Convergence and prediction of principal component scores in high-dimensional settings

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 paper, we demonstrate that naive approaches to PC score prediction can be substantially biased toward 0 in the analysis of large matrices. This phenomenon is largely related to known inconsistency results for sample eigenvalues and eigenvectors as both dimensions of the matrix increase. For the spiked eigenvalue model for random matrices, we expand the generality of these results, and propose bias-adjusted PC score prediction. In addition, we compute the asymptotic correlation coefficient between PC scores from sample and population eigenvectors. Simulation and real data examples from the genetics literature show the improved bias and numerical properties of our estimators.

arXiv commentPublished in at http://dx.doi.org/10.1214/10-AOS821 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (5)

• PCA consistency in high dimension, low sample size context
• Finite sample approximation results for principal component analysis: a matrix perturbation approach
• Central limit theorems for eigenvalues in a spiked population model
• Singular Value Decomposition and Principal Component Analysis
• Statistical eigen-inference from large Wishart matrices

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• Notes on asymptotics of sample eigenstructure for spiked covariance models with non-Gaussian data
• Joint CLT for several random sesquilinear forms with applications to large-dimensional spiked population models
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• Computation of ancestry scores with mixed families and unrelated individuals