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Effective PCA for high-dimension, low-sample-size data with singular value decomposition of cross data matrix

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  • Yata, Kazuyoshi
  • Aoshima, Makoto

Abstract

In this paper, we propose a new methodology to deal with PCA in high-dimension, low-sample-size (HDLSS) data situations. We give an idea of estimating eigenvalues via singular values of a cross data matrix. We provide consistency properties of the eigenvalue estimation as well as its limiting distribution when the dimension d and the sample size n both grow to infinity in such a way that n is much lower than d. We apply the new methodology to estimating PC directions and PC scores in HDLSS data situations. We give an application of the findings in this paper to a mixture model to classify a dataset into two clusters. We demonstrate how the new methodology performs by using HDLSS data from a microarray study of prostate cancer.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jmvana:v:101:y:2010:i:9:p:2060-2077
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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. 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.
    4. 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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    Citations

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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. 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.
    3. Ishii, Aki & Yata, Kazuyoshi & Aoshima, Makoto, 2022. "Geometric classifiers for high-dimensional noisy data," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    4. 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).
    5. Makoto Aoshima & Kazuyoshi Yata, 2014. "A distance-based, misclassification rate adjusted classifier for multiclass, high-dimensional data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(5), pages 983-1010, October.
    6. 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.
    7. Yata, Kazuyoshi & Aoshima, Makoto, 2013. "Correlation tests for high-dimensional data using extended cross-data-matrix methodology," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 313-331.
    8. Yata, Kazuyoshi & Aoshima, Makoto, 2016. "High-dimensional inference on covariance structures via the extended cross-data-matrix methodology," Journal of Multivariate Analysis, Elsevier, vol. 151(C), pages 151-166.
    9. 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.
    10. Jonathan Gillard & Emily O’Riordan & Anatoly Zhigljavsky, 2023. "Polynomial whitening for high-dimensional data," Computational Statistics, Springer, vol. 38(3), pages 1427-1461, September.
    11. 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.
    12. Wang, Shao-Hsuan & Huang, Su-Yun, 2022. "Perturbation theory for cross data matrix-based PCA," Journal of Multivariate Analysis, Elsevier, vol. 190(C).
    13. Makoto Aoshima & Kazuyoshi Yata, 2015. "Asymptotic Normality for Inference on Multisample, High-Dimensional Mean Vectors Under Mild Conditions," Methodology and Computing in Applied Probability, Springer, vol. 17(2), pages 419-439, June.
    14. Aki Ishii & Kazuyoshi Yata & Makoto Aoshima, 2021. "Hypothesis tests for high-dimensional covariance structures," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 73(3), pages 599-622, June.
    15. Hong, Zhaoping & Lian, Heng, 2013. "Sparse-smooth regularized singular value decomposition," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 163-174.

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