On Adaptive Sparse Principal Component Analysis

In this simple note, we attempt to further improve the sparse principal component analysis (SPCA) of Zou et al. (2006) on the following two aspects. First, we replace the traditional lasso penalty utilized in the original SPCA by the most recently developed adaptive lasso penalty (Zou, 2006; Wang et al., 2006). By doing so, adaptive amounts of shrinkage can be applied to different loading coefficients according to their estimated relative importance, which naturally improves the efficiency of the estimation and selection results. Secondly, a BIC-type tuning parameter selector is developed for a consistent selection of the tuning parameters. We refer to our approach as the adaptive sparse principal component analysis (ASPCA). Asymptotic theory is established to support such a proposal and numerical studies are conducted to further demonstrate its usefulness.