Functional quadratic regression. (English) Zbl 1183.62113

Yao, Fang; Müller, Hans-Georg

Biometrika 97, No. 1, 49-64 (2010).

Summary: We extend the common linear functional regression model to the case where the dependency of a scalar response on a functional predictor is of polynomial rather than linear nature. Focusing on the quadratic case, we demonstrate the usefulness of the polynomial functional regression model, which encompasses linear functional regression as a special case. Our approach works under mild conditions for the case of densely spaced observations and also can be extended to the important practical situation where the functional predictors are derived from sparse and irregular measurements, as is the case in many longitudinal studies.
A key observation is the equivalence of the functional polynomial model with a regression model that is a polynomial of the same order in the functional principal component scores of the predictor processes. Theoretical analysis as well as practical implementations are based on this equivalence and on basis representations of predictor processes. We also obtain an explicit representation of the regression surface that defines quadratic functional regression and provide functional asymptotic results for an increasing number of model components as the number of subjects in the study increases. The improvements that can be gained by adopting quadratic as compared to linear functional regression are illustrated with a case study that includes absorption spectra as functional predictors.

MSC:

62J02	General nonlinear regression
62H25	Factor analysis and principal components; correspondence analysis
62P10	Applications of statistics to biology and medical sciences; meta analysis
62J05	Linear regression; mixed models

Keywords:

absorption spectra; asymptotics; functional data analysis; polynomial regression; prediction; principal component