Multivariate Time Series Forecasting With GARCH Models on Graphs

Data that house topological information is manifested as relationships between multiple variables via a graph formulation. Various methods have been developed for analyzing time series on the nodes of graphs but research works on graph signals with volatility are limited. In this article, we propose a graph framework of multivariate Generalized Autoregressive Conditional Heteroscedasticity (GARCH) models from the spectral perspective with the Laplacian matrix. We introduce three graphical GARCH models: one symmetric Graph GARCH model and two asymmetric models namely Graph Exponential GARCH and Graph GJR-GARCH. Assuming that graph signals and their residuals are graph stationary, this framework can decompose the multivariate GARCH models into a linear combination of several univariate GARCH processes in the graph spectral domain. Moreover, it is possible to reduce the number of parameters with the graph topology information and further reduce the estimation cost by utilizing the principal components of the graph signal in the frequency domain. These proposed models are tested on synthetic data and on two real applications for weather prediction and wind power forecasting. With the data and GARCH model residuals being graph stationary, the experiment results demonstrate that these three graphical models can make multi-step predictions more accurately than non-graph GARCH models and Graph Vector Autoregressive Moving Average model.