Bayesian identification of early warning signals for long-range dependent climatic time series

Detecting early warning signals in climatic time series is essential for anticipating critical transitions and tipping points. Common statistical indicators include increased variance and lag-one autocorrelation prior to bifurcation points. However, these indicators are sensitive to observational noise, long-term mean trends, and long-memory dependence, all of which are prevalent in climatic time series. Such effects can easily obscure genuine signals or generate spurious detections. To address these challenges, we employ a flexible Bayesian framework for modelling time-varying autocorrelation in long-range dependent time series, also accounting for time-varying variance. The approach uses a mixture of two fractional Gaussian noise processes with a time-dependent weight function to represent fractional Gaussian noise with a time-varying Hurst exponent. Inference is performed via integrated nested Laplace approximation, enabling joint estimation of mean trends and handling of irregularly sampled observations. The strengths and limitations of detecting changes in the autocorrelation is investigated in extensive simulations. Applied to real climatic data sets, we find evidence of early warning signals in a reconstructed Atlantic multidecadal variability index, while dismissing such signals for paleoclimate records spanning the Dansgaard-Oeschger events.

💡 Research Summary

The paper tackles a central problem in climate risk assessment: how to reliably detect early‑warning signals (EWS) of critical transitions in time series that exhibit long‑range dependence, observational noise, irregular sampling, and non‑stationary variance. Classical EWS—rising variance and lag‑1 autocorrelation—are known to be highly sensitive to these complications, often producing false positives or missing genuine signals. To overcome these limitations, the authors develop a flexible Bayesian hierarchical model that directly incorporates time‑varying autocorrelation and variance within a single inference framework.

Model construction
The core idea is to represent a long‑memory process with a time‑varying Hurst exponent H(t) by mixing two independent fractional Gaussian noise (fGn) processes, x₁(t) and x₂(t), with fixed Hurst exponents H₁ and H₂. A deterministic weight function w(t) (linear in the paper) interpolates between the two components:

ε(t) = τ⁻¹ᐟ²