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Boundary behavior in High Dimension, Low Sample Size asymptotics of PCA

Title:

Boundary behavior in High Dimension, Low Sample Size asymptotics of PCA

Jung, Sungkyu, Sen, Arusharka and Marron, J.S. (2012) Boundary behavior in High Dimension, Low Sample Size asymptotics of PCA. Journal of Multivariate Analysis, 109 . pp. 190-203. ISSN 0047259X

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Official URL: http://dx.doi.org/10.1016/j.jmva.2012.03.005

Abstract

In High Dimension, Low Sample Size (HDLSS) data situations, where the dimension d is much larger than the sample size n, principal component analysis (PCA) plays an important role in statistical analysis. Under which conditions does the sample PCA well reflect the population covariance structure? We answer this question in a relevant asymptotic context where d grows and n is fixed, under a generalized spiked covariance model. Specifically, we assume the largest population eigenvalues to be of the order dα, where α<, =, or >1. Earlier results show the conditions for consistency and strong inconsistency of eigenvectors of the sample covariance matrix. In the boundary case, α=1, where the sample PC directions are neither consistent nor strongly inconsistent, we show that eigenvalues and eigenvectors do not degenerate but have limiting distributions. The result smoothly bridges the phase transition represented by the other two cases, and thus gives a spectrum of limits for the sample PCA in the HDLSS asymptotics. While the results hold under a general situation, the limiting distributions under Gaussian assumption are illustrated in greater detail. In addition, the geometric representation of HDLSS data is extended to give three different representations, that depend on the magnitude of variances in the first few principal components.

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Article
Refereed:Yes
Authors:Jung, Sungkyu and Sen, Arusharka and Marron, J.S.
Journal or Publication:Journal of Multivariate Analysis
Date:2012
Digital Object Identifier (DOI):10.1016/j.jmva.2012.03.005
Keywords:Principal component analysis; High Dimension Low Sample Size; Geometric representation; ρ-mixing; Consistency and strong inconsistency; Spiked covariance model
ID Code:976820
Deposited By: DANIELLE DENNIE
Deposited On:29 Jan 2013 13:28
Last Modified:18 Jan 2018 17:43

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