Estimation of functional regression model via functional dimension reduction

• Published in: Journal of computational and applied mathematics Vol. 379; p. 112948
• Main Authors: Wang, Guochang, Zhang, Baoxue, Liao, Wenhui, Xie, Baojian
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.12.2020
• Subjects: Functional data analysis; Functional linear model; Functional sufficient dimension reduction; Functional linear model; Functional sufficient dimension reduction; Functional data analysis
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2020.112948

The functional linear regression model corresponding to a scalar response and a functional predictor is becoming increasingly common. Since the predictor is infinite dimensional, some form of dimension reduction is essential. There are two popular dimension reduction methods such as expanding the functional predictor or regression parameter function on the functional principal component basis or on a fixed bases (such as B-spline, Wavelet). In the present paper, we estimate the functional linear model by using the functional sufficient dimension reduction (FSDR) basis. Compared to the existing methods, the proposed method is appealing because the FSDR basis is related to both the functional predictor and the response variable, whereas the functional principal component basis is only related to the functional predictor and the fixed basis (B-spline, Wavelet) is independent from both the functional predictor and the response variable. Our techniques involve methods for giving a new expansion for the predictor, for giving a specific expression for the regression parameter function, for estimating the FSDR space by a new method and some asymptotical properties about the regression parameter function and the prediction for the test samples. Numerical studies, including both simulation studies and applications on real-life data, are presented to demonstrate the accuracy of the proposed method.