arXiv subjects

Justin Lien

Publications and source records attributed to Justin Lien.

5 recordsLinked to original sources

arXiv2025

Beyond Gaussian Assumptions: A Nonlinear Generalization of Linear Inverse Modeling

The Linear Inverse Model (LIM) is a class of data-driven methods that construct approximate linear stochastic models to represent complex observational data. The stochastic forcing can be modeled using either Gaussian white noise or Ornstein-Uhlenbeck colored noise; the corresponding models are called White-LIM and Colored-LIM, respectively. Although LIMs are widely applied in climate sciences, they inherently approximate observed distributions as Gaussian, limiting their ability to capture asymmetries. In this study, we extend LIMs to incorporate nonlinear dynamics, introducing White-nLIM and Colored-nLIM which allow for a more flexible and accurate representation of complex dynamics from observations. The proposed methods not only account for the nonlinear nature of the underlying system but also effectively capture the skewness of the observed distribution. Moreover, we apply these methods to a lower-dimensional representation of ENSO and demonstrate that both White-nLIM and Colored-nLIM successfully capture its nonlinear characteristic.

math.NA↗

arXiv2024

A Liang-Kleeman Causality Analysis based on Linear Inverse Modeling

Causality analysis is a powerful tool for determining cause-and-effect relationships between variables in a system by quantifying the influence of one variable on another. Despite significant advancements in the field, many existing studies are constrained by their focus on unidirectional causality or Gaussian external forcing, limiting their applicability to complex real-world problems. This study proposes a novel data-driven approach to causality analysis for complex stochastic differential systems, integrating the concepts of Liang-Kleeman information flow and linear inverse modeling. Our method models environmental noise as either memoryless Gaussian white noise or memory-retaining Ornstein-Uhlenbeck colored noise, and allows for self and mutual causality, providing a more realistic representation and interpretation of the underlying system. Moreover, this LIM-based approach can identify the individual contribution of dynamics and correlation to causality. We apply this approach to re-examine the causal relationships between the El Ni\~{n}o-Southern Oscillation (ENSO) and the Indian Ocean Dipole (IOD), two major climate phenomena that significantly influence global climate patterns. In general, regardless of the type of noise used, the causality between ENSO and IOD is mutual but asymmetric, with the causality map reflecting an ENSO-like pattern consistent with previous studies. Notably, in the case of colored noise, the noise memory map reveals a hotspot in the Ni\~no 3 region, which is further related to the information flow. This suggests that our approach offers a more comprehensive framework and provides deeper insights into the causal inference of global climate systems.

math.NA↗

arXiv2024

On the Cyclostationary Linear Inverse Models: A Mathematical Insight and Implication

Cyclostationary linear inverse models (CS-LIMs), generalized versions of the classical (stationary) LIM, are advanced data-driven techniques for extracting the first-order time-dependent dynamics and random forcing relevant information from complex non-linear stochastic processes. Though CS-LIMs lead to a breakthrough in climate sciences, their mathematical background and properties are worth further exploration. This study focuses on the mathematical perspective of CS-LIMs and introduces two variants: e-CS-LIM and l-CS-LIM. The former refines the original CS-LIM using the interval-wise linear Markov approximation, while the latter serves as an analytic inverse model for the linear periodic stochastic systems. Although relying on approximation, e-CS-LIM converges to l-CS-LIM under specific conditions and shows noise-robust performance. Numerical experiments demonstrate that each CS-LIM reveals the temporal structure of the system. The e-CS-LIM optimizes the original model for better dynamics performance, while l-CS-LIM excels in diffusion estimation due to reduced approximation reliance. Moreover, CS-LIMs are applied to real-world ENSO data, yielding a consistent result aligning with observations and current ENSO understanding.

math.NA↗

arXiv2024

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

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.

math.NA↗

arXiv2023

Hypergraph Echo State Network

A hypergraph as a generalization of graphs records higher-order interactions among nodes, yields a more flexible network model, and allows non-linear features for a group of nodes. In this article, we propose a hypergraph echo state network (HypergraphESN) as a generalization of graph echo state network (GraphESN) designed for efficient processing of hypergraph-structured data, derive convergence conditions for the algorithm, and discuss its versatility in comparison to GraphESN. The numerical experiments on the binary classification tasks demonstrate that HypergraphESN exhibits comparable or superior accuracy performance to GraphESN for hypergraph-structured data, and accuracy increases if more higher-order interactions in a network are identified.

cs.LG↗