# Forecasting Time Series with VARMA Recursions on Graphs

arXiv: https://arxiv.org/abs/1810.08581 · HTML: https://mlsanity.com/paper/1810.08581/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Oct 2018.

## Built on by (most cited first, 27 of 27)

- [Spectral Temporal Graph Neural Network for Multivariate Time-series Forecasting](https://mlsanity.com/paper/2103.07719/) (2021)
- [Taming Local Effects in Graph-based Spatiotemporal Forecasting](https://mlsanity.com/paper/2302.04071/) (2023)
- [Graph Deep Learning for Time Series Forecasting](https://mlsanity.com/paper/2310.15978/) (2023)
- [Sparse Graph Learning from Spatiotemporal Time Series](https://mlsanity.com/paper/2205.13492/) (2022)
- [Synthesizing Decentralized Controllers with Graph Neural Networks and Imitation Learning](https://mlsanity.com/paper/2012.14906/) (2020)
- [Graph-Time Convolutional Neural Networks: Architecture and Theoretical Analysis](https://mlsanity.com/paper/2206.15174/) (2022)
- [GSP-KalmanNet: Tracking Graph Signals via Neural-Aided Kalman Filtering](https://mlsanity.com/paper/2311.16602/) (2023)
- [Spatiotemporal Covariance Neural Networks](https://mlsanity.com/paper/2409.10068/) (2024)
- [Graph-Time Convolutional Neural Networks](https://mlsanity.com/paper/2103.01730/) (2021)
- [Continuous Product Graph Neural Networks](https://mlsanity.com/paper/2405.18877/) (2024)
- [Joint Estimation and Prediction of City-wide Delivery Demand: A Large Language Model Empowered Graph-based Learning Approach](https://mlsanity.com/paper/2408.17258/) (2024)
- [Graph filtering over expanding graphs](https://mlsanity.com/paper/2203.08058/) (2022)
- [AZ-whiteness test: a test for uncorrelated noise on spatio-temporal graphs](https://mlsanity.com/paper/2204.11135/) (2022)
- [Graph Kalman Filters](https://mlsanity.com/paper/2303.12021/) (2023)
- [Structural Knowledge Informed Continual Multivariate Time Series Forecasting](https://mlsanity.com/paper/2402.12722/) (2024)
- [A Robust and Efficient Multi-Scale Seasonal-Trend Decomposition](https://mlsanity.com/paper/2109.08800/) (2021)
- [LLM Online Spatial-temporal Signal Reconstruction Under Noise](https://mlsanity.com/paper/2411.15764/) (2024)
- [Guaranteed Multidimensional Time Series Prediction via Deterministic Tensor Completion Theory](https://mlsanity.com/paper/2501.15388/) (2025)
- [GKNet: Graph Kalman Filtering and Model Inference via Model-based Deep Learning](https://mlsanity.com/paper/2506.22004/) (2025)
- [Learning Graph ARMA Processes from Time-Vertex Spectra](https://mlsanity.com/paper/2302.06887/) (2023)
- [Spatiotemporal Forecasting Meets Efficiency: Causal Graph Process Neural Networks](https://mlsanity.com/paper/2405.18879/) (2024)
- [Online Graph Topology Learning via Time-Vertex Adaptive Filters: From Theory to Cardiac Fibrillation](https://mlsanity.com/paper/2411.01567/) (2024)
- [Statistical Testing on Directed Graphs by Surrogate Data Generation](https://mlsanity.com/paper/2606.00758/) (2026)
- [Forecasting Graph Signals with Recursive MIMO Graph Filters](https://mlsanity.com/paper/2210.15258/) (2022)
- [Locally Stationary Graph Processes](https://mlsanity.com/paper/2309.01657/) (2023)
- [Second-Order Tensorial Partial Differential Equations on Graphs](https://mlsanity.com/paper/2509.02015/) (2025)
- [Role-Specific Predictive Geometries for Nonstationary Multivariate Graph-Signal Forecasting](https://mlsanity.com/paper/2609.06519/) (2026)

## Abstract (arXiv)

Graph-based techniques emerged as a choice to deal with the dimensionality issues in modeling multivariate time series. However, there is yet no complete understanding of how the underlying structure could be exploited to ease this task. This work provides contributions in this direction by considering the forecasting of a process evolving over a graph. We make use of the (approximate) time-vertex stationarity assumption, i.e., timevarying graph signals whose first and second order statistical moments are invariant over time and correlated to a known graph topology. The latter is combined with VAR and VARMA models to tackle the dimensionality issues present in predicting the temporal evolution of multivariate time series. We find out that by projecting the data to the graph spectral domain: (i) the multivariate model estimation reduces to that of fitting a number of uncorrelated univariate ARMA models and (ii) an optimal low-rank data representation can be exploited so as to further reduce the estimation costs. In the case that the multivariate process can be observed at a subset of nodes, the proposed models extend naturally to Kalman filtering on graphs allowing for optimal tracking. Numerical experiments with both synthetic and real data validate the proposed approach and highlight its benefits over state-of-the-art alternatives.