Locally Adaptive Smoothing for Functional Data

Despite increasing accessibility to function data, effective methods for flexibly estimating underlying trend structures are still scarce. We thereby develop locally adaptive smoothing methods for both functional time series and spatial data by extending trend filtering, a powerful nonparametric trend estimation technique for scalar data. We formulate the functional version of trend filtering by introducing $L_2$-norm of the differences of adjacent trend functions. Through orthonormal basis expansion, we simplify the objective function to squared loss for coefficient vectors with grouped fused lasso penalty, and develop an efficient iteration algorithm for optimization. The tuning parameter in the proposed method is selected via cross validation. We also consider an extension of the proposed algorithm to spatial functional data. The proposed methods are demonstrated by simulation studies and an application to two real world datasets.