The estimation of functional networks through functional covariance and graphical models have recently attracted increasing attention in settings with high dimensional functional data, where the number of functional variables p is comparable to, and maybe larger than, the number of subjects. In this paper, we first reframe the functional covariance model estimation as a tuning-free problem of simultaneously testing p(p-1)/2 hypotheses for cross-covariance functions. Our procedure begins by constructing a Hilbert-Schmidt-norm-based test statistic for each pair, and employs normal quantile transformations for all test statistics, upon which a multiple testing step is proposed. We then explore the multiple testing procedure under a general error-contamination framework and establish that our procedure can control false discoveries asymptotically. Additionally, we demonstrate that our proposed methods for two concrete examples: the functional covariance model with partial observations and, importantly, the more challenging functional graphical model, can be seamlessly integrated into the general error-contamination framework, and, with verifiable conditions, achieve theoretical guarantees on effective false discovery control. Finally, we showcase the superiority of our proposals through extensive simulations and functional connectivity analysis of two neuroimaging datasets.

翻译：在高维功能数据背景下，功能协方差与图模型的功能网络估计近年来受到广泛关注，其中功能变量数量p与样本量相当甚至更大。本文首先将功能协方差模型估计重构为对p(p-1)/2个交叉协方差函数进行同步检验的无调参问题。我们的方法首先为每对变量构建基于希尔伯特-施密特范数的检验统计量，并对所有统计量进行正态分位数变换，在此基础上提出多重检验步骤。随后，我们在广义误差污染框架下探究该多重检验过程，并证明该方法能渐近控制错误发现。此外，我们通过两个具体案例——部分观测的功能协方差模型以及更具挑战性的功能图模型——论证所提方法可无缝融入广义误差污染框架，并在可验证条件下实现有效错误控制的理论保证。最后，通过大量模拟实验及两个神经影像数据集的功能连接分析，我们展示了所提方案的优越性。