Dynamic Principal Component Analysis in High Dimensions

Principal component analysis is a versatile tool to reduce dimensionality which has wide applications in statistics and machine learning. It is particularly useful for modeling data in high-dimensional scenarios where the number of variables <i>p</i> is comparable to, or much larger than the sample size <i>n</i>. Despite an extensive literature on this topic, researchers have focused on modeling static principal eigenvectors, which are not suitable for stochastic processes that are dynamic in nature. To characterize the change in the entire course of high-dimensional data collection, we propose a unified framework to directly estimate dynamic eigenvectors of covariance matrices. Specifically, we formulate an optimization problem by combining the local linear smoothing and regularization penalty together with the orthogonality constraint, which can be effectively solved by manifold optimization algorithms. We show that our method is suitable for high-dimensional data observed under both common and irregular designs, and theoretical properties of the estimators are investigated under lq(0≤q≤1) sparsity. Extensive experiments demonstrate the effectiveness of the proposed method in both simulated and real data examples.

主成分分析（Principal Component Analysis）是一种应用广泛的降维工具，在统计学与机器学习领域均有诸多应用。其在变量数<i>p</i>与样本量<i>n</i>相当或远大于样本量的高维场景下的数据建模中尤为实用。尽管该领域已有大量研究文献，但现有研究多聚焦于静态主特征向量的建模，而此类方法并不适用于本质上具有动态性的随机过程。为刻画高维数据采集全流程中的变化规律，本文提出一种可直接估计协方差矩阵动态特征向量的统一框架。具体而言，本文将局部线性平滑、正则化惩罚项与正交性约束相结合，构建了一个优化问题，该问题可通过流形优化算法高效求解。本文证明所提方法适用于常规设计与非规则设计下的高维数据，并在0≤q≤1的lq稀疏性假设下分析了估计量的理论性质。大量仿真与真实数据实验均验证了所提方法的有效性。