一种针对顺序到达高维数据的三阶段PCA程序
A Three-Stage PCA Procedure for Sequentially Arriving High-Dimensional Data
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- Florida State University（佛罗里达州立大学）
- Krea University（克雷亚大学）
- Indian Institute of Technology Jodhpur（印度理工学院乔德普尔分校）
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中文总结 AI 辅助
提出一种三阶段自适应PCA程序，平衡压缩损失与采样成本，通过中间阶段更新避免依赖未知特征值，实现一阶和二阶效率，并在模拟和癌症基因数据中验证其性能。
中文摘要 AI 辅助
我们开发了一种三阶段自适应程序，用于在高维观测数据顺序收集且额外采样产生成本时进行主成分分析（PCA）。该程序在通过预设的解释方差准则选择保留维度的同时，平衡PCA压缩损失与采样成本。从一个试点样本出发，中间阶段在确定最终样本量之前更新PCA量，从而避免依赖未知的总体特征值。在适当的正则性条件下，我们建立了相对于总体神谕的一阶和二阶效率。与相应的两阶段规则相比，额外的重新校准产生了更精确的二阶控制，并减少了试点阶段对最终采样决策的影响。该理论允许环境维度在适当的协方差和谱条件下超过样本量。模拟研究证明了该程序在增加维度和多种密集协方差结构下的强有限样本性能。作为实际数据应用，我们对癌症基因组图谱中32个癌症类型队列的基因表达数据进行了回顾性研究，展示了成本有效的提前停止以及建议额外观测的设置。
英文摘要
We develop a three-stage adaptive procedure for principal component analysis (PCA) when high-dimensional observations are collected sequentially and additional sampling incurs a cost. The procedure balances PCA compression loss against sampling cost while selecting the retained dimension through a prescribed explained-variance criterion. Starting from a pilot sample, an intermediate stage updates the PCA quantities before determining the final sample size, thereby avoiding reliance on unknown population eigenvalues. Under suitable regularity conditions, we establish both first- and second-order efficiency relative to the population oracle. Comparison with the corresponding two-stage rule shows that the additional recalibration yields sharper second-order control and reduces the influence of the pilot stage on the final sampling decision. The theory allows the ambient dimension to exceed the sample size under appropriate covariance and spectral conditions. Simulation studies demonstrate the strong finite-sample performance of the procedure across increasing dimensions and several dense covariance structures. As a real-data application, we conduct a retrospective study of gene-expression data from 32 cancer-type cohorts in The Cancer Genome Atlas, illustrating both cost-effective early stopping and settings in which additional observations are recommended.
