论文标题
高维度稀疏协方差函数估计的自适应功能阈值
Adaptive Functional Thresholding for Sparse Covariance Function Estimation in High Dimensions
论文作者
论文摘要
协方差函数估计是多元功能数据分析中的一项基本任务，并且在许多应用中出现。在本文中，我们考虑为高维函数数据估算稀疏协方差函数，其中随机函数p的数量P可与样本量n相当，甚至大于样本量n。在Hilbert-Schmidt功能规范的帮助下，我们引入了一类新的功能阈值操作员，它们结合了阈值和收缩的功能版本，并通过将样品共价函数单个条目的差异纳入功能阈值。为了处理在某些错误观察到的曲线的实际情况，我们还开发了一种非参数平滑方法，以获得平滑的自适应功能阈值估计器及其binned实现以加速计算。当p在完全和部分观察到的功能场景下与n呈指数增长时，我们研究提案的理论特性。最后，我们证明了提出的自适应功能阈值估计值通过广泛的模拟和两个神经成像数据集的功能连通性分析显着优于竞争对手。
Covariance function estimation is a fundamental task in multivariate functional data analysis and arises in many applications. In this paper, we consider estimating sparse covariance functions for high-dimensional functional data, where the number of random functions p is comparable to, or even larger than the sample size n. Aided by the Hilbert--Schmidt norm of functions, we introduce a new class of functional thresholding operators that combine functional versions of thresholding and shrinkage, and propose the adaptive functional thresholding estimator by incorporating the variance effects of individual entries of the sample covariance function into functional thresholding. To handle the practical scenario where curves are partially observed with errors, we also develop a nonparametric smoothing approach to obtain the smoothed adaptive functional thresholding estimator and its binned implementation to accelerate the computation. We investigate the theoretical properties of our proposals when p grows exponentially with n under both fully and partially observed functional scenarios. Finally, we demonstrate that the proposed adaptive functional thresholding estimators significantly outperform the competitors through extensive simulations and the functional connectivity analysis of two neuroimaging datasets.
