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：本研究关注于同时具有时间和空间依赖性的数据的非参数回归函数估计问题。在此背景下，我们研究了趋势滤波方法——一种由\cite{mammen1997locally}和\cite{rudin1992nonlinear}提出的非参数估计器。在单变量设定下，我们假设信号具有有界全变差的k阶弱导数，从而允许一般化的平滑度。在多变量场景中，我们研究了类似\cite{padilla2018adaptive}中基于$K$-近邻融合套索的估计器，采用ADMM算法，适用于具有有界变差且满足分段Lipschitz连续准则的信号。通过与下界对齐，验证了估计器的极小极大最优性。通过分析，我们发现了趋势滤波研究中前所未有的独特相变现象。仿真研究和真实数据应用均表明，与现有文献中的成熟技术相比，我们的方法具有更优的性能表现。