Forecasting Time Series with VARMA Recursions on Graphs
For practitioners dealing with high-dimensional time series on graphs, this paper provides a principled way to reduce model complexity while maintaining performance, though it is an incremental extension of existing VARMA and graph signal processing ideas.
This work addresses the dimensionality curse in multivariate time series forecasting by leveraging graph structure under a time-vertex stationarity assumption. The proposed graph-based VARMA model reduces estimation to fitting uncorrelated univariate ARMA models in the spectral domain, achieving lower computational cost and competitive accuracy on synthetic and real data.
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.