Covariance operators are fundamental concepts and modelling tools for many functional data analysis methods, such as functional principal component analysis. However, the empirical (or estimated) covariance operator becomes too costly to compute when the functional dataset gets big. This paper studies a randomized algorithm for covariance operator estimation. The algorithm works by sampling and rescaling observations from the large functional data collection to form a sketch of much smaller size and performs computation on the sketch to obtain the subsampled empirical covariance operator. The proposed algorithm is theoretically justified via nonasymptotic bounds between the subsampled and the full‐sample empirical covariance operator in terms of the Hilbert‐Schmidt norm and the operator norm. It is shown that the optimal sampling probability that minimizes the expected squared Hilbert‐Schmidt norm of the subsampling error is determined by the norm of each function. Simulated and real data examples are used to illustrate the effectiveness of the proposed algorithm.

中文翻译：

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通过二次采样对函数协方差算子进行随机估计

协方差算子是许多功能数据分析方法（例如功能主成分分析）的基本概念和建模工具。但是，当功能数据集变大时，经验（或估计）协方差算子的计算成本太高。本文研究了一种用于协方差算子估计的随机算法。该算法的工作原理是对来自大型功能数据集合的观察值进行采样和重新缩放，以形成尺寸更小的草图，并对草图进行计算以获得二次采样的经验协方差算子。该算法在理论上是通过Hilbert-Schmidt范数和算子范数在子样本和全样本经验协方差算子之间的非渐近边界来证明的。结果表明，最小化二次抽样误差的希尔伯特-施密特平方期望范数的最佳抽样概率由每个函数的范数确定。仿真和实际数据示例用于说明所提出算法的有效性。