  Principal component analysis (PCA) is a longstanding and well-studied
approach for dimension reduction. It rests upon the assumption that the
underlying signal in the data 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
data analyses, to the best of our knowledge the notion of inference on the PVE
remains unexplored.
  In this paper, we consider inference on the PVE. We first introduce a new
population quantity for the PVE with respect to an unknown matrix mean.
Critically, 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 conditional on the sample singular vectors. We
show that it is possible to conduct inference, in the sense of confidence
intervals, p-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 the proposed approach in
simulation and in an application to a gene expression dataset.
