Temporal-spatial model via Trend Filtering

TL;DR

This paper develops a novel trend filtering approach for non-parametric regression with spatial-temporal data, demonstrating minimax optimality and revealing a new phase transition phenomenon, with superior empirical performance.

Contribution

Introduces a new spatial-temporal trend filtering model with theoretical optimality and uncovers a novel phase transition in the estimation process.

Findings

Method achieves minimax optimality.

Reveals a new phase transition phenomenon.

Outperforms existing techniques in simulations and real data.

Abstract

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.