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Some properties of canonical correlations and variates in infinite dimensions

Author

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  • Cupidon, J.
  • Eubank, R.
  • Gilliam, D.
  • Ruymgaart, F.

Abstract

In this paper the notion of functional canonical correlation as a maximum of correlations of linear functionals is explored. It is shown that the population functional canonical correlation is in general well defined, but that it is a supremum rather than a maximum, so that a pair of canonical variates may not exist in the spaces considered. Also the relation with the maximum eigenvalue of an associated pair of operators and the corresponding eigenvectors is not in general valid. When the inverses of the operators involved are regularized, however, all of the above properties are restored. Relations between the actual population quantities and their regularized versions are also established. The sample functional canonical correlations can be regularized in a similar way, and consistency is shown at a fixed level of the regularization parameter.

Suggested Citation

  • Cupidon, J. & Eubank, R. & Gilliam, D. & Ruymgaart, F., 2008. "Some properties of canonical correlations and variates in infinite dimensions," Journal of Multivariate Analysis, Elsevier, vol. 99(6), pages 1083-1104, July.
    • Handle: RePEc:eee:jmvana:v:99:y:2008:i:6:p:1083-1104
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    References listed on IDEAS

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    1. Vinod, H. D., 1976. "Canonical ridge and econometrics of joint production," Journal of Econometrics, Elsevier, vol. 4(2), pages 147-166, May.
    2. Ruymgaart, Frits H. & Yang, Song, 1997. "Some Applications of Watson's Perturbation Approach to Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 60(1), pages 48-60, January.
    3. Dauxois, J. & Pousse, A. & Romain, Y., 1982. "Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference," Journal of Multivariate Analysis, Elsevier, vol. 12(1), pages 136-154, March.
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    2. Alvarez, Agustín & Boente, Graciela & Kudraszow, Nadia, 2019. "Robust sieve estimators for functional canonical correlation analysis," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 46-62.
    3. Konrad Menzel, 2023. "Transfer Estimates for Causal Effects across Heterogeneous Sites," Papers 2305.01435, arXiv.org, revised May 2024.
    4. Zhou, Yang & Lin, Shu-Chin & Wang, Jane-Ling, 2018. "Local and global temporal correlations for longitudinal data," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 1-14.
    5. Shin, Hyejin & Lee, Seokho, 2015. "Canonical correlation analysis for irregularly and sparsely observed functional data," Journal of Multivariate Analysis, Elsevier, vol. 134(C), pages 1-18.
    6. Zhu, Hanbing & Li, Rui & Zhang, Riquan & Lian, Heng, 2020. "Nonlinear functional canonical correlation analysis via distance covariance," Journal of Multivariate Analysis, Elsevier, vol. 180(C).

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