An algorithm for locally adaptive bivariate penalized splines

This paper introduces a penalized spline model for bivariate smoothing that locally adjusts to gridded data set affected by noise varying across different areas of the domain. As for all penalized spline methods, the approach requires the definition of suitable penalty terms and the selection of the regularization parameters. In our model, the regularization parameters are chosen anisotropically using a data-driven approach that adapts the roughness amount to the noise level. The proposed approach alternates the construction of univariate penalized spline based on B-spline basis functions along both coordinates, and uses a tensor product structure to capture interactions between the two dimensions. Numerical experiments confirm the efficacy of the approach, the anisotropy of the model, and the ability to locally adapt the amount of regression to different noise levels in different areas. The model is compared with two state-of-the-art smoothers, for which we also provide an original reformulation highlighting their construction as penalized splines.