Role-Specific Predictive Geometries for Nonstationary Multivariate Graph-Signal Forecasting

Forecasting multivariate graph signals is challenging when node-level trajectories are nonstationary but stable relations persist across nodes and features. In an error-correction representation, long-run equilibrium restoration and short-run transient propagation represent different predictive roles and need not share a common cross-feature geometry. We introduce role-specific predictive geometries in which directed Long relations act on estimated equilibrium coordinates, whereas directed Short relations act on lagged differences. Matrix-valued Long responses mix equilibrium coordinates before graph propagation, while Short responses use graph-filtered transient designs; a direct multi-horizon estimator couples forecast corrections across adjacent horizons. Temporal cross-fitting and Frisch-Waugh-Lovell partialling-out give selected edges a conditional predictive interpretation relative to a graph-temporal backbone. The Long operator remains right-factorized through the equilibrium subspace and therefore annihilates source common-trend directions. Controlled experiments recover all planted Long relations (20/20), all planted Short relations (20/20), and both role families in every Dual realization (10/10). Across four real-world benchmarks, the proposed predictor improves on the G-VARMA backbone in three datasets, with all 25 fold-horizon comparisons favorable on the five-fold financial benchmark.