Convergence and Prediction of Principal Component Scores in High-Dimensional and Ultra-High Dimensional Settings

• Activity Number: 451
• Type: Topic Contributed
• Date/Time: Wednesday, August 3, 2011 : 8:30 AM to 10:20 AM
• Sponsor: Section on Statistical Learning and Data Mining
• Author(s): Seunggeun Lee*+
• Companies: Harvard University
• Keywords: PCA ; PC scores ; Random Matrix

Abstract

A number of settings arise in which it is of interest to predict Principal Component (PC) scores for new observations using data from an initial sample. In this talk, we demonstrate that naive approaches to PC score prediction can be substantially biased towards 0 in the analysis of large matrices. This phenomenon is largely related to known inconsistency results for sample eigenvalues and eigenvectors as both dimensions (p) and sample sizes (n) increases. For the spiked eigenvalue model for random matrices, we expand the generality of these results, and propose bias-adjusted PC score prediction. Simulation and real data examples from the genetics literature show the improved bias and numerical properties of our estimators. In addition, we discuss asymptotic behaviors of sample eigenvalues, eigenvectors and PC scores under high dimensional (p/n < 8) and ultra high dimensional (p/n ? 8) settings.