Functional Regression for Densely Observed Data With Novel Regularization

Smoothness penalty is an efficient regularization method in functional data analysis. However, for a spiky coefficient function which may arise when densely observed spiky functional data are involved, the traditional smoothness penalty could be too strong and lead to an over-smoothed estimate. In this article, we propose a new family of smoothness penalties which are expressed using wavelet coefficients. Some of them are as strong as the traditional smoothness penalty, while others are weaker and more appropriate for a spiky coefficient function. We adaptively select an appropriate penalty from this family by cross-validation. Equipped with these new penalties, we propose new estimation methods for scalar-on-function and function-on-function regression models, respectively. Simulation studies and real data applications illustrate that the new methods perform well for various coefficient functions with different smoothness levels. When the coefficient function is smooth, the new regularization has similar performance as the traditional smoothness penalty, and when the coefficient function is spiky, the new regularization has better performance. The proposed new regression methods have been implemented in the $R$ package FRegSigCom. Supplementary materials for this article are available online.