Comparison of nonlinear Granger causality extensions for low-dimensional systems

Abstract

The identification of hidden interdependences among the parts of a complex system is a fundamental issue. Typically, the objective is, given only a sequence of scalar measurements, to infer as much as possible about the internal dynamics of the system and about the interactions between its subsystems. In general, such interactions are not only nonlinear but also asymmetric. Constraints on the estimation of hidden relationships are further posed by noise and by the length of signals sampled from real world systems. The focus of this paper is causal dependences between bivariate time series. We especially focus on the nonlinear extension of Granger causality with polynomial terms of the conventional embedding vector. In this paper, we study the performance of this measure in comparison with three alternative methods proposed recently in low-dimensional and low-order-nonlinearity systems. Those methods are tested with three different artificial chaotic maps with several noise contamination setups. As a result, we find that the polynomial embedding technique successfully detects asymmetric (causal) dependences between bivariate time series in many low-dimensional cases.