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On the distribution of the largest eigenvalue in principal components analysis (2001)
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| Venue: | ANN. STATIST |
| Citations: | 404 - 4 self |
BibTeX
@ARTICLE{Johnstone01onthe,
author = {Iain M. Johnstone},
title = {On the distribution of the largest eigenvalue in principal components analysis},
journal = {ANN. STATIST},
year = {2001},
volume = {29},
number = {2},
pages = {295--327}
}
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Abstract
Let x �1 � denote the square of the largest singular value of an n × p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x �1 � is the largest principal component variance of the covariance matrix X ′ X, or the largest eigenvalue of a p-variate Wishart distribution on n degrees of freedom with identity covariance. Consider the limit of large p and n with n/p = γ ≥ 1. When centered by µ p = � √ n − 1 + √ p � 2 and scaled by σ p = � √ n − 1 + √ p��1 / √ n − 1 + 1 / √ p � 1/3 � the distribution of x �1 � approaches the Tracy–Widom lawof order 1, which is defined in terms of the Painlevé II differential equation and can be numerically evaluated and tabulated in software. Simulations showthe approximation to be informative for n and p as small as 5. The limit is derived via a corresponding result for complex Wishart matrices using methods from random matrix theory. The result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts.
Keyphrases
principal component analysis tracy widom identity covariance independent standard gaussian variate large multivariate distribution theory principal component variance complex wishart matrix simulation showthe approximation random matrix theory corresponding result painlev ii differential equation covariance matrix singular value fixed counterpart p-variate wishart distribution
