Inference on the Proportion of Variance Explained in Principal Component Analysis

Principal component analysis (PCA) is a longstanding approach for dimension reduction. It rests upon the assumption that the underlying signal has low rank, and thus can be well-summarized using a small number of dimensions. The output of PCA is typically represented using a scree plot, which displays the proportion of variance explained (PVE) by each principal component. While the PVE is extensively reported in routine analyses, to the best of our knowledge the notion of <i>inference</i> on the PVE remains unexplored. We consider inference on a new population quantity for the PVE with respect to an unknown matrix mean. Our interest lies in the PVE of the sample principal components (as opposed to unobserved population principal components); thus, the population PVE that we introduce is defined <i>conditional</i> on the sample singular vectors. We show that it is possible to conduct inference, in the sense of confidence intervals, <i>p</i>-values, and point estimates, on this population quantity. Furthermore, we can conduct valid inference on the PVE of a subset of the principal components, even when the subset is selected using a data-driven approach such as the elbow rule. We demonstrate our approach in simulation and in an application to gene expression data. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

主成分分析（PCA）是一类应用广泛的经典降维方法。其理论基础为：潜在信号具备低秩特性，因此可通过少量维度实现高效概括与表征。主成分分析的输出通常以碎石图（scree plot）呈现，该图展示了各主成分对应的方差解释比例（Proportion of Variance Explained, PVE）。尽管方差解释比例（PVE）在常规数据分析中被广泛报道，但据我们所知，针对PVE的统计推断（inference）这一研究方向仍未得到充分探索。
本文针对未知矩阵均值的场景，构建了用于PVE分析的全新总体统计量，并将研究聚焦于样本主成分的PVE（而非未观测的总体主成分）。据此，本文提出的总体PVE是基于样本奇异向量的条件定义的。研究表明，可针对该总体量开展完整的统计推断，包括构造置信区间、计算p值与点估计。此外，即便主成分子集是通过肘部准则（elbow rule）这类数据驱动方法选取的，我们仍可对该子集的PVE开展有效的统计推断。我们通过模拟实验与基因表达数据的应用案例验证了所提方法的有效性。本文的补充材料可在线获取，其中包含可用于复现研究工作的标准化材料说明。