IDEAS home Printed from
   My bibliography  Save this article

Ordering and selecting components in multivariate or functional data linear prediction


  • Peter Hall
  • You-Jun Yang


The problem of component choice in regression-based prediction has a long history. The main cases where important choices must be made are functional data analysis, and problems in which the explanatory variables are relatively high dimensional vectors. Indeed, principal component analysis has become the basis for methods for functional linear regression. In this context the number of components can also be interpreted as a smoothing parameter, and so the viewpoint is a little different from that for standard linear regression. However, arguments for and against conventional component choice methods are relevant to both settings and have received significant recent attention. We give a theoretical argument, which is applicable in a wide variety of settings, justifying the conventional approach. Although our result is of minimax type, it is not asymptotic in nature; it holds for each sample size. Motivated by the insight that is gained from this analysis, we give theoretical and numerical justification for cross-validation choice of the number of components that is used for prediction. In particular we show that cross-validation leads to asymptotic minimization of mean summed squared error, in settings which include functional data analysis. Copyright (c) 2010 Royal Statistical Society.

Suggested Citation

  • 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.
  • Handle: RePEc:bla:jorssb:v:72:y:2010:i:1:p:93-110

    Download full text from publisher

    File URL:
    File Function: link to full text
    Download Restriction: Access to full text is restricted to subscribers.

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    1. Boente, Graciela & Fraiman, Ricardo, 2000. "Kernel-based functional principal components," Statistics & Probability Letters, Elsevier, vol. 48(4), pages 335-345, July.
    2. He, Guozhong & Müller, Hans-Georg & Wang, Jane-Ling, 2003. "Functional canonical analysis for square integrable stochastic processes," Journal of Multivariate Analysis, Elsevier, vol. 85(1), pages 54-77, April.
    3. Peter Hall & Mohammad Hosseini-Nasab, 2006. "On properties of functional principal components analysis," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(1), pages 109-126.
    4. Maruyama, Yuzo, 1998. "A Unified and Broadened Class of Admissible Minimax Estimators of a Multivariate Normal Mean," Journal of Multivariate Analysis, Elsevier, vol. 64(2), pages 196-205, February.
    5. Hervé Cardot, 2007. "Conditional Functional Principal Components Analysis," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 34(2), pages 317-335.
    6. Fang Yao & Thomas C. M. Lee, 2006. "Penalized spline models for functional principal component analysis," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(1), pages 3-25.
    Full references (including those not matched with items on IDEAS)


    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.

    Cited by:

    1. Lee, Eun Ryung & Park, Byeong U., 2012. "Sparse estimation in functional linear regression," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 1-17.
    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.

    More about this item


    Access and download statistics


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:bla:jorssb:v:72:y:2010:i:1:p:93-110. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Wiley-Blackwell Digital Licensing) or (Christopher F. Baum). General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.