A data-driven approach for estimating functions in a multivariate nonparametric regression model based on B-splines with an application to geoscience

In this paper, we will outline a novel data-driven method for estimating functions in a multivariate nonparametric regression model based on an adaptive knot selection for B-splines. The underlying idea of our approach for selecting knots is to apply the generalized lasso, since the knots of the B-spline basis can be seen as locations of the changes in the derivatives of the function to be estimated. This method was then extended to functions depending on several variables by processing each dimension independently, thus reducing the problem to a univariate setting. The regularization parameters were chosen by means of a criterion based on the Extended Bayesian Information Criterion. The nonparametric estimator was obtained using a multivariate B-spline regression with the corresponding selected knots. Our procedure was validated through numerical experiments by varying the number of observations, the level of noise and the observation sampling to investigate its behavior under such conditions. Our method was applied to two distinct classical geochemical cases. For each different example considered in this paper, our approach performed better than state-of-the-art methods. Our completely data-driven method is implemented in the glober R package which is available on the Comprehensive R Archive Network.