带有警示注释的时间序列数据主成分分析的渐近理论
Asymptotic Theory of Principal Component Analysis for Time Series Data with Cautionary Comments
摘要：
主成分分析（PCA）是数据科学中几乎所有分支中最常用的统计工具。然而，与许多其他统计工具一样，有时存在误用甚至滥用的风险。在本文中，我们强调可能的陷阱，在使用PCA的理论结果的基础上假设独立的数据时，数据是时间序列。对于后者，我们陈述并证明了特征值和特征向量（载荷）的中心极限定理，给出了它们的渐近协方差的直接和自助估计，并通过模拟评估了它们的有效性。具体来说，我们注意到的变异的比例，决定了主成分（PC）的数量，和负载，这有助于解释PC的意义。我们的研究结果是，虽然变化的比例是相当强大的不同的依赖性假设，PC负载的推断需要仔细注意。我们开始和结束我们的调查与实证的例子投资组合管理，其中PC负荷发挥了突出的作用。这是正确使用主成分分析方法处理时间序列数据的一个范例。
英文摘要：
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.
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