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

Justin Lien, Yan-Ning Kuo, Hiroyasu Ando, Shoichiro Kido

TL;DR

Colored-LIM addresses the limitation of white-noise assumptions in classical LIM by incorporating temporally correlated stochastic forcing through OU noise. It derives a correlation-function–based inverse problem that recovers the linear dynamics and diffusion from data, with a one-dimensional tau selection to determine noise memory when tau is unknown. The method is shown to differ from the white-noise LIM in the tau→0 limit and is connected to Dynamic Mode Decomposition as a colored-noise extension. Validation on linear and nonlinear models, plus real-world applications to ENSO forecasting and a campus electricity network, demonstrates improved interpretability and forecasting skill when memory effects are accounted. The approach offers a versatile framework for dynamical-mode discovery and stochastic parameterization in complex systems with memory.

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 Niño-Southern Oscillation and the electricity network of Tohoku University.

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

TL;DR

Colored-LIM addresses the limitation of white-noise assumptions in classical LIM by incorporating temporally correlated stochastic forcing through OU noise. It derives a correlation-function–based inverse problem that recovers the linear dynamics and diffusion from data, with a one-dimensional tau selection to determine noise memory when tau is unknown. The method is shown to differ from the white-noise LIM in the tau→0 limit and is connected to Dynamic Mode Decomposition as a colored-noise extension. Validation on linear and nonlinear models, plus real-world applications to ENSO forecasting and a campus electricity network, demonstrates improved interpretability and forecasting skill when memory effects are accounted. The approach offers a versatile framework for dynamical-mode discovery and stochastic parameterization in complex systems with memory.

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 Niño-Southern Oscillation and the electricity network of Tohoku University.
Paper Structure (18 sections, 4 theorems, 62 equations, 6 figures, 2 tables, 2 algorithms)

This paper contains 18 sections, 4 theorems, 62 equations, 6 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. A Review of the Classical LIM
  3. The method of transition function
  4. The fluctuation-dissipation relation (FDR)
  5. The algorithm of the Classical LIM
  6. The Mathematics of Colored-LIM
  7. Mathematical background
  8. The determination of the noise correlation time
  9. From an application perspective
  10. Connections with dynamic mode decomposition
  11. Validation
  12. Solving a linear problem
  13. Solving a non-linear problem: SIS model and network identification
  14. Applications to Real-world Complex Systems
  15. ENSO: prediction skill
  16. ...and 3 more sections

Key Result

Theorem 1

Let the correlation function be given by ${\bf K}(s) \coloneqq \langle {\bf x}(\cdot+s) {\bf x}(\cdot)^T \rangle$. For the linear stochastic process satisfying Eq. (Eq:WienerProcess), the dynamical matrix ${\bf A}$ satisfies for any $s>0$, where the $\log$ denotes the matrix logarithm. In particular, the correlation function is the exponential function of the form for $s \ge 0$, where ${{\bf C}_

Figures (6)

  • Figure 1: The $(i,j)$-entry of the correlation functions where $i$ and $j$ index the rows and columns, respectively. Each subplot corresponds to a different combination of $i$-th and $j$-th variables. The ground truth ${\bf K}$ for Eq. (\ref{['Eq:Linear_Problem']}) is represented by the solid red line, the observation ${K_\text{obs}}$ is marked by a black circle, and the one $K$ constructed by Colored-LIM is shown by blue.
  • Figure 2: The underlying network of the network-level SIS model in the numerical experiment in section \ref{['Chap:Network-identification']}. The time series shown at each node is a segment of the prevalence of infected individuals (input data). The orange dashed line indicates the analytic mean value (steady state).
  • Figure 3: The observation of SST and D20 from 1979 to 2022, and a trajectory of Colored-LIM and Classical-LIM processes of length $43$ years.
  • Figure 4: The $(i,j)$-entries of the observed correlation function of the normalized SST and D20 anomalies symbolized as $1$, and $2$, respectively, along with the corresponding Colored-LIM and Classical-LIM correlation functions. For clarity, the insets provide a zoomed-in view near the origin.
  • Figure 5: The ECR of SST and D20 for Colored-LIM and Classical LIM. The shaded area specifies the interval $(0.9,1.1)$, indicating a fair ensemble forecasting skill.
  • ...and 1 more figures

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Theorem 2: Classical Fokker-Plank equation
  • Corollary 2.1: The classical fluctuation-dissipation relation
  • proof
  • Theorem 3
  • proof