In causal discovery, non-Gaussianity has been used to characterize the complete configuration of a Linear Non-Gaussian Acyclic Model (LiNGAM), encompassing both the causal ordering of variables and their respective connection strengths. However, LiNGAM can only deal with the finite-dimensional case. To expand this concept, we extend the notion of variables to encompass vectors and even functions, leading to the Functional Linear Non-Gaussian Acyclic Model (Func-LiNGAM). Our motivation stems from the desire to identify causal relationships in brain-effective connectivity tasks involving, for example, fMRI and EEG datasets. We demonstrate why the original LiNGAM fails to handle these inherently infinite-dimensional datasets and explain the availability of functional data analysis from both empirical and theoretical perspectives. {We establish theoretical guarantees of the identifiability of the causal relationship among non-Gaussian random vectors and even random functions in infinite-dimensional Hilbert spaces.} To address the issue of sparsity in discrete time points within intrinsic infinite-dimensional functional data, we propose optimizing the coordinates of the vectors using functional principal component analysis. Experimental results on synthetic data verify the ability of the proposed framework to identify causal relationships among multivariate functions using the observed samples. For real data, we focus on analyzing the brain connectivity patterns derived from fMRI data.

翻译：在因果发现中，非高斯性已被用于刻画线性非高斯无环模型（LiNGAM）的完整配置，包括变量的因果顺序及其各自的连接强度。然而，LiNGAM仅能处理有限维情形。为拓展这一概念，我们将变量的定义范围扩展至向量乃至函数，从而提出函数线性非高斯无环模型（Func-LiNGAM）。我们的研究动机源于在脑有效连接任务（例如涉及fMRI和EEG数据集）中识别因果关系的需求。我们论证了原始LiNGAM为何无法处理这些本质上的无限维数据集，并从经验与理论两个角度阐释了函数型数据分析的可行性。{我们建立了在无限维希尔伯特空间中，非高斯随机向量乃至随机函数间因果关系可辨识性的理论保证。}为应对本质无限维函数型数据中离散时间点的稀疏性问题，我们提出利用函数型主成分分析法优化向量坐标。合成数据实验结果验证了所提出框架能够通过观测样本识别多元函数间的因果关系。在真实数据方面，我们重点分析了基于fMRI数据导出的脑连接模式。