Shape-preserving, multiscale fitting of univariate data by cubic L1 smoothing splines

Bivariate cubic L1 smoothing splines are introduced. The coefficients of a cubic L1 smoothing spline are calculated by minimizing the weighted sum of the L1 norms of second derivatives of the spline and the 1 norm of the residuals of the data-fitting equations. Cubic L1 smoothing splines are compared with conventional cubic smoothing splines based on the L2 and 2 norms. Computational results for fitting a challenging data set consisting of discontinuously connected flat and quadratic areas by Csmooth Sibson-element splines on a tensor-product grid are presented. In these computational results, the cubic L1 smoothing splines preserve the shape of the data while cubic L2 smoothing splines do not.