Non-Stationary Dynamic Mode Decomposition

Abstract

Many physical processes display complex high-dimensional time-varying behavior, from global weather patterns to brain activity. An outstanding challenge is to express high dimensional data in terms of a dynamical model that reveals their spatiotemporal structure. Dynamic Mode Decomposition is a means to achieve this goal, allowing the identification of key spatiotemporal modes through the diagonalization of a finite dimensional approximation of the Koopman operator. However, DMD methods apply best to time-translationally invariant or stationary data, while in many typical cases, dynamics vary across time and conditions. To capture this temporal evolution, we developed a method, Non-Stationary Dynamic Mode Decomposition (NS-DMD), that generalizes DMD by fitting global modulations of drifting spatiotemporal modes. This method accurately predicts the temporal evolution of modes in simulations and recovers previously known results from simpler methods. To demonstrate its properties, the method is applied to multi-channel recordings from an awake behaving non-human primate performing a cognitive task.

Data-Driven Modal Decomposition Methods as Feature Detection Techniques for Flow Fields in Hydraulic Machinery: A Mini Review

“…In 2023, Rosenfeld et al [30] addressed limitations in DMD by removing the Koopman operator, achieving theoretical objectives not realized in other contexts. To capture temporal variation, Ferre et al [31] developed non-stationary dynamic mode decomposition (NS-DMD), accurately predicting temporal evolution in simulations. Anzaki et al [32] proposed dynamic mode decomposition with memory (DMDm) to analyze time-series data, overcoming the constraints of existing DMD methods.…”