Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Beyond Gaussian Assumptions: A Nonlinear Generalization of Linear Inverse Modeling

Justin Lien, Hiroyasu Ando

TL;DR

This work extends the Linear Inverse Model framework by introducing White-nLIM and Colored-nLIM, nonlinear stochastic models that incorporate quadratic terms and constant offsets to better capture non-Gaussian and skewed dynamics observed in real systems. The approach links dynamics to higher-order correlations (K, M, S) and their derivatives, using fluctuation-dissipation relations that generalize LIM to include both white and OU colored noise, with stationarity considerations and relationships to traditional LIM. Validation on synthetic 2D and chaotic Lorenz-63 systems shows robustness and limits of the method under sampling and conditioning, while application to ENSO (El Niño 3.4) data demonstrates improved fit to observed correlation structures and skewed distributions compared with classical LIM. Overall, nLIM offers a principled, data-driven way to capture nonlinear, memory-bearing stochastic dynamics in geophysical time series, with implications for improved representation and forecasting of complex climate variability.

Abstract

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.

Beyond Gaussian Assumptions: A Nonlinear Generalization of Linear Inverse Modeling

TL;DR

This work extends the Linear Inverse Model framework by introducing White-nLIM and Colored-nLIM, nonlinear stochastic models that incorporate quadratic terms and constant offsets to better capture non-Gaussian and skewed dynamics observed in real systems. The approach links dynamics to higher-order correlations (K, M, S) and their derivatives, using fluctuation-dissipation relations that generalize LIM to include both white and OU colored noise, with stationarity considerations and relationships to traditional LIM. Validation on synthetic 2D and chaotic Lorenz-63 systems shows robustness and limits of the method under sampling and conditioning, while application to ENSO (El Niño 3.4) data demonstrates improved fit to observed correlation structures and skewed distributions compared with classical LIM. Overall, nLIM offers a principled, data-driven way to capture nonlinear, memory-bearing stochastic dynamics in geophysical time series, with implications for improved representation and forecasting of complex climate variability.

Abstract

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.

Paper Structure

This paper contains 15 sections, 42 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Review of Linear Inverse Modelings
  3. White-LIM
  4. Colored-LIM
  5. Mathematical Foundations of Non-Linear Inverse Modelings
  6. White-nLIM
  7. Colored-nLIM
  8. Stationarity
  9. Relationships among models
  10. Validation
  11. A 2-dimensional example
  12. Stochastic Lorenz 63 system
  13. Modeling El Nino 3.4 index using LIMs
  14. Concluding Remarks
  15. Proof of nLIMs

Figures (2)

  • Figure 1: The entries of the observed correlation function ${K^{\text{obs}}}$ and ${M^{\text{obs}}}$ of the preprocessed SST and D20 anomalies symbolized as 1, and 2, respectively, along with the correlation functions determined by White-LIM, Colored-LIM, White-nLIM, and Colored-nLIM. For LIMs, the correlation functions are obtained from Eqs. (\ref{['Eq:ST-K']}) and (\ref{['Eq:ST-Colored-K']}), while for nLIMs, they are determined by a 1000-year realization of the approximate systems.
  • Figure 2: The observed distributions of preprocessed SST and D20, along with the probability distributions of the approximate systems determined by White-LIM, Colored-LIM, White-nLIM, and Colored-nLIM. White-LIM and Colored-LIM produce Gaussian distributions whose covariances are the diagonal entries of ${K^{\text{obs}}}(0)$ and they are indistinguishable. For White-nLIM and Colored-nLIM, the distributions are determined by a 1000-year realization of the approximate systems.