Spatio-temporal model via Locally Adaptive Regression Splines

This paper focuses on the estimation of a non-parametric regression function in the presence of data with spatio-temporal dependencies. In such a context, we study Locally Adaptive Regression Splines, a nonparametric estimator introduced by Mammen and Van De Geer (1997) and Rudin et al. (1992). To the best of our knowledge, this estimator has not previously been examined in a similar context. 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 setting, we study a variant of the K-Nearest Neighbor fused lasso estimator studied by Padilla et al. (2018a). For this case, the function is required to have bounded variation and satisfy a property that extends a piecewise Lipschitz continuity criterion, or the function is assumed to be piecewise Lipschitz. We develop an ADMM algorithm for practical computation. By aligning with lower bounds, the minimax optimality of our univariate and multivariate estimators is shown. A unique phase transition phenomenon, previously unprecedented in Locally Adaptive Regression Splines 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.