CONVERGENCE AND PREDICTION OF PRINCIPAL COMPONENT SCORES IN HIGH-DIMENSIONAL SETTINGS

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....

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Bibliographic Details
Published in:The Annals of statistics 2010-12, Vol.38 (6), p.3605-3629
Main Authors: Lee, Seunggeun, Zou, Fei, Wright, Fred A.
Format: Article
Language:English
Subjects:
15A18
62H25
Bias
Consistent estimators
Convergence
Covariance matrices
Datasets
Eigenvalues
Eigenvectors
Estimate reliability
Estimators
Exact sciences and technology
General topics
Inference from stochastic processes
time series analysis
Mathematics
Matrices
Multivariate analysis
PC regression
PC scores
PCA
Principal components analysis
Probability and statistics
Probability theory and stochastic processes
random matrix
Sciences and techniques of general use
Simulation training
Statistics
Stochastic processes
Studies
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Summary: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.
ISSN:0090-5364
2168-8966
DOI:10.1214/10-aos821