Extended Online DMD and Weighted Modifications for Streaming Data Analysis

Abstract

We present novel methods for computing the online dynamic mode decomposition (online DMD) for streaming datasets. We propose a framework that allows incremental updates to the DMD operator as data become available. Due to its ability to work on datasets with lower ranks, the proposed method is more advantageous than existing ones. A noteworthy feature of the method is that it is entirely data-driven and does not require knowledge of any underlying governing equations. Additionally, we present a modified version of our proposed approach that utilizes a weighted alternative to online DMD. The suggested techniques are demonstrated using several numerical examples.

New Robust Streaming DMD with Forecasting

“…The key idea of this approach is to use the Sherman-Morrison-Woodbury formula that is used for fast update of the pseudoinverse X † = X T (XX T ) -1 . Some approaches to the streaming DMD use techniques for updating the SVD of X, see [30], [38]. Since our main interest are the cases of large dimensional data, the SVD based updating is less attractive and we focus on [19], [54].…”

“…In this concrete case, it means updating the Cholesky factor of G x instead of updating its inverse. If G x = R T x R x is the Cholesky factorization, then the updating formula (38) can be used, but z = P x x new is computed as R -1…”

New Robust Streaming DMD with Forecasting

“…The key idea of this approach is to use the Sherman-Morrison-Woodbury formula that is used for fast update of the pseudoinverse X † = X T (XX T ) -1 . Some approaches to the streaming DMD use techniques for updating the SVD of X, see [30], [38]. Since our main interest are the cases of large dimensional data, the SVD based updating is less attractive and we focus on [19], [54].…”

“…In this concrete case, it means updating the Cholesky factor of G x instead of updating its inverse. If G x = R T x R x is the Cholesky factorization, then the updating formula (38) can be used, but z = P x x new is computed as R -1…”

Online physics-informed dynamic mode decomposition: theory and applications

“…Recent efforts to optimize DMD for online applications have primarily focused on a common scenario in which the system evolves slowly over time while data arrives continuously [33,37]. Notable progress in wind-tunnel testing, electroencephalography data examination and wind-farm forecasting demonstrates the enhanced efficiency and real-world relevance of these methods [38–42]. Despite these advancements, integrating physical constraints into DMD’s online learning algorithms remains an unaddressed challenge.…”

“…electroencephalography data examination and wind-farm forecasting demonstrates the enhanced efficiency and real-world relevance of these methods [38][39][40][41][42]. Despite these advancements, integrating physical constraints into DMD's online learning algorithms remains an unaddressed challenge.…”

New Robust Streaming DMD with Forecasting

“…The key idea of this approach is to use the Sherman-Morrison-Woodbury formula that is used for fast update of the pseudoinverse X † = X T (XX T ) -1 . Some approaches to the streaming DMD use techniques for updating the SVD of X, see [30], [38]. Since our main interest are the cases of large dimensional data, the SVD based updating is less attractive and we focus on [19], [54].…”

“…In this concrete case, it means updating the Cholesky factor of G x instead of updating its inverse. If G x = R T x R x is the Cholesky factorization, then the updating formula (38) can be used, but z = P x x new is computed as R -1…”

Online physics-informed dynamic mode decomposition: theory and applications

“…Recent efforts to optimize DMD for online applications have primarily focused on a common scenario in which the system evolves slowly over time while data arrives continuously [33,37]. Notable progress in wind-tunnel testing, electroencephalography data examination and wind-farm forecasting demonstrates the enhanced efficiency and real-world relevance of these methods [38–42]. Despite these advancements, integrating physical constraints into DMD’s online learning algorithms remains an unaddressed challenge.…”

“…electroencephalography data examination and wind-farm forecasting demonstrates the enhanced efficiency and real-world relevance of these methods [38][39][40][41][42]. Despite these advancements, integrating physical constraints into DMD's online learning algorithms remains an unaddressed challenge.…”

New Robust Streaming DMD with Forecasting

“…The key idea of this approach is to use the Sherman-Morrison-Woodbury formula that is used for fast update of the pseudoinverse X † = X T (XX T ) -1 . Some approaches to the streaming DMD use techniques for updating the SVD of X, see [30], [38]. Since our main interest are the cases of large dimensional data, the SVD based updating is less attractive and we focus on [19], [54].…”

“…In this concrete case, it means updating the Cholesky factor of G x instead of updating its inverse. If G x = R T x R x is the Cholesky factorization, then the updating formula (38) can be used, but z = P x x new is computed as R -1…”

Online physics-informed dynamic mode decomposition: theory and applications

“…Recent efforts to optimize DMD for online applications have primarily focused on a common scenario in which the system evolves slowly over time while data arrives continuously [33,37]. Notable progress in wind-tunnel testing, electroencephalography data examination and wind-farm forecasting demonstrates the enhanced efficiency and real-world relevance of these methods [38–42]. Despite these advancements, integrating physical constraints into DMD’s online learning algorithms remains an unaddressed challenge.…”

“…electroencephalography data examination and wind-farm forecasting demonstrates the enhanced efficiency and real-world relevance of these methods [38][39][40][41][42]. Despite these advancements, integrating physical constraints into DMD's online learning algorithms remains an unaddressed challenge.…”