Surrogate-Referenced Directionality in Granger-Type Coupling Estimates: A Conditional q-Generalized Index Evaluated on Chaotic Benchmarks and Cardiorespiratory Data

Directional coupling between time series is summarized by an asymmetry index built from Granger-type measures. We show that such an index can be strongly biased by the pair of signals without any coupling and propose to reference it to a surrogate null obtained by cyclically shifting the putative driver. We evaluate two estimators, classical Granger causality on signal levels, and a conditional, q-generalized Granger index on scale-k increments, which reduces to ordinary Granger causality on increments at q=2. On unidirectionally coupled Lorenz–Rössler systems simulated in both driving directions, the uncalibrated index of the conditional estimator was +0.54 and −0.62 without coupling, pointing from Lorenz to Rössler in both configurations; after calibration, false-detection rates were compatible with the nominal level. A control crossing estimator type with signal representation showed that the complementarity of the estimators separates single-scale estimators on levels from scale-k estimators, not classical from conditional ones. In respiration and RR-interval recordings from the Fantasia database (15 young, 16 elderly), all estimators showed stronger respiration→RR coupling in the young cohort (for the conditional index at q≤4); its profile across q was nearly a uniform factor, so the q-generalization added no detectable information, and the difference was present in all scale ranges for 1≤q≤3. Elderly records, however, contained a higher fraction of corrected beats (median 6.5% versus 0.9%), a difference in data quality that the present analyses do not separate from age. Conclusions about directionality depended on the RR time-stamp convention. We recommend surrogate-referenced reporting of directionality indices and explicit convention-sensitivity checks.