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.

中文翻译：

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高维时间序列中格兰杰因果关系估计的多项式回归模型降维


多元时间序列的因果分析和复杂网络的形成依赖于对从一个观察变量到另一个观察变量的直接因果关系的估计，并考虑到其他观察变量的存在。这种影响通过条件格兰杰因果关系指数 (CGCI) 进行量化，该指数由线性向量自回归模型 (VAR) 导出。已经开发出降维方法来限制 VAR 模型，例如给出受限 CGCI (rCGCI) 的改进的时间向后选择方法 (mBTS)。 rCGCI 仅限于线性系统。在本研究中，我们将 mBTS 降维方案扩展到多项式 VAR，以便对非线性动力学和因果关系进行建模，并导出称为限制多项式 CGCI (rpCGCI) 的格兰杰因果关系测度。解决了 mBTS 适应受限多项式 VAR 和格兰杰因果关系指数构造的复杂性，还涉及 mBTS 步骤中的随机化显着性检验。 rpCGCI 相对于其他非线性格兰杰因果关系度量的优势在于它可以应用于高维度的短时间序列（许多观测变量）。对不同类型的多元随机过程和不同长度的生成时间序列的模拟研究表明，与 CGCI、rCGCI、pCGCI（使用多项式 VAR）和其他非线性因果度量相比，所提出的 rpCGCI 具有优越性。此外，rpCGCI 与心率变异性、呼吸和血液中氧浓度信号以及包含癫痫样放电的癫痫患者的多通道头皮脑电图 (EEG) 记录的其他测量指标进行比较。