2026/04/08 by Bernd von Mallinckrodt · 1 voice
Environmental Science · Physics and Astronomy · Computer Science · #Ecosystem dynamics and resilience #Chaos control and synchronization #Anomaly Detection Techniques and Applications
paper · doi:10.5281/zenodo.19468810
openalex publication_date 2026/04/08 · openalex created_date 2026/04/09 · openalex updated_date 2026/07/01
This paper introduces Structural–Dynamic Separability (SDS) as a formal interpretability condition for composite early warning indicators (EWS) of the form T = R / Φ, where R denotes an AR(1)-based recovery rate and Φ the spectral-entropy effective rank of a rolling covariance matrix. Composite indicators that combine structural and dynamic components are widely used to detect critical transitions in complex systems. However, when these components are not statistically independent, the resulting index becomes an unidentified mixture, and its values cannot be uniquely attributed to structural change or dynamical slowing. SDS is proposed as a diagnostic gate that determines whether such a composite can be meaningfully interpreted. Using a graded coupling sweep in a bivariate stochastic system, I show that increasing inter-variable coupling leads to a monotonic collapse of structural dimensionality (Φ) and a corresponding inflation of the composite index (T), even in the absence of a genuine approach to a dynamical critical transition. This demonstrates that composite EWS can produce misleading signals when structural–dynamic separability is violated. A key finding is that naive SDS implementations based on unconditional correlation between components may fail in the presence of shared system drivers. I therefore introduce a conditional formulation of SDS, interpreted as a conditional identifiability criterion rather than a simple independence test. This reframing aligns SDS with established statistical principles and provides a practical path toward robust application, including residualization and surrogate-based testing. The results position SDS not as a performance enhancement, but as a validity condition for composite indicators. Without satisfying SDS, composite EWS values should not be interpreted as indicators of proximity to critical transitions, but rather as mixtures of structurally and dynamically confounded signals. Primary keywords: early warning signals critical transitions composite indicators interpretability identifiability Method / technical: spectral entropy effective rank AR(1) covariance structure structural–dynamic separability CRTI complex systems tipping points multivariate time series