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

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  • Jung, Sungkyu
  • Sen, Arusharka
  • Marron, J.S.
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    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.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 109 (2012)
    Issue (Month): C ()
    Pages: 190-203

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    Handle: RePEc:eee:jmvana:v:109:y:2012:i:c:p:190-203

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    Related research

    Keywords: Principal component analysis; High Dimension Low Sample Size; Geometric representation; ρ-mixing; Consistency and strong inconsistency; Spiked covariance model;

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    Cited by:
    1. Yata, Kazuyoshi & Aoshima, Makoto, 2013. "PCA consistency for the power spiked model in high-dimensional settings," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 334-354.

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