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Sparse Principal Component Analysis in Hilbert Space

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  • Xin Qi
  • Ruiyan Luo

Abstract

type="main" xml:id="sjos12106-abs-0001"> Technical advances in many areas have produced more complicated high-dimensional data sets than the usual high-dimensional data matrix, such as the fMRI data collected in a period for independent trials, or expression levels of genes measured in different tissues. Multiple measurements exist for each variable in each sample unit of these data. Regarding the multiple measurements as an element in a Hilbert space, we propose Principal Component Analysis (PCA) in Hilbert space. The principal components (PCs) thus defined carry information about not only the patterns of variations in individual variables but also the relationships between variables. To extract the features with greatest contributions to the explained variations in PCs for high-dimensional data, we also propose sparse PCA in Hilbert space by imposing a generalized elastic-net constraint. Efficient algorithms to solve the optimization problems in our methods are provided. We also propose a criterion for selecting the tuning parameter.

Suggested Citation

  • Xin Qi & Ruiyan Luo, 2015. "Sparse Principal Component Analysis in Hilbert Space," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(1), pages 270-289, March.
    • Handle: RePEc:bla:scjsta:v:42:y:2015:i:1:p:270-289
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    File URL: http://hdl.handle.net/10.1111/sjos.12106
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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.
    2. Shen, Haipeng & Huang, Jianhua Z., 2008. "Sparse principal component analysis via regularized low rank matrix approximation," Journal of Multivariate Analysis, Elsevier, vol. 99(6), pages 1015-1034, July.
    3. Qi, Xin & Luo, Ruiyan & Zhao, Hongyu, 2013. "Sparse principal component analysis by choice of norm," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 127-160.
    4. Trendafilov, Nickolay T. & Jolliffe, Ian T., 2006. "Projected gradient approach to the numerical solution of the SCoTLASS," Computational Statistics & Data Analysis, Elsevier, vol. 50(1), pages 242-253, January.
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    Cited by:

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    3. Cristinel Petrisor Constantin & Ioana Bianca Chitu & Evelina Gradinaru & Lavinia Dovleac & Gabriel Bratucu, 2019. "Approaching the Bioeconomy in Terms of Increasing the Energy Efficiency of Households in Romania," The AMFITEATRU ECONOMIC journal, Academy of Economic Studies - Bucharest, Romania, vol. 21(50), pages 1-90, February.

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