Dynamic mode decomposition for large and streaming datasets, Physics of Fluids, 26 (2014), 111701.
doi: 10.1063/1.4901016. |
[14] |
T. K. Huckle,
Efficient computation of sparse approximate inverses, Numer. Linear Algebra Appl., 5 (1998), 57-71.
doi: 10.1002/(SICI)1099-1506(199801/02)5:1<57::AID-NLA129>3.0.CO;2-C. |
[15] |
R. Isermann and M. Münchhof, Identification of Dynamic Systems: An Introduction with Applications, Springer, Heidelberg, 2011.
doi: 10.1007/978-3-540-78879-9. |
[16] |
S. M. Kay, Fundamentals of Statistical Signal Processing, Prentice Hall PTR, 1993. |
[17] |
B. O. Koopman,
Hamiltonian systems and transformation in Hilbert space, Proceedings of the National Academy of Sciences, 17 (1931), 315-318.
doi: 10.1073/pnas.17.5.315. |
[18] |
M. Korda and I. Mezić,
Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control, Automatica J. IFAC, 93 (2018), 149-160.
doi: 10.1016/j.automatica.2018.03.046. |
[19] |
J. N. Kutz, S. L. Brunton, B. W. Brunton and J. L. Proctor, Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems, vol. 149, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2016.
doi: 10.1137/1.9781611974508. |
[20] |
S. Maćešić, N. Črnjarić-Žic and I. Mezić, Koopman operator family spectrum for nonautonomous systems-part 1, arXiv preprint, arXiv: 1703.07324. |
[21] |
D. Matsumoto and T. Indinger, On-the-fly algorithm for dynamic mode decomposition using incremental singular value decomposition and total least squares, arXiv preprint, arXiv: 1703.11004. |
[22] |
C. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, Philadelphia, PA, 2000.
doi: 10.1137/1.9780898719512. |
[23] |
I. Mezić,
Spectral properties of dynamical systems, model reduction and decompositions, Nonlinear Dynam., 41 (2005), 309-325.
doi: 10.1007/s11071-005-2824-x. |
[24] |
I. Mezić,
Analysis of fluid fows via spectral properties of the Koopman operator, Annual Review of Fluid Mechanics, 45 (2013), 357-378.
doi: 10.1146/annurev-fluid-011212-140652. |
[25] |
I. Mezić and A. Banaszuk,
Comparison of systems with complex behavior, Phys. D, 197 (2004), 101-133.
doi: 10.1016/j.physd.2004.06.015. |
[26] |
G. M. Oxberry, T. Kostova-Vassilevska, W. Arrighi and K. Chand,
Limited-memory adaptive snapshot selection for proper orthogonal decomposition, Internat. J. Numer. Methods Engrg., 109 (2017), 198-217.
doi: 10.1002/nme.5283. |
[27] |
J. L. Proctor, S. L. Brunton and J. N. Kutz,
Dynamic mode decomposition with control, SIAM J. Appl. Dyn. Syst., 15 (2016), 142-161.
doi: 10.1137/15M1013857. |
[28] |
J. L. Proctor and P. A. Eckhoff,
Discovering dynamic patterns from infectious disease data using dynamic mode decomposition, International Health, 7 (2015), 139-145.
doi: 10.1093/inthealth/ihv009. |
[29] |
C. W. Rowley, I. Mezić, S. Bagheri, P. Schlatter and D. S. Henningson,
Spectral analysis of nonlinear flows, J. Fluid Mech., 641 (2009), 115-127.
doi: 10.1017/S0022112009992059. |
[30] |
P. J. Schmid,
Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech., 656 (2010), 5-28.
doi: 10.1017/S0022112010001217. |
[31] |
J. Sherman and W. J. Morrison,
Adjustment of an inverse matrix corresponding to a change in one element of a given matrix, Ann. Math. Statistics, 21 (1950), 124-127.
doi: 10.1214/aoms/1177729893. |
[32] |
A. Surana, Koopman operator based nonlinear dynamic textures, in 2015 American Control Conference (ACC), IEEE, 2015, 1333–1338.
doi: 10.1109/ACC.2015.7170918. |
[33] |
J. H. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. Brunton and J. N. Kutz,
On dynamic mode decomposition: Theory and applications, J. Comput. Dyn., 1 (2014), 391-421.
doi: 10.3934/jcd.2014.1.391. |
[34] |
M. O. Williams, I. G. Kevrekidis and C. W. Rowley,
A data–driven approximation of the Koopman operator: Extending dynamic mode decomposition, J. Nonlinear Sci., 25 (2015), 1307-1346.
doi: 10.1007/s00332-015-9258-5. |
[35] |
H. Zhang, C. W. Rowley, E. A. Deem and L. N. Cattafesta,
Online dynamic mode decomposition for time-varying systems, SIAM J. Appl. Dyn. Syst., 18 (2019), 1586-1609.
doi: 10.1137/18M1192329. |















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