  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 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 EBIC. 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 and the level of noise to investigate its robustness. The
influence of observation sampling was also assessed and our method was applied
to a chemical system commonly used in geoscience. For each different framework
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 (CRAN).
