We study spline fitting with a roughness penalty that adapts to spatial heterogeneity in the regression function. Our estimates are pth degree piecewise polynomials with p− 1 continuous derivatives. A large and fixed number of knots is used and smoothing is achieved by putting a quadratic penalty on the jumps of the pth derivative at the knots. To be spatially adaptive, the logarithm of the pen...

In the first part of this paper we apply a saddle point theorem from convex analysis to show that various constrained minimization problems are equivalent to the problem of smoothing by spline functions. In particular, we show that near-interpolants are smoothing splines with weights that arise as Lagrange multipliers corresponding to the constraints in the problem of near-interpolation. In the...

In this paper we focus on nonparametric estimation of a monotone regression function using general spline smoothing techniques. We propose to construct a new class of monotone splines by using some tools that have been recently developed in the context of image warping to compute smooth and bijective deformations. Our idea is to adapt these tools to construct a monotone spline that can be used ...

Smoothing splines provide a powerful and flexible means for nonparametric estimation inference. With cubic time complexity, fitting smoothing spline models to large data is computationally prohibitive. In this paper, we use the theoretical optimal eigenspace derive low-rank approximation of estimates. We develop method approximate eigensystem when it unknown error bounds The proposed methods ar...

Topographic maps that illustrate variations in the density of different neuronal sub-types across the retina are valuable tools for understanding the adaptive significance of retinal specialisations in different species of vertebrates. To date, such maps have been created from raw count data that have been subjected to only limited analysis (linear interpolation) and, in many cases, have been p...

This paper introduces a new method for curve smoothing which is specific for isobathymetric line generalization. Isobaths are modeled with B-spline curves that can maintain the smooth feature of the curves. Smoothing is performed by keeping the curve points always on the deepest side of the original curve so that the result is consistent with the safety constraint. It is done by using a snake m...