An incremental approach to online dynamic mode decomposition for time-varying systems with applications to EEG data modeling

Abstract

Dynamic Mode Decomposition (DMD) is a data-driven technique to identify a low dimensional linear time invariant dynamics underlying highdimensional 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 [35] to scenarios in which high dimensional data are governed by low dimensional time-varying dynamics. We consider two classes of algorithms that rely on (i) a discount factor on previous observations, and (ii) a sliding window of observations. Our algorithms leverage existing techniques for incremental singular value decomposition and allow us to determine an appropriately reduced model at each time and are applicable even if data matrix is singular. We apply the developed algorithms for autonomous systems to Electroencephalographic (EEG) data and demonstrate their effectiveness in terms of reconstruction and prediction. Our algorithms for non-autonomous systems are illustrated using randomly generated linear time-varying systems.

Adaptive Stochastic Reduced Order Modeling for Autonomous Ocean Platforms

“…There exist various works that address this such as the streaming DMD [23], online DMD [85], and incremental DMD [1], which all aim to incrementally update the DMD model with new state vectors. A more detailed comparison between the streaming DMD and online DMD is provided in [25].…”

“…A more detailed comparison between the streaming DMD and online DMD is provided in [25]. The novel incremental Low-Rank Dynamic Mode Decomposition (iLRDMD) [70,69] introduced in this work uses the incremental DMD method [1] in conjunction with the low-rank DMD derived in Section 3.1.1 to efficiently update the DMD model for onboard forecasting. The second limitation of DMD is that it is a locally linear of the dynamical system given in Eq.…”

“…In this work, we obtain efficient and accurate compression of the full-dimensional ocean forecasts by projecting it onto a set of POD basis. These compressed state vectors Z and Z ′ are defined as 1) . For communication disadvantaged platforms, receiving compressed states in the form of POD coefficients results in a significant reduction in bandwidth requirements.…”

“…To update the onboard DMD model, we utilize the incremental DMD (iDMD) method [1]. The low-rank DMD model in Section 3.1.1 can now be computed and updated using incremental methods with the compressed state vectors.…”

“…The low-rank DMD model in Section 3.1.1 can now be computed and updated using incremental methods with the compressed state vectors. The following analytical derivation of the incremental DMD follows closely from [1].…”