Estimation of Functional Response Model by Supervised Functional Principal Component Analysis

Authors: Hyon I. Paek, Sang Rim Kim, Hyon A. Ryu

Abstract:

In functional linear regression, one typical problem is to reduce dimension. Compared with multivariate linear regression, functional linear regression is regarded as an infinite-dimensional case, and the main task is to reduce dimensions of functional response and functional predictors. One common approach is to adapt functional principal component analysis (FPCA) on functional predictors and then use a few leading functional principal components (FPC) to predict the functional model. The leading FPCs estimated by the typical FPCA explain a major variation of the functional predictor, but these leading FPCs may not be mostly correlated with the functional response, so they may not be significant in the prediction for response. In this paper, we propose a supervised FPCA method for a functional response model with FPCs obtained by considering the correlation of the functional response. Our method would have a better prediction accuracy than the typical FPCA method.

Keywords: Supervised, functional principal component analysis, functional response, functional linear regression.

References:

[1] Ramsay, J. O., Silverman, B. W., 2005. Functional Data Analysis (Second edition). New York: Springer.
[2] Ramsay, J. O., Hooker, G., Graves, S., 2009. Functional Data Analysis with R and MATLAB. Use R! Springer New York.
[3] Kokoszka, P., Reimherr, M., 2018. Introduction to functional data analysis. CRC Press.
[4] Ratcliffe, S. J., Heller, G. Z., Leader, L. R., 2002. Functional data analysis with application to periodically stimulated foetal heart rate data. ii: Functional logistic regression. Statistics in medicine 21(8), 1115–1127.
[5] Muller, H.-G., Stadtmuller, U., 2005. Generalized functional linear models. Annals of Statistics, 774–805.
[6] Cardot, H., Faivre, R., Goulard, M., 2003. Functional approaches for predicting land use with the temporal evolution of coarse resolution remote sensing data. Journal of Applied Statistics 30(10), 1185–1199.
[7] Tang, Q.G., Cheng, L.S., 2013. Partial functional linear quantile regression. Science China Press and Springer-Verlag Berlin Heidelberg 57, 1-20.
[8] Bair, E., Hastie, T., Paul, D., Tibshirani, R., 2006. Prediction by supervised principal components. Journal of the American Statistical Association 101(473).
[9] Li, G., Yang, D., Nobel, A. B., Shen, H., 2015. Supervised singular value decomposition and its asymptotic properties. Journal of Multivariate Analysis.
[10] Kawano, S., Fujisawa, H., Takada, T., Shiroishi, T., 2018. Sparse principal component regression for generalized linear models. Computational Statistics and Data Analysis 124, 180-196
[11] Yang, W., Muller, H.-G., 2011. Functional singular component analysis. Journal of the Royal Statistical Society 3(73), 303-324.
[12] Li, G., Shen, H., Huang, J. Z., 2016. Supervised sparse and functional principal component analysis. Journal of Computational and Graphical Statistics 25(3), 859–878.
[13] Zhang, X.Y., Sun, Q., Kong, D., 2023. Supervised principal component regression for functional responses with high dimensional predictors. arXiv:2103.11567v4.