This research focuses on the estimation of a non-parametric regression function designed for data with simultaneous time and space dependencies. In such a context, we study the Trend Filtering, a nonparametric estimator introduced by \cite{mammen1997locally} and \cite{rudin1992nonlinear}. For univariate settings, the signals we consider are assumed to have a kth weak derivative with bounded total variation, allowing for a general degree of smoothness. In the multivariate scenario, we study a $K$-Nearest Neighbor fused lasso estimator as in \cite{padilla2018adaptive}, employing an ADMM algorithm, suitable for signals with bounded variation that adhere to a piecewise Lipschitz continuity criterion. By aligning with lower bounds, the minimax optimality of our estimators is validated. A unique phase transition phenomenon, previously uncharted in Trend Filtering studies, emerges through our analysis. Both Simulation studies and real data applications underscore the superior performance of our method when compared with established techniques in the existing literature.

翻译：本研究聚焦于为具有时间和空间同步依赖性的数据设计非参数回归函数的估计方法。在此背景下，我们研究了由文献\cite{mammen1997locally}和\cite{rudin1992nonlinear}提出的非参数估计量——趋势滤波。对于单变量情形，假设所考虑的信号具有k阶弱导数且总变差有界，从而允许一定程度的平滑性。在多变量情形中，我们参照文献\cite{padilla2018adaptive}研究了基于K近邻融合套索的估计量，并采用ADMM算法，该估计量适用于满足分段Lipschitz连续性准则的有界变差信号。通过与下界对齐，验证了所提估计量的极小化最优性。分析中揭示了一种在趋势滤波研究中未曾报道的独特相变现象。模拟实验与真实数据应用均表明，与已有文献中的经典方法相比，本方法展现出更优的性能。