New methods 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 ran- dom 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 re- lationship 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.