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Canonical correlation for stochastic processes

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  • Eubank, R.L.
  • Hsing, Tailen

Abstract

A general notion of canonical correlation is developed that extends the classical multivariate concept to include function-valued random elements X and Y. The approach is based on the polar representation of a particular linear operator defined on reproducing kernel Hilbert spaces corresponding to the random functions X and Y. In this context, canonical correlations and variables are limits of finite-dimensional subproblems thereby providing a seamless transition between Hotelling's original development and infinite-dimensional settings. Several infinite-dimensional treatments of canonical correlations that have been proposed for specific problems are shown to be special cases of this general formulation. We also examine our notion of canonical correlation from a large sample perspective and show that the asymptotic behavior of estimators can be tied to that of estimators from standard, finite-dimensional, multivariate analysis.

Suggested Citation

  • Eubank, R.L. & Hsing, Tailen, 2008. "Canonical correlation for stochastic processes," Stochastic Processes and their Applications, Elsevier, vol. 118(9), pages 1634-1661, September.
    • Handle: RePEc:eee:spapps:v:118:y:2008:i:9:p:1634-1661
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    References listed on IDEAS

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    1. Anderson, T. W., 1999. "Asymptotic Theory for Canonical Correlation Analysis," Journal of Multivariate Analysis, Elsevier, vol. 70(1), pages 1-29, July.
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    2. King, David, 2010. "A note on the cross-covariance operator and on congruence relations for Hilbert space valued stochastic processes," Statistics & Probability Letters, Elsevier, vol. 80(5-6), pages 361-365, March.
    3. 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.
    4. Cho, Haeran & Goude, Yannig & Brossat, Xavier & Yao, Qiwei, 2013. "Modeling and forecasting daily electricity load curves: a hybrid approach," LSE Research Online Documents on Economics 49634, London School of Economics and Political Science, LSE Library.
    5. Hans-Georg Müller & Wenjing Yang, 2010. "Dynamic relations for sparsely sampled Gaussian processes," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 19(1), pages 1-29, May.
    6. S. Barahona & P. Centella & X. Gual-Arnau & M. V. Ibáñez & A. Simó, 2020. "Supervised classification of geometrical objects by integrating currents and functional data analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(3), pages 637-660, September.
    7. 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.
    8. Robert T. Krafty, 2016. "Discriminant Analysis of Time Series in the Presence of Within-Group Spectral Variability," Journal of Time Series Analysis, Wiley Blackwell, vol. 37(4), pages 435-450, July.
    9. 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).
    10. Ruzong Fan & Hong-Bin Fang, 2022. "Stochastic functional linear models and Malliavin calculus," Computational Statistics, Springer, vol. 37(2), pages 591-611, April.

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