Functional data usually consist of a sample of functions, with each function observed on a discrete grid. The key idea of functional data analysis is to consider each function as a single, structured object rather than a collection of data points. To represent and investigate functional data, curve registration and functional regression are two important techniques. Curve registration is used to align random curves that display time variations. This procedure, known as functional convex averaging, leads to phase-variance adjusted mean functions. Therefore, compared to a simple averaged mean function, phase-variance adjusted mean function by functional convex averaging is a more accurate representation of the inherent function from which the functional data arise. Several curve registration methods are reviewed in this work, including landmark, self-warping and Bayesian hierarchical curve registration (BHCR). For BHCR, when the number of random curves is large or the sampling grid is intensive, the computational cost increases dramatically. To solve this problem, we introduce an accelerated BHCR algorithm via a predictive process model (PPM), known as PPM-BHCR. Tested by a simulation study and real data, this new method is demonstrated to save large amounts of computing time, without a large sacrifice of accuracy.

Functional regression is used to explore the relationship between the outcome and the predictor, where either or both of them are functional. In this work, several functional regression methods are reviewed according to the function-on-scalar, scalar-on-function and function-on-function categories. Registration is traditionally performed as a data preprocessing step before regression. In this work, we introduce a new method called warped functional regression (WFR), which integrates curve registration and functional regression into one joint model. Therefore, we are able to provide prediction based on an unwarped predictor using this new model. The proposed method is evaluated by simulation studies and demonstrates high accuracy. Several case studies illustrate the key contributions of the proposed method in addressing complex scientific questions.