The Annals of Statistics

PCA consistency in high dimension, low sample size context

Sungkyu Jung and J. S. Marron

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Principal Component Analysis (PCA) is an important tool of dimension reduction especially when the dimension (or the number of variables) is very high. Asymptotic studies where the sample size is fixed, and the dimension grows [i.e., High Dimension, Low Sample Size (HDLSS)] are becoming increasingly relevant. We investigate the asymptotic behavior of the Principal Component (PC) directions. HDLSS asymptotics are used to study consistency, strong inconsistency and subspace consistency. We show that if the first few eigenvalues of a population covariance matrix are large enough compared to the others, then the corresponding estimated PC directions are consistent or converge to the appropriate subspace (subspace consistency) and most other PC directions are strongly inconsistent. Broad sets of sufficient conditions for each of these cases are specified and the main theorem gives a catalogue of possible combinations. In preparation for these results, we show that the geometric representation of HDLSS data holds under general conditions, which includes a ρ-mixing condition and a broad range of sphericity measures of the covariance matrix.

Article information

Ann. Statist. Volume 37, Number 6B (2009), 4104-4130.

First available in Project Euclid: 23 October 2009

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Zentralblatt MATH identifier

Primary: 62H25: Factor analysis and principal components; correspondence analysis 34L20: Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions
Secondary: 62F12: Asymptotic properties of estimators

Principal component analysis sample covariance matrix ρ-mixing high dimension low sample size data nonstandard asymptotics consistency and strong inconsistency spiked population model


Jung, Sungkyu; Marron, J. S. PCA consistency in high dimension, low sample size context. Ann. Statist. 37 (2009), no. 6B, 4104--4130. doi:10.1214/09-AOS709.

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