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Shape-Preserving, Multi-Scale Fitting of Bivariate Data by Cubic L1 Smoothing Splines



David E. Gilsinn, J E. Lavery


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 ofsecond derivatives of the spline and the l1 norm of the residuals of the data-fitting equations. Cubic L1 smoothing splines are compared with conventional cubic smoothingsplines based on the L2 and l2 norms. Computational results for fitting a challengingdata set consisting of discontinuously connected flat and quadratic areas by C1-smooth 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.
Journal of Approximation Theory


bivariate approximation, least absolute regression, least squares regression, sibson elements, smoothing splines, tensor-product grids


Gilsinn, D. and Lavery, J. (2002), Shape-Preserving, Multi-Scale Fitting of Bivariate Data by Cubic L<sub>1</sub> Smoothing Splines, Journal of Approximation Theory, [online], (Accessed May 23, 2024)


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Created May 10, 2002, Updated June 2, 2021