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：本研究聚焦于针对同时具有时间与空间依赖性的数据设计非参数回归函数的估计方法。在此框架下，我们深入研究了由\cite{mammen1997locally}和\cite{rudin1992nonlinear}提出的非参数估计器——趋势滤波。针对单变量场景，我们假设所考虑的信号具有k阶弱导数且总变差有界，从而允许一般化的光滑度水平。在多变量场景中，我们研究了\cite{padilla2018adaptive}提出的基于$K$-近邻融合套索估计器，采用ADMM算法，该估计器适用于满足分段Lipschitz连续准则的有界变差信号。通过与下界对齐，我们的估计器的极小极大最优性得到了验证。分析中揭示了一种先前在趋势滤波研究中未涉及的唯一相变现象。仿真实验和实际数据应用均表明，与现有文献中的成熟技术相比，我们的方法具有更优越的性能。