Asymptotic Theory of Principal Component Analysis for Time Series Data with Cautionary Comments
研究了时间序列数据中使用主成分分析（PCA）时，基于独立数据假设的理论结果可能被误用的问题，给出了特征值和特征向量的中心极限定理及其协方差估计，并通过投资组合管理实例说明正确用法。
Abstract Principal component analysis (PCA) is a most frequently used statistical tool in almost all branches of data science. However, like many other statistical tools, there is sometimes the risk of misuse or even abuse. In this paper, we highlight possible pitfalls in using the theoretical results of PCA based on the assumption of independent data when the data are time series. For the latter, we state with proof a central limit theorem of the eigenvalues and eigenvectors (loadings), give direct and bootstrap estimation of their asymptotic covariances, and assess their efficacy via simulation. Specifically, we pay attention to the proportion of variation, which decides the number of principal components (PCs), and the loadings, which help interpret the meaning of PCs. Our findings are that while the proportion of variation is quite robust to different dependence assumptions, the inference of PC loadings requires careful attention. We initiate and conclude our investigation with an empirical example on portfolio management, in which the PC loadings play a prominent role. It is given as a paradigm of correct usage of PCA for time series data.