Colored-LIM: A Data-Driven Method for Studying Dynamical Systems with Temporally Correlated Stochasticity

Abstract

In real-world problems, environmental noise is often idealized as Gaussian white noise, despite potential temporal dependencies. The Linear Inverse Model (LIM) is a class of data-driven methods that extract dynamic and stochastic information from finite time-series data of complex systems. In this study, we introduce a new variant of LIM, called Colored-LIM, which models stochasticity using Ornstein-Uhlenbeck colored noise. Despite the non-trivial correlation between observable and colored noise, we show that Colored-LIM unveils the desired information merely from the correlation function of the observable. Therefore, this approach not only accounts for the memory effects of environmental noise, traditionally represented by time-uncorrelated white noise in the Classical LIM framework, but does so using the same observation dataset without requiring additional data. Furthermore, we show that Colored-LIM does not reduce to Classical LIM in the white noise limit, underscoring the importance of temporal dependencies in stochastic systems. In this paper, we rigorously develop the Colored-LIM, explore its connections with the Classical LIM and Dynamic Mode Decomposition, and validate its effectiveness on both ideal linear and nonlinear systems. In addition, we illustrate the potential applications and implications of Colored-LIM for real-world problems, including the El Ni\~no-Southern Oscillation and the electricity network of Tohoku University.