Forecasting Time Series With VARMA Recursions on Graphs

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 paper 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., time-varying 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 vector autoregressive and vector autoregressive moving average 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 the multivariate model estimation reduces to that of fitting a number of uncorrelated univariate autoregressive-moving-average models and 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 he 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.