Eigenvector rotation precedes eigenvalue-based early-warning signals: a TVP-Kalman approach to detecting critical transitions

Abstract

Early-warning signals (EWS) for critical transitions are predominantly based on changes in the dominant eigenvalue of the system's Jacobian-rising variance and lag-1 autocorrelation (AR(1)). However, eigenvalue-based EWS have $O(delta theta^2)$ sensitivity to perturbations, limiting their lead time. We introduce a complementary EWS based on eigenvector rotation, measured by the time-varying elasticity $beta(t) = d log y / d log x$ estimated via a TVP-Kalman filter in log-log space. Since eigenvector sensitivity is $O(delta theta)$, $beta$ is predicted to precede eigenvalue-based signals. We test this hypothesis on 24 years of monthly NASA AIRS data (2002--2026, 284 observations) across three climatically distinct regions (Arctic 65-90N, Tropics 10S-10N, Indian Monsoon), using temperature ($T$) and specific humidity ($q$) as the coupled variables. $beta$ is orthogonal to AR(1) in all regions (Pearson $r approx 0$, n.s.), confirming the distinct information content. Systematic lead-lag analysis reveals that $beta$ precedes AR(1) by 14--24 months, consistent with the $O(delta theta) > O(delta theta^2)$ mechanism. Six simulated systems with known tipping points (Stommel AMOC model, fold bifurcation, logistic map, critical slowing down) further validate that $beta$ leads AR(1) by 39-153 timesteps when the transition involves coupling degradation. The dimensionless nature of $beta$ (scale-free log-log exponent) suggests it may serve as a universal, cross-system EWS, analogous to scaling exponents in critical phenomena.