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.

翻译：尽管功能数据越来越容易获得,但灵活估计基本趋势结构的有效方法仍然很少,因此,我们通过扩展趋势过滤法,为功能时间序列和空间数据制定适应当地情况的平滑方法,这是对星标数据的强大非参数趋势估计技术;我们通过采用相邻趋势功能差异的低温,来制定趋势过滤功能的功能版本;通过正态基础扩展,我们简化目标功能,将系数矢量的损耗平方化成组合引信,并开发高效的迭代算法以优化。拟议方法的调控参数是通过交叉验证选定的;我们还考虑将拟议算法扩大到空间功能数据;通过模拟研究和对两个真实的世界数据集的应用来证明拟议方法。