Dimension Reduction of Polynomial Regression Models for the Estimation of Granger Causality in High-Dimensional Time Series

The study of complex systems has gained great interest recently due to the abundance of big data and the advances of the methodology for their analysis. For complex dynamical systems, the observed variables can be large resulting in high-dimensional time series. To make sense of the dependencies among the observed variables and form complex networks, Granger causality has been the pioneering concept from which many inter-dependence (connectivity) measures have been originated. Particularly, the conditional Granger causality index (CGCI) is derived by vector autoregressive models (VAR) to estimate the direct cause-effect from one variable to another accounting for the presence of other observed variables. However, when there are many observed time series of relatively small length the estimation of VAR may not be stable. For this, in our recent study [1] we proposed a dimension reduction approach to restrict VAR models for the estimation of CGCI (rCGCI), using the backward-in-time selection method (BTS) [2]. The rCGCI was found to estimate accurately the direct cause-effect relationships even in high-dimensional linear stochastic processes, as compared to the classical VAR models and restricted VAR models from other dimension reduction techniques such as LASSO. Nevertheless, the rCGCI is limited to linear systems.
In this study, we extend the BTS dimension reduction scheme to polynomial VAR, so as to model nonlinear dynamics and cause-effect relationships and derive the Granger causality measure termed rpCGCI. The complexity and computational cost of the new setting is increased, but it is still manageable, and most importantly it can be applied to short time series from high-dimensional complex systems. A simulation study is conducted to evaluate the rpCGCI and compare it to other nonlinear Granger causality measures, using nonlinear multivariate dynamical systems and stochastic processes and different lengths of generated time series.