从稀疏到密集设计的函数线性回归：池化-岭方法与极小极大最优性
Functional linear regression from sparse to dense designs: a pooling-ridge method and minimax optimality
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中文总结 AI 辅助
针对离散观测数据的函数线性回归最优估计难题，提出池化-岭方法，实现从稀疏到密集设计下两类回归模型的极小极大最优性，经模拟与实际数据验证。
中文摘要 AI 辅助
函数数据分析是将数据视为随机函数处理的重要统计领域。在实践中，随机函数往往未被完全观测，而是在离散时间点进行测量。虽然均值和协方差估计等更简单的问题已针对离散观测数据被广泛研究，但针对该数据类型的线性回归最优估计问题已悬而未决超过二十年。为解决这一基础挑战，我们提出一种名为池化-岭估计的新方法，该方法结合了池化策略与基于再生核希尔伯特空间（RKHS）方法的优势，通过整合所有受试者离散观测测量值的算子无偏估计实现。该统一估计框架使我们能够在从稀疏到密集设计的任意采样方案下，对标量-函数回归和函数-函数回归模型均达到预测风险的极小极大最优性。这类方法学与理论进展为首次获得，且准确揭示了离散采样的影响。对于标量-函数回归，仅发生一次相变，将收敛行为分为两个不同区域；值得注意的是，对于函数-函数回归，最多可能发生三次相变，由预测/响应函数的采样频率决定。最后，模拟实验和两个实际数据示例为所提方法提供了实证支持。
英文摘要
Functional data analysis is an important statistical field that treats data as random functions. In practice, the random functions are often not fully observed but instead measured at discrete times. While simpler problems, such as mean and covariance estimation, have been widely studied for discretely observed data, optimal estimation of linear regression for this data type has remained unsolved for over two decades. To tackle this fundamental challenge, we propose a novel approach, referred to as pooling ridge estimation, which combines the advantages of pooling strategy and RKHS-based method by incorporating the unbiased estimation of operators based on discretely observed measurements from all subjects. This unified estimation framework enables us to achieve minimax optimality in prediction risk in arbitrary sampling schemes ranging from sparse to dense designs, for both scalar-on-function and function-on-function regression models. Such methodological and theoretical advances are obtained for the first time and accurately reveal the influence of discrete sampling. For scalar-on-function regression, the phase transition occurs once, separating the convergence behavior into two distinct regimes. Remarkably, for function-on-function regression, up to three phase transitions may occur, determined by the sampling frequencies of the predictor/response functions. Finally, simulation experiments and two real data examples provide empirical support for the proposed methods.
发表机构
- Tsinghua University（清华大学）
- Peking University（北京大学）
机构由 AI 辅助整理，请以论文原文为准。
