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Asymptotic properties of principal component analysis and shrinkage-bias adjustment under the generalized spiked population model

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  • Dey, Rounak
  • Lee, Seunggeun

Abstract

With the development of high-throughput technologies, principal component analysis (PCA) in the high-dimensional regime is of great interest. Most of the existing theoretical and methodological results for high-dimensional PCA are based on the spiked population model in which all the population eigenvalues are equal except for a few large ones. Due to the presence of local correlation among features, however, this assumption may not be satisfied in many real-world datasets. To address this issue, we investigate the asymptotic behavior of PCA under the generalized spiked population model. Based on our theoretical results, we propose a series of methods for the consistent estimation of population eigenvalues, angles between the sample and population eigenvectors, correlation coefficients between the sample and population principal component (PC) scores, and the shrinkage bias adjustment for the predicted PC scores. Using numerical experiments and real data examples from the genetics literature, we show that our methods can greatly reduce bias and improve prediction accuracy.

Suggested Citation

  • Dey, Rounak & Lee, Seunggeun, 2019. "Asymptotic properties of principal component analysis and shrinkage-bias adjustment under the generalized spiked population model," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 145-164.
    • Handle: RePEc:eee:jmvana:v:173:y:2019:i:c:p:145-164
    DOI: 10.1016/j.jmva.2019.02.007
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    References listed on IDEAS

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    1. 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.
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    3. Bai, Zhidong & Yao, Jianfeng, 2012. "On sample eigenvalues in a generalized spiked population model," Journal of Multivariate Analysis, Elsevier, vol. 106(C), pages 167-177.
    4. Silverstein, J. W. & Choi, S. I., 1995. "Analysis of the Limiting Spectral Distribution of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 54(2), pages 295-309, August.
    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. Seunggeun Lee & Fei Zou & Fred A. Wright, 2014. "Convergence of sample eigenvalues, eigenvectors, and principal component scores for ultra-high dimensional data," Biometrika, Biometrika Trust, vol. 101(2), pages 484-490.
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