New method and models in functional data analysis
Functional data analysis (FDA) plays an important role in analyzing function-valued data such as growth curves, medical images and electromagnetic spectrum profiles, etc. Since dimension reduction can be achieved for infinite-dimensional functional data via functional principal component analysis (FPCA), this technique has attracted substantial attention. We develop an easy-to-implement algorithm to perform FPCA and find that this algorithm compares favorably with traditional methods in numerous applications. Knowing how random functions interact is critical to studying mechanisms like gene regulations and event-related brain activation. A new approach is proposed to calibrate dynamical correlations of random functions and we apply this approach to quantify functional connectivity from medical images. Scalar-on-function regression, which is widely used to characterize the relationship between a functional covariate and a scalar response, is an important ingredient of FDA. We propose several new scalar-on-function regression models and investigate their properties from both theoretical and practical perspectives.