Temporal-spatial model via Trend Filtering

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.