Granger causality (GC) is undoubtedly the most widely used method to infer cause-effect relations from observational time series. Several nonlinear alternatives to GC have been proposed based on kernel methods. We generalize kernel Granger causality by considering the variables' cross-relations explicitly in Hilbert spaces. The framework is shown to generalize the linear and kernel GC methods and comes with tighter bounds of performance based on Rademacher complexity. We successfully evaluate its performance in standard dynamical systems, as well as to identify the arrow of time in coupled Rössler systems, and it is exploited to disclose the El Niño–Southern Oscillation phenomenon footprints on soil moisture globally.

中文翻译：

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内核希尔伯特空间中的明确格兰杰因果关系

格兰杰因果关系（GC）无疑是从观测时间序列推断因果关系的最广泛使用的方法。基于核方法已提出了几种非线性的GC替代方法。我们通过在希尔伯特空间中明确考虑变量的交叉关系来概括内核格兰杰因果关系。该框架已被证明可以推广线性和内核GC方法，并且基于Rademacher复杂性具有更严格的性能界限。我们成功地评估了其在标准动力系统中的性能，并确定了耦合的Rössler系统中的时间箭头，并利用它来揭示全球土壤湿度上的厄尔尼诺-南方涛动现象足迹。