Sackler Centre for Consciousness Science and School of Informatics, University of Sussex, Brighton BN1 9QJ, UK.
J Neurosci Methods. 2014 Feb 15;223:50-68. doi: 10.1016/j.jneumeth.2013.10.018. Epub 2013 Nov 5.
Wiener-Granger causality ("G-causality") is a statistical notion of causality applicable to time series data, whereby cause precedes, and helps predict, effect. It is defined in both time and frequency domains, and allows for the conditioning out of common causal influences. Originally developed in the context of econometric theory, it has since achieved broad application in the neurosciences and beyond. Prediction in the G-causality formalism is based on VAR (vector autoregressive) modelling.
The MVGC Matlab© Toolbox approach to G-causal inference is based on multiple equivalent representations of a VAR model by (i) regression parameters, (ii) the autocovariance sequence and (iii) the cross-power spectral density of the underlying process. It features a variety of algorithms for moving between these representations, enabling selection of the most suitable algorithms with regard to computational efficiency and numerical accuracy.
In this paper we explain the theoretical basis, computational strategy and application to empirical G-causal inference of the MVGC Toolbox. We also show via numerical simulations the advantages of our Toolbox over previous methods in terms of computational accuracy and statistical inference.
COMPARISON WITH EXISTING METHOD(S): The standard method of computing G-causality involves estimation of parameters for both a full and a nested (reduced) VAR model. The MVGC approach, by contrast, avoids explicit estimation of the reduced model, thus eliminating a source of estimation error and improving statistical power, and in addition facilitates fast and accurate estimation of the computationally awkward case of conditional G-causality in the frequency domain.
The MVGC Toolbox implements a flexible, powerful and efficient approach to G-causal inference.
Wiener-Granger 因果关系（“G 因果关系”）是一种适用于时间序列数据的因果关系统计概念，其中原因先于结果，并有助于预测结果。它在时域和频域都有定义，并允许排除共同因果影响。最初在计量经济学理论的背景下开发，此后已在神经科学等领域得到广泛应用。G 因果关系形式的预测基于 VAR（向量自回归）模型。
MVGC Matlab©工具箱方法用于 G 因果推断的方法是基于 VAR 模型的多种等效表示形式，包括 (i) 回归参数、(ii) 自协方差序列和 (iii) 基础过程的互功率谱密度。它具有多种在这些表示形式之间移动的算法，能够根据计算效率和数值精度选择最合适的算法。
本文解释了 MVGC 工具箱的理论基础、计算策略以及对经验 G 因果推断的应用。我们还通过数值模拟展示了我们的工具箱在计算精度和统计推断方面相对于以前的方法的优势。
计算 G 因果关系的标准方法涉及对完整和嵌套（简化）VAR 模型的参数进行估计。相比之下，MVGC 方法避免了简化模型的显式估计，从而消除了估计误差的来源，并提高了统计功效，此外还促进了在频域中计算困难的条件 G 因果关系的快速准确估计。
MVGC 工具箱实现了一种灵活、强大且高效的 G 因果推断方法。
