Abstract
In functional linear regression, one conventional approach is to first perform functional principal component analysis (FPCA) on the functional predictor and then use the first few leading functional principal component (FPC) scores to predict the response variable. The leading FPCs estimated by the conventional FPCA stand for the major source of variation of the functional predictor, but these leading FPCs may not be mostly correlated with the response variable, so the prediction accuracy of the functional linear regression model may not be optimal. In this paper, we propose a supervised version of FPCA by considering the correlation of the functional predictor and response variable. It can automatically estimate leading FPCs, which represent the major source of variation of the functional predictor and are simultaneously correlated with the response variable. Our supervised FPCA method is demonstrated to have a better prediction accuracy than the conventional FPCA method by using one real application on electroencephalography (EEG) data and three carefully designed simulation studies.
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Acknowledgements
The authors would like to thank the Editor, the Associated Editor, and two reviewers for their very constructive suggestions and comments on the manuscript. They are extremely helpful for us to improve our work. The authors are also very grateful to Prof. Gen Li and Prof. Xiaoyan Leng for kindly providing us with their data and their computing codes. This research was supported by Nie’s Postgraduate Scholarship-Doctorial (PGS-D) from the Natural Sciences and Engineering Research Council of Canada (NSERC), and the NSERC Discovery grants of Wang and Cao.
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Supplementary Document: This file contains some additional figures for simulation studies in Section 5, two additional simulation studies with an arbitrary coefficient function and a large number of FPCs related to the outcome, respectively. We also include another real data application analyzing the time course yeast gene expression data. (supplementary.pdf, PDF file), ESM 1 (R 3KB), ESM 2 (PDF 377KB)
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Nie, Y., Wang, L., Liu, B. et al. Supervised functional principal component analysis. Stat Comput 28, 713–723 (2018). https://doi.org/10.1007/s11222-017-9758-2
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DOI: https://doi.org/10.1007/s11222-017-9758-2