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

  • Jung, Sungkyu
  • Sen, Arusharka
  • Marron, J.S.
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    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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    File URL: http://www.sciencedirect.com/science/article/pii/S0047259X12000747
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    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
    DOI: 10.1016/j.jmva.2012.03.005
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    1. Yata, Kazuyoshi & Aoshima, Makoto, 2010. "Effective PCA for high-dimension, low-sample-size data with singular value decomposition of cross data matrix," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 2060-2077, October.
    2. Peter Hall & J. S. Marron & Amnon Neeman, 2005. "Geometric representation of high dimension, low sample size data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(3), pages 427-444.
    3. Hanwen Huang & Yufeng Liu & J. S. Marron, 2012. "Bidirectional discrimination with application to data visualization," Biometrika, Biometrika Trust, vol. 99(4), pages 851-864.
    4. Qiao, Xingye & Zhang, Hao Helen & Liu, Yufeng & Todd, Michael J. & Marron, J. S., 2010. "Weighted Distance Weighted Discrimination and Its Asymptotic Properties," Journal of the American Statistical Association, American Statistical Association, vol. 105(489), pages 401-414.
    5. Baik, Jinho & Silverstein, Jack W., 2006. "Eigenvalues of large sample covariance matrices of spiked population models," Journal of Multivariate Analysis, Elsevier, vol. 97(6), pages 1382-1408, July.
    6. Jeongyoun Ahn & J. S. Marron & Keith M. Muller & Yueh-Yun Chi, 2007. "The high-dimension, low-sample-size geometric representation holds under mild conditions," Biometrika, Biometrika Trust, vol. 94(3), pages 760-766.
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