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 是近期提出的一种强大框架，能有效控制变量选择中的错误发现率（FDR）。然而，现有的所有 Knockoffs 方法均无法直接处理多变量或高维功能数据，而此类数据在各种科学应用中正日益普遍。本文提出了一种专为稀疏高维功能模型设计的新型功能模型-X Knockoffs 选择框架，并证明我们的方法能在任意样本量下实现有效的 FDR 控制。此外，我们通过常用的功能线性加性回归模型和功能图模型示例，阐述了所提出的功能模型-X Knockoffs 选择程序及其在 FDR 控制和渐近功效方面的相关理论保证。在构建功能 Knockoffs 时，我们整合了相关算子矩阵、Karhunen-Loève 展开和半定规划等核心组件，并开发了可执行算法。通过大量模拟实验和对两个脑成像数据集的分析，我们证明了所提方法相较于现有方法的优越性。