Adaptive Ridge-Penalized Functional Local Linear Regression

Abstract: We introduce an original method of multidimensional ridge penalization in functional local linear regressions. The nonparametric regression of functional data is extended from its multivariate counterpart, and is known to be sensitive to the choice of $J$, where $J$ is the dimension of the projection subspace of the data. Under multivariate setting, a roughness penalty is helpful for variance reduction. However, among the limited works covering roughness penalty under the functional setting, most only use a single scalar for tuning. Our new approach proposes a class of data-adaptive ridge penalties, meaning that the model automatically adjusts the structure of the penalty according to the data sets. This structure has $J$ free parameters and enables a quadratic programming search for optimal tuning parameters that minimize the estimated mean squared error (MSE) of prediction, and is capable of applying different roughness penalty levels to each of the $J$ basis. The strength of the method in prediction accuracy and variance reduction with finite data is demonstrated through multiple simulation scenarios and two real-data examples. Its asymptotic performance is proved and compared to the unpenalized functional local linear regressions.