Selecting the number of principal components: Estimation of the true rank of a noisy matrix
Yunjin Choi, Jonathan Taylor, Robert Tibshirani
The Annals of Statistics · 2017 · Volume 45 · Issue 6
Abstract
Principal component analysis (PCA) is a well-known tool in multivariate statistics. One significant challenge in using PCA is the choice of the number of principal components. In order to address this challenge, we propose distribution-based methods with exact type 1 error controls for hypothesis testing and construction of confidence intervals for signals in a noisy matrix with finite samples. Assuming Gaussian noise, we derive exact type 1 error controls based on the conditional distribution of the singular values of a Gaussian matrix by utilizing a post-selection inference framework, and extending the approach of [Taylor, Loftus and Tibshirani (2013)] in a PCA setting. In simulation studies, we find that our proposed methods compare well to existing approaches.
Keywords
62F03, 62J05, 62J07, exact $p$-value, hypothesis test, principal components
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