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

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 \(C_p\) 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.

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Acknowledgements

The authors would like to thank the editor, the associated editor and two anonymous referees for their constructive suggestions and comments that led to the improvement of an early manuscript.

Funding

The research described herewith was supported by a Grant (HKBU12303419) from The University Grants Council of Hong Kong, and a grant from The National Natural Science Foundation of China (NSFC11671042).

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Correspondence to Lixing Zhu.

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Zheng, J., Zhu, L. Determining the number of canonical correlation pairs for high-dimensional vectors. Ann Inst Stat Math (2021). https://doi.org/10.1007/s10463-020-00776-x

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Keywords

  • Canonical correlation matrix
  • Eigenvalue-based ridge ratios
  • High dimensionality
  • The number of canonical correlation pairs