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.

利用 Functional Linear Non-Gaussian Acyclic Model（Func-LiNGAM）扩展线性非高斯无环模型（LiNGAM）的概念，以应对涉及功能性磁共振成像和脑电图数据的大脑有效连接任务中的因果关系识别问题。在理论和实证角度上，我们对非高斯随机向量和无限维希尔伯特空间中甚至随机函数之间的因果关系可辨识性建立了理论保证。通过优化向量的坐标使用功能主成分分析，我们解决了内在无限维度功能性数据中离散时间点的稀疏性问题。在合成数据方面的实验结果验证了该框架利用观测样本能够识别多变量函数之间的因果关系。而在真实数据方面，我们专注于分析来自 fMRI 数据的脑连接模式。