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.
ABSTRACT
The problem of temporal localization and directional mapping of the dynamic interdependencies between parts of a complex system is addressed. We present a technique that weights the sampled values so as to minimize the mutual prediction error between pairs of measured signals. The reliability of the detected intermittent causal interactions is maximized by (a) smoothing the weight landscape through regularization, and (b) using a nonlinear (polynomial) variant of the conventional embedding vector. The effectiveness of the proposed technique is demonstrated by studying three numerical examples of dynamically coupled chaotic maps and by comparing it with two other measures of causal dependency.