# An Incremental Approach to Online Dynamic Mode Decomposition for Time-Varying Systems with Applications to EEG Data Modeling

## TL;DR

This paper introduces incremental algorithms for online dynamic mode decomposition tailored to time-varying systems, with applications to EEG data modeling, enabling real-time analysis of evolving dynamics.

## Contribution

It develops novel incremental DMD algorithms for systems with time-varying dynamics, extending existing methods to handle high-dimensional, non-stationary data efficiently.

## Key findings

- Effective reconstruction and prediction of EEG data.
- Algorithms successfully handle singular data matrices.
- Demonstrated applicability to both autonomous and non-autonomous systems.

## Abstract

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 [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.