A generalized information criterion for high-dimensional PCA rank selection

Abstract

Principal component analysis (PCA) is a commonly used statistical tool for dimension reduction. An important issue in PCA is to determine the rank, which is the number of dominant eigenvalues of the covariance matrix. Among information-based criteria, the Akaike information criterion (AIC) and the Bayesian information criterion (BIC) are the two most common ones. Both use the number of free parameters for assessing model complexity, which requires the validity of the simple spiked covariance model. As a result, AIC and BIC may suffer from the problem of model misspecification when the tail eigenvalues do not follow the simple spiked model assumption. To alleviate this difficulty, we adopt the idea of the generalized information criterion (GIC) to propose a model complexity measure for PCA rank selection. The proposed model complexity takes into account the sizes of eigenvalues and, hence, is more robust to model misspecification. Asymptotic properties of our GIC are established under the high-dimensional setting, where $n\rightarrow \infty$ and $p/n\rightarrow c >0$. Our asymptotic results show that GIC is better than AIC in excluding noise eigenvalues, and is more sensitive than BIC in detecting signal eigenvalues. Numerical studies and a real data example are presented.