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Determining the number of canonical correlation pairs for high-dimensional vectors

Author

Listed:
  • Jiasen Zheng

    (Renmin University of China)

  • Lixing Zhu

    (Hong Kong Baptist University
    Beijing Normal University)

Abstract

For two random vectors whose dimensions are both proportional to the sample size, we in this paper propose two ridge ratio criteria to determine the number of canonical correlation pairs. The criteria are, respectively, based on eigenvalue difference-based and centered eigenvalue-based ridge ratios. Unlike existing methods, the criteria make the ratio at the index we want to identify stick out to show a visualized “valley-cliff” pattern and thus can adequately avoid the local optimal solutions that often occur in the eigenvalues multiplicity cases. The numerical studies also suggest its advantage over existing scree plot-based method that is not a visualization method and more seriously underestimates the number of pairs than the proposed ones and the AIC and $$C_p$$ C p criteria that often extremely over-estimate the number, and the BIC criterion that has very serious underestimation problem. A real data set is analyzed for illustration.

Suggested Citation

  • Jiasen Zheng & Lixing Zhu, 2021. "Determining the number of canonical correlation pairs for high-dimensional vectors," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 73(4), pages 737-756, August.
    • Handle: RePEc:spr:aistmt:v:73:y:2021:i:4:d:10.1007_s10463-020-00776-x
    DOI: 10.1007/s10463-020-00776-x
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    1. Christopher R Cabanski & Yuan Qi & Xiaoying Yin & Eric Bair & Michele C Hayward & Cheng Fan & Jianying Li & Matthew D Wilkerson & J S Marron & Charles M Perou & D Neil Hayes, 2010. "SWISS MADE: Standardized WithIn Class Sum of Squares to Evaluate Methodologies and Dataset Elements," PLOS ONE, Public Library of Science, vol. 5(3), pages 1-13, March.
    2. Fujikoshi, Yasunori & Sakurai, Tetsuro, 2009. "High-dimensional asymptotic expansions for the distributions of canonical correlations," Journal of Multivariate Analysis, Elsevier, vol. 100(1), pages 231-242, January.
    3. Gunderson, Brenda K. & Muirhead, Robb J., 1997. "On Estimating the Dimensionality in Canonical Correlation Analysis," Journal of Multivariate Analysis, Elsevier, vol. 62(1), pages 121-136, July.
    4. Headrick, Todd C., 2002. "Fast fifth-order polynomial transforms for generating univariate and multivariate nonnormal distributions," Computational Statistics & Data Analysis, Elsevier, vol. 40(4), pages 685-711, October.
    5. Zhu, Xuehu & Guo, Xu & Wang, Tao & Zhu, Lixing, 2020. "Dimensionality determination: A thresholding double ridge ratio approach," Computational Statistics & Data Analysis, Elsevier, vol. 146(C).
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