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

Abstract

The causality analysis of multivariate time series and formation of complex networks relies on the estimation of the direct cause-effect from one observed variable to another accounting for the presence of other observed variables. Such effects are quantified by the conditional Granger causality index (CGCI), derived by linear vector autoregressive models (VAR). Dimension reduction approaches have been developed to restrict VAR models, such as the modified backward-in-time selection method (mBTS) giving the restricted CGCI (rCGCI). The rCGCI is limited to linear systems. In this study, we extend the mBTS dimension reduction scheme to polynomial VAR, so as to model nonlinear dynamics and cause-effect relationships and derive the Granger causality measure termed restricted polynomial CGCI (rpCGCI). The complications in the adaptation of mBTS to the restricted polynomial VAR and the construction of the Granger causality index are addressed, involving also a randomization significance test in the steps of mBTS. The rpCGCI has the advantage against other nonlinear Granger causality measures that it can be applied to short time series of high dimension (many observed variables). The simulation study on different types of multivariate stochastic processes and different lengths of generated time series showed the superiority of the proposed rpCGCI as compared to CGCI, rCGCI, pCGCI (using polynomial VAR) and other nonlinear causality measures. Further, rpCGCI was compared favorably to the other measures on signals of heart rate variability, respiration, and oxygen concentration in the blood, as well as multi-channel scalp electroencephalogram (EEG) recordings of epileptic patients containing epileptiform discharges.