Granger causal inference is a contentious but widespread method used in fields ranging from economics to neuroscience. The original definition addresses the notion of causality in time series by establishing functional dependence conditional on a specified model. Adaptation of Granger causality to nonlinear data remains challenging, and many methods apply in-sample tests that do not incorporate out-of-sample predictability, leading to concerns of model overfitting. To allow for out-of-sample comparison, a measure of functional connectivity is explicitly defined using permutations of the covariate set. Artificial neural networks serve as featurizers of the data to approximate any arbitrary, nonlinear relationship, and consistent estimation of the variance for each permutation is shown under certain conditions on the featurization process and the model residuals. Performance of the permutation method is compared to penalized variable selection, naive replacement, and omission techniques via simulation, and it is applied to neuronal responses of acoustic stimuli in the auditory cortex of anesthetized rats. Targeted use of the Granger causal framework, when prior knowledge of the causal mechanisms in a dataset are limited, can help to reveal potential predictive relationships between sets of variables that warrant further study.

翻译：格兰杰因果推断是一个具有争议但在经济学到神经科学等多个领域被广泛使用的方法。原始定义通过建立依赖于指定模型的功能依赖关系来阐述时间序列中的因果概念。将格兰杰因果关系适配到非线性数据仍具挑战性，许多方法采用不纳入样本外可预测性的样本内检验，导致模型过拟合的担忧。为实现样本外比较，我们利用协变量集的置换显式定义了功能连通性度量。人工神经网络作为数据特征提取器以逼近任意非线性关系，并在特征化过程和模型残差的特定条件下证明了每个置换方差的相合估计。通过仿真将该置换方法与惩罚变量选择、朴素替换及遗漏技术进行比较，并将其应用于麻醉大鼠听觉皮层中声音刺激的神经元响应。当对数据集中因果机制的先验知识有限时，有针对性地使用格兰杰因果框架有助于揭示变量集合间值得进一步研究的潜在预测关系。