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On principal components regression with Hilbertian predictors

Author

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  • Ben Jones

    (Cardiff University)

  • Andreas Artemiou

    (Cardiff University)

Abstract

We demonstrate that, in a regression setting with a Hilbertian predictor, a response variable is more likely to be more highly correlated with the leading principal components of the predictor than with trailing ones. This is despite the extraction procedure being unsupervised. Our results are established under the conditional independence model, which includes linear regression and single-index models as special cases, with some assumptions on the regression vector. These results are a generalisation of earlier work which showed that this phenomenon holds for predictors which are real random vectors. A simulation study is used to quantify the phenomenon.

Suggested Citation

  • Ben Jones & Andreas Artemiou, 2020. "On principal components regression with Hilbertian predictors," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(2), pages 627-644, April.
    • Handle: RePEc:spr:aistmt:v:72:y:2020:i:2:d:10.1007_s10463-018-0702-9
    DOI: 10.1007/s10463-018-0702-9
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    References listed on IDEAS

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    1. Peter Hall & You‐Jun Yang, 2010. "Ordering and selecting components in multivariate or functional data linear prediction," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(1), pages 93-110, January.
    2. Artemiou, Andreas & Li, Bing, 2013. "Predictive power of principal components for single-index model and sufficient dimension reduction," Journal of Multivariate Analysis, Elsevier, vol. 119(C), pages 176-184.
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    Cited by:

    1. Jones, Ben & Artemiou, Andreas, 2021. "Revisiting the predictive power of kernel principal components," Statistics & Probability Letters, Elsevier, vol. 171(C).

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