Smoothing noisy data for irregular regions using penalized bivariate splines on triangulations

The penalized spline method has been widely used for estimating univariate smooth functions based on noisy data. This paper studies its extension to the two-dimensional case. To accommodate the need of handling data distributed on irregular regions, we consider bivariate splines defined on triangulations. Penalty functions based on the second-order derivatives are employed to regularize the spline fit and generalized cross-validation is used to select the penalty parameters. A simulation study shows that the penalized bivariate spline method is competitive to some well-established two-dimensional smoothers. The method is also illustrated using a real dataset on Texas temperature.