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, we explicitly define a measure of functional connectivity using permutations of the covariate set. Artificial neural networks serve as featurizers of the data to approximate any arbitrary, nonlinear relationship, and under certain conditions on the featurization process and the model residuals, we prove consistent estimation of the variance for each permutation. Performance of the permutation method is compared to penalized objective, naive replacement, and omission techniques via simulation, and we investigate its application to neuronal responses of acoustic stimuli in the auditory cortex of anesthetized rats. We contend that 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种颇具争议但广泛应用的方法，涵盖从经济学到神经科学等多个领域。其原始定义通过建立基于特定模型的条件函数依赖关系，阐释时间序列中的因果概念。将格兰杰因果关系扩展到非线性数据仍面临挑战，许多方法采用不包含样本外可预测性的样本内检验，导致模型过拟合问题。为实现样本外比较，我们通过协变量集的排列显式定义了一种功能连接性度量。人工神经网络作为数据的特征提取器，近似任意非线性关系，在特征化过程及模型残差的特定条件下，我们证明了每个排列方差的相合估计。通过仿真实验，将该排列方法与惩罚目标、简单替换及遗漏技术进行性能比较，并探究其在麻醉大鼠听觉皮层声刺激神经元反应中的应用。我们认为，当数据集中因果机制的先验知识有限时，针对性使用格兰杰因果框架有助于揭示变量集之间值得进一步研究的潜在预测关系。