The knockoffs is a recently proposed powerful framework that effectively controls the false discovery rate (FDR) for variable selection. However, none of the existing knockoff solutions are directly suited to handle multivariate or high-dimensional functional data, which has become increasingly prevalent in various scientific applications. In this paper, we propose a novel functional model-X knockoffs selection framework tailored to sparse high-dimensional functional models, and show that our proposal can achieve the effective FDR control for any sample size. Furthermore, we illustrate the proposed functional model-X knockoffs selection procedure along with the associated theoretical guarantees for both FDR control and asymptotic power using examples of commonly adopted functional linear additive regression models and the functional graphical model. In the construction of functional knockoffs, we integrate essential components including the correlation operator matrix, the Karhunen-Lo\`eve expansion, and semidefinite programming, and develop executable algorithms. We demonstrate the superiority of our proposed methods over the competitors through both extensive simulations and the analysis of two brain imaging datasets.

翻译：Knockoffs是一种新近提出的强大框架，能有效控制变量选择中的错误发现率。然而，现有Knockoffs方法均无法直接处理多变量或高维函数型数据，而此类数据在各科学领域的应用日益普遍。本文针对稀疏高维函数型模型，提出了一种新颖的函数型Model-X Knockoffs选择框架，并证明该框架能在任意样本量下实现有效的错误发现率控制。进一步地，我们通过常用函数型线性可加回归模型和函数型图模型的实例，阐述了所提出的函数型Model-X Knockoffs选择流程及其在错误发现率控制和渐近功效方面的理论保证。在函数型Knockoffs的构建中，我们整合了相关算子矩阵、Karhunen-Loève展开和半定规划等核心要素，并开发了可执行算法。通过大量模拟实验和两个脑影像数据集的分析，我们证明了所提方法相较于现有方案的优越性。