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

arXiv: https://arxiv.org/abs/2609.06519 · HTML: https://mlsanity.com/paper/2609.06519/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 27 references that are in the map.
- First posted on arXiv 6 Sep 2026.

## Builds on (most cited first, 16 of 16)

- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [Learning graphs from data: A signal representation perspective](https://mlsanity.com/paper/1806.00848/) (2018)
- [Connecting the Dots: Identifying Network Structure via Graph Signal Processing](https://mlsanity.com/paper/1810.13066/) (2018)
- [TimeFilter: Patch-Specific Spatial-Temporal Graph Filtration for Time Series Forecasting](https://mlsanity.com/paper/2501.13041/) (2025)
- [Network Topology Inference from Spectral Templates](https://mlsanity.com/paper/1608.03008/) (2016)
- [TimeBridge: Non-Stationarity Matters for Long-term Time Series Forecasting](https://mlsanity.com/paper/2410.04442/) (2024)
- [Stationary Graph Processes and Spectral Estimation](https://mlsanity.com/paper/1603.04667/) (2016)
- [Stationary signal processing on graphs](https://mlsanity.com/paper/1601.02522/) (2016)
- [Signal Processing on Graphs: Causal Modeling of Unstructured Data](https://mlsanity.com/paper/1503.00173/) (2015)
- [Forecasting Time Series with VARMA Recursions on Graphs](https://mlsanity.com/paper/1810.08581/) (2018)
- [A Time-Vertex Signal Processing Framework](https://mlsanity.com/paper/1705.02307/) (2017)
- [Online Topology Identification from Vector Autoregressive Time Series](https://mlsanity.com/paper/1904.01864/) (2019)
- [Stationary time-vertex signal processing](https://mlsanity.com/paper/1611.00255/) (2016)
- [Learning product graphs from multidomain signals](https://mlsanity.com/paper/1911.07411/) (2019)
- [Forecasting Multi-Dimensional Processes over Graphs](https://mlsanity.com/paper/2004.08260/) (2020)
- [Spatiotemporal Forecasting Meets Efficiency: Causal Graph Process Neural Networks](https://mlsanity.com/paper/2405.18879/) (2024)

## Abstract (arXiv)

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.