Consistency of Principal Component Scores in High-Dimensional Data

• Activity Number: 349
• Type: Contributed
• Date/Time: Tuesday, August 6, 2013 : 10:30 AM to 12:20 PM
• Sponsor: Biometrics Section
• Author(s): Kristoffer Hellton*+ and Magne Thoresen
• Companies: Department of Biostatistics, University of Oslo and Department of Biostatistics, Institute of Basic Medical Sciences, University of Oslo
• Keywords: Principal component analysis ; Principal component regression ; Genetic data ; Component scores ; Consistency

Abstract

Principal component analysis (PCA) is widely used to reduce high-dimensional data to a small set of component scores. In applications, such as genomics, these scores are used in conventional classification and regression methods or to visualize main features.

However, in the high-dimensional setting, PCA is in general not asymptotically consistent, as the population eigenvalues and -vectors are not consistently estimated by the sample eigenvalues and -vectors. In this work, we show why the visual information given by component scores is still valid under certain assumptions on the population structure. The asymptotic bias in eigenvalues and -vectors has been investigated by several authors for fixed population eigenvalues, among others Paul (2007). In a different asymptotic setting, starting with Jung and Marron (2009), the sample size is fixed and the largest eigenvalues depend on the number of variables. We will argue that it is realistic to assume the largest population eigenvalues to depend linearly on the data dimension and illustrate this by genetic data. The implication of this assumption will be investigated in terms of the ratio between estimated and true scores.