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PCA consistency for the power spiked model in high-dimensional settings

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  • Yata, Kazuyoshi
  • Aoshima, Makoto

Abstract

In this paper, we propose a general spiked model called the power spiked model in high-dimensional settings. We derive relations among the data dimension, the sample size and the high-dimensional noise structure. We first consider asymptotic properties of the conventional estimator of eigenvalues. We show that the estimator is affected by the high-dimensional noise structure directly, so that it becomes inconsistent. In order to overcome such difficulties in a high-dimensional situation, we develop new principal component analysis (PCA) methods called the noise-reduction methodology and the cross-data-matrix methodology under the power spiked model. We show that the new PCA methods can enjoy consistency properties not only for eigenvalues but also for PC directions and PC scores in high-dimensional settings.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jmvana:v:122:y:2013:i:c:p:334-354
    DOI: 10.1016/j.jmva.2013.08.003
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    References listed on IDEAS

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    1. 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, June.
    2. Johnstone, Iain M. & Lu, Arthur Yu, 2009. "On Consistency and Sparsity for Principal Components Analysis in High Dimensions," Journal of the American Statistical Association, American Statistical Association, vol. 104(486), pages 682-693.
    3. Jung, Sungkyu & Sen, Arusharka & Marron, J.S., 2012. "Boundary behavior in High Dimension, Low Sample Size asymptotics of PCA," Journal of Multivariate Analysis, Elsevier, vol. 109(C), pages 190-203.
    4. 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.
    5. Yata, Kazuyoshi & Aoshima, Makoto, 2012. "Effective PCA for high-dimension, low-sample-size data with noise reduction via geometric representations," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 193-215.
    6. 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.
    7. Chen, Song Xi & Qin, Yingli, 2010. "A Two Sample Test for High Dimensional Data with Applications to Gene-set Testing," MPRA Paper 59642, University Library of Munich, Germany.
    8. 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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    Cited by:

    1. Makoto Aoshima & Kazuyoshi Yata, 2019. "Distance-based classifier by data transformation for high-dimension, strongly spiked eigenvalue models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(3), pages 473-503, June.
    2. Huang, Shih-Hao & Huang, Su-Yun, 2021. "On the asymptotic normality and efficiency of Kronecker envelope principal component analysis," Journal of Multivariate Analysis, Elsevier, vol. 184(C).
    3. Wang, Shao-Hsuan & Huang, Su-Yun & Chen, Ting-Li, 2020. "On asymptotic normality of cross data matrix-based PCA in high dimension low sample size," Journal of Multivariate Analysis, Elsevier, vol. 175(C).
    4. Okudo, Michiko & Komaki, Fumiyasu, 2021. "Shrinkage priors for single-spiked covariance models," Statistics & Probability Letters, Elsevier, vol. 176(C).
    5. Jonathan Gillard & Emily O’Riordan & Anatoly Zhigljavsky, 2023. "Polynomial whitening for high-dimensional data," Computational Statistics, Springer, vol. 38(3), pages 1427-1461, September.
    6. Bando, Takuma & Sei, Tomonari & Yata, Kazuyoshi, 2022. "Consistency of the objective general index in high-dimensional settings," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    7. Kazuyoshi Yata & Makoto Aoshima, 2020. "Geometric consistency of principal component scores for high‐dimensional mixture models and its application," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 47(3), pages 899-921, September.

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