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Most-predictive design points for functional data predictors

Author

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  • F. Ferraty
  • P. Hall
  • P. Vieu

Abstract

We suggest a way of reducing the very high dimension of a functional predictor, X, to a low number of dimensions chosen so as to give the best predictive performance. Specifically, if X is observed on a fine grid of design points t 1 ,…, t r , we propose a method for choosing a small subset of these, say t i 1 ,…, t i k , to optimize the prediction of a response variable, Y. The values t i j are referred to as the most predictive design points, or covariates, for a given value of k, and are computed using information contained in a set of independent observations (X i , Y i ) of (X, Y). The algorithm is based on local linear regression, and calculations can be accelerated using linear regression to preselect the design points. Boosting can be employed to further improve the predictive performance. We illustrate the usefulness of our ideas through simulations and examples drawn from chemometrics, and we develop theoretical arguments showing that the methodology can be applied successfully in a range of settings. Copyright 2010, Oxford University Press.

Suggested Citation

  • F. Ferraty & P. Hall & P. Vieu, 2010. "Most-predictive design points for functional data predictors," Biometrika, Biometrika Trust, vol. 97(4), pages 807-824.
  • Handle: RePEc:oup:biomet:v:97:y:2010:i:4:p:807-824
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    File URL: http://hdl.handle.net/10.1093/biomet/asq058
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    Citations

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    Cited by:

    1. Guochang Wang & Jianjun Zhou & Wuqing Wu & Min Chen, 2017. "Robust functional sliced inverse regression," Statistical Papers, Springer, vol. 58(1), pages 227-245, March.
    2. G. Aneiros & P. Vieu, 2016. "Sparse nonparametric model for regression with functional covariate," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 28(4), pages 839-859, October.
    3. Karl Mosler & Pavlo Mozharovskyi, 2017. "Fast DD-classification of functional data," Statistical Papers, Springer, vol. 58(4), pages 1055-1089, December.
    4. Aldo Goia & Philippe Vieu, 2015. "A partitioned Single Functional Index Model," Computational Statistics, Springer, vol. 30(3), pages 673-692, September.
    5. Matsui, Hidetoshi & Konishi, Sadanori, 2011. "Variable selection for functional regression models via the L1 regularization," Computational Statistics & Data Analysis, Elsevier, vol. 55(12), pages 3304-3310, December.
    6. Zhang, Tao & Zhang, Qingzhao & Wang, Qihua, 2014. "Model detection for functional polynomial regression," Computational Statistics & Data Analysis, Elsevier, vol. 70(C), pages 183-197.
    7. Aneiros, Germán & Novo, Silvia & Vieu, Philippe, 2022. "Variable selection in functional regression models: A review," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    8. Park, So Young & Xiao, Luo & Willbur, Jayson D. & Staicu, Ana-Maria & Jumbe, N. L’ntshotsholé, 2018. "A joint design for functional data with application to scheduling ultrasound scans," Computational Statistics & Data Analysis, Elsevier, vol. 122(C), pages 101-114.
    9. James Taylor & Jochen Einbeck, 2013. "Challenging the curse of dimensionality in multivariate local linear regression," Computational Statistics, Springer, vol. 28(3), pages 955-976, June.
    10. Matsui, Hidetoshi, 2014. "Variable and boundary selection for functional data via multiclass logistic regression modeling," Computational Statistics & Data Analysis, Elsevier, vol. 78(C), pages 176-185.
    11. Zhu, Hanbing & Zhang, Riquan & Yu, Zhou & Lian, Heng & Liu, Yanghui, 2019. "Estimation and testing for partially functional linear errors-in-variables models," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 296-314.
    12. Marco Stefanucci & Laura M. Sangalli & Pierpaolo Brutti, 2018. "PCA‐based discrimination of partially observed functional data, with an application to AneuRisk65 data set," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 72(3), pages 246-264, August.
    13. Bodnar, Taras & Okhrin, Ostap & Parolya, Nestor, 2019. "Optimal shrinkage estimator for high-dimensional mean vector," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 63-79.
    14. A. Pini & S. Vantini, 2017. "Interval-wise testing for functional data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 29(2), pages 407-424, April.
    15. Diego Vidaurre & Concha Bielza & Pedro Larrañaga, 2012. "Lazy lasso for local regression," Computational Statistics, Springer, vol. 27(3), pages 531-550, September.
    16. Hao Ji & Hans-Georg Müller, 2017. "Optimal designs for longitudinal and functional data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(3), pages 859-876, June.
    17. Belli, Edoardo, 2022. "Smoothly adaptively centered ridge estimator," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    18. Han Lin Shang, 2014. "Bayesian bandwidth estimation for a functional nonparametric regression model with mixed types of regressors and unknown error density," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 26(3), pages 599-615, September.
    19. Berrendero, José R. & Bueno-Larraz, Beatriz & Cuevas, Antonio, 2019. "An RKHS model for variable selection in functional linear regression," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 25-45.
    20. Pedro Delicado & Philippe Vieu, 2017. "Choosing the most relevant level sets for depicting a sample of densities," Computational Statistics, Springer, vol. 32(3), pages 1083-1113, September.
    21. Geenens, Gery, 2015. "Moments, errors, asymptotic normality and large deviation principle in nonparametric functional regression," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 369-377.
    22. Fraiman, Ricardo & Gimenez, Yanina & Svarc, Marcela, 2016. "Feature selection for functional data," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 191-208.
    23. Philip T. Reiss & Jeff Goldsmith & Han Lin Shang & R. Todd Ogden, 2017. "Methods for Scalar-on-Function Regression," International Statistical Review, International Statistical Institute, vol. 85(2), pages 228-249, August.
    24. Germán Aneiros & Philippe Vieu, 2015. "Partial linear modelling with multi-functional covariates," Computational Statistics, Springer, vol. 30(3), pages 647-671, September.

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