Functional graphical models have undergone extensive development during the recent years, leading to a variety models such as the functional Gaussian graphical model, the functional copula Gaussian graphical model, the functional Bayesian graphical model, the nonparametric functional additive graphical model, and the conditional functional graphical model. These models rely either on some parametric form of distributions on random functions, or on additive conditional independence, a criterion that is different from probabilistic conditional independence. In this paper we introduce a nonparametric functional graphical model based on functional sufficient dimension reduction. Our method not only relaxes the Gaussian or copula Gaussian assumptions, but also enhances estimation accuracy by avoiding the ``curse of dimensionality''. Moreover, it retains the probabilistic conditional independence as the criterion to determine the absence of edges. By doing simulation study and analysis of the f-MRI dataset, we demonstrate the advantages of our method.

翻译：近年来，函数图模型得到了广泛发展，衍生出多种模型，如函数高斯图模型、函数Copula高斯图模型、函数贝叶斯图模型、非参数函数可加图模型以及条件函数图模型。这些模型要么依赖于随机函数的某种参数化分布形式，要么依赖于加性条件独立性——这是一种不同于概率条件独立性的判定准则。本文提出了一种基于函数充分降维的非参数函数图模型。我们的方法不仅放宽了高斯或Copula高斯的假设，还通过避免“维数灾难”提高了估计精度。此外，该方法保留了概率条件独立性作为判定边缺失的准则。通过模拟研究和f-MRI数据集分析，我们验证了所提方法的优势。