On Functional Principal Component Regression
Functional data analysis has been increasingly used in biomedical studies, where the basic unit of measurement is a function, curve, or image. For example, in mobile health (mHealth) studies, wearable sensors collect high-resolution trajectories of physiological and behavioral signals over time. Functional linear regression models are useful tools for quantifying the association between functional covariates and scalar/functional responses, where a popular approach is via functional principal component analysis. In this talk, we consider inference for functional principal component regression, using scalar-on-function regression as an illustrating example. We will first briefly review challenges arising from choosing the number of principal components and demonstrate that ordering the components by eigenvalues might not be the best choice for regression problems. We then present an associate-variation index and propose to use it to rank the principal components instead. Both asymptotic and finite sample properties of the proposed procedure were investigated. It is shown more robust to the choice of tuning parameters that determine the number of principal components than standard functional principal component regression. The proposed approach was also applied to the Objective Physical Activity and Cardiovascular Health Study and a diffusing tensor imaging study.