Selecting the Number of Principal Components: Estimation of the True Rank of a Noisy Matrix

• Activity Number: 698
• Type: Contributed
• Date/Time: Thursday, August 13, 2015 : 10:30 AM to 12:20 PM
• Sponsor: Section on Statistical Learning and Data Mining
• Author(s): Yunjin Choi* and Rob Tibshirani and Jonathan Taylor
• Companies: Stanford University and Stanford University and Stanford University
• Keywords: Principal Component Analysis ; Hypothesis Testing ; Sequential Test

Abstract

Principal component analysis (PCA) is a well-known tool in multivariate statistics. One big challenge in using the method is the choice of the number of components. In this paper, we propose an exact distribution-based method for this purpose: our approach is related to the adaptive regression framework of Taylor et al. (2013). Assuming Gaussian noise, we use the conditional distribution of the eigenvalues of a Wishart matrix as our test statistic, and derive exact hypothesis tests and confidence intervals for the true singular values. In simulation studies we find that our proposed method compares well to the proposal of Kritchman & Nadler (2008), which uses the asymptotic distribution of singular values based on the Tracy-Widom laws.