Bernd von Mallinckrodt 🦋Apr 8
What if many composite early warning signals don’t add information at all … because their components never truly separate? #Structural–DynamicSeparability ( #SDS) turns this into a testable condition within #CRTI. doi.org/10.5281/zeno... 🖖

Structural–Dynamic Separabilit...
Structural–Dynamic Separability as an Interpretability Condition for Composite Early Warning Indicators

This work introduces a methodological framework for interpreting composite early warning indicators (EWS) in complex dynamical systems. While classical EWS such as variance and autocorrelation are well established for detecting fold-type critical transitions, composite indicators combining structural and dynamic components have lacked a formal criterion for interpretability.   I formalize this problem as one of identifiability and introduce the Structural–Dynamic Separability (SDS) condition as a necessary criterion for interpreting composite indicators as joint diagnostics. SDS requires that structural and dynamic components vary sufficiently independently over the observation window; when this condition is violated, composite indicators collapse to functions of a single dominant component and do not provide additional information.   As a concrete instantiation, I define the Compression–Response Transition Index (CRTI), combining spectral compression of the rolling covariance matrix—operationalized via effective rank—with a recovery rate estimate derived from autoregressive dynamics. The commonly used ratio form is presented as a baseline within a parameterized family of composite indicators rather than as a canonical formulation.   The framework is explicitly scoped to fold bifurcations in low-to-moderate dimensional systems under approximately stationary noise. Four boundary conditions are characterized: isotropic noise regimes, variance-driven transitions, high-dimensional estimation bias, and anisotropic noise with time-varying structure. Synthetic validation demonstrates that the composite indicator provides additional diagnostic value relative to individual components when and only when the SDS condition is satisfied. Application to ecological time series illustrates how SDS partitions observation windows into interpretable and non-interpretable regimes without discarding data.   The central contribution is the formalization of interpretability conditions for composite early warning indicators. The SDS criterion is independent of the specific choice of structural or dynamic components and can, in principle, be applied to a broad class of composite diagnostics in complex systems.     early warning signals, critical transitions, fold bifurcation, critical slowing down, composite indicators, identifiability, structural-dynamic separability, spectral entropy, effective rank, covariance matrix, autoregressive processes, AR(1), complex systems, nonlinear dynamics, random matrix theory, ecological transitions, tipping points

Zenodo
Bernd von Mallinckrodt 🦋Apr 8
When do composite early warning signals #EWS actually tell us something new … rather than just repackage the same signal twice? I introduce #Structural–DynamicSeparability ( #SDS) as an identifiability condition and test it within the #CRTI framework … doi.org/10.5281/zeno... 🖖

Structural–Dynamic Separabilit...
Structural–Dynamic Separability as an Interpretability Condition for Composite Early Warning Indicators

This work introduces a methodological framework for interpreting composite early warning indicators (EWS) in complex dynamical systems. While classical EWS such as variance and autocorrelation are well established for detecting fold-type critical transitions, composite indicators combining structural and dynamic components have lacked a formal criterion for interpretability.   I formalize this problem as one of identifiability and introduce the Structural–Dynamic Separability (SDS) condition as a necessary criterion for interpreting composite indicators as joint diagnostics. SDS requires that structural and dynamic components vary sufficiently independently over the observation window; when this condition is violated, composite indicators collapse to functions of a single dominant component and do not provide additional information.   As a concrete instantiation, I define the Compression–Response Transition Index (CRTI), combining spectral compression of the rolling covariance matrix—operationalized via effective rank—with a recovery rate estimate derived from autoregressive dynamics. The commonly used ratio form is presented as a baseline within a parameterized family of composite indicators rather than as a canonical formulation.   The framework is explicitly scoped to fold bifurcations in low-to-moderate dimensional systems under approximately stationary noise. Four boundary conditions are characterized: isotropic noise regimes, variance-driven transitions, high-dimensional estimation bias, and anisotropic noise with time-varying structure. Synthetic validation demonstrates that the composite indicator provides additional diagnostic value relative to individual components when and only when the SDS condition is satisfied. Application to ecological time series illustrates how SDS partitions observation windows into interpretable and non-interpretable regimes without discarding data.   The central contribution is the formalization of interpretability conditions for composite early warning indicators. The SDS criterion is independent of the specific choice of structural or dynamic components and can, in principle, be applied to a broad class of composite diagnostics in complex systems.     early warning signals, critical transitions, fold bifurcation, critical slowing down, composite indicators, identifiability, structural-dynamic separability, spectral entropy, effective rank, covariance matrix, autoregressive processes, AR(1), complex systems, nonlinear dynamics, random matrix theory, ecological transitions, tipping points

Zenodo