PCA Consistency for High-Dimension, Low Sample Size Context

• Activity Number: 237
• Type: Contributed
• Date/Time: Monday, August 2, 2010 : 2:00 PM to 3:50 PM
• Sponsor: Biometrics Section
• Author(s): Sungkyu Jung*+ and J. S. Marron
• Companies: The University of North Carolina at Chapel Hill and The University of North Carolina at Chapel Hill
• Keywords: Principal component analysis ; high dimension low sample size ; consistency ; spiked population model

Abstract

Principal Component Analysis (PCA) is an important tool of dimension reduction especially when the dimension (or the number of variables) is very high. Asymptotic studies where the sample size is fixed, and the dimension grows [i.e., High Dimension, Low Sample Size (HDLSS)] are becoming increasingly relevant. We investigate the consistency or strong inconsistency of the Principal Component directions through HDLSS asymptotics. Moreover, some limiting distributions are revealed under neither consistent nor strongly inconsistent scenarios. Broad sets of sufficient conditions for each of these cases are specified, and we show that the geometric representation of HDLSS data holds under general conditions, which includes a ?-mixing condition and a broad range of sphericity measures of the covariance matrix.