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Tracy–Widom statistic for the largest eigenvalue of autoscaled real matrices
Author(s) -
Saccenti Edoardo,
Smilde Age K.,
Westerhuis Johan A.,
Hendriks Margriet M. W. B.
Publication year - 2011
Publication title -
journal of chemometrics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.47
H-Index - 92
eISSN - 1099-128X
pISSN - 0886-9383
DOI - 10.1002/cem.1411
Subject(s) - eigenvalues and eigenvectors , mathematics , normalization (sociology) , covariance matrix , covariance , statistics , physics , quantum mechanics , sociology , anthropology
Eigenanalysis is common practice in biostatistics, and the largest eigenvalue of a data set contains valuable information about the data. However, to make inferences about the size of the largest eigenvalue, its distribution must be known. Johnstone's theorem states that the largest eigenvalues l 1 of real random covariance matrices are distributed according to the Tracy–Widom distribution of order 1 when properly normalized toL 1 =l 1 − η npξ np, where η np and ξ np are functions of the data matrix dimensions n and p . Very often, data are expressed in terms of correlations (autoscaling) for which case Johnstone's theorem does not work because the normalizing parameters η np and ξ np are not theoretically known. In this paper we propose a semi‐empirical method based on test‐equating theory to numerically approximate the normalization parameters in the case of autoscaled matrices. This opens the way of making inferences regarding the largest eigenvalue of an autoscaled data set. The method is illustrated by means of application to two real‐life data sets. Copyright © 2011 John Wiley & Sons, Ltd.