The asymptotic properties of smoothing parameter estimates for smoothing splines are developed. We consider a variety of estimates including Generalized Cross Validation, Generalized Maximum Likelihood, and more generally Type II ML estimates and the properties of the marginal posterior mode. Under the usual Sobolov space frequentist assumptions on the function to be estimated , consistency and...

In this paper, we forecast ground level ozone concentrations over the USA, using past spatially distributed measurements and the functional linear regression model. We employ bivariate splines defined over triangulations of the relevant region of the USA to implement this functional data approach in which random surfaces represent ozone concentrations. We compare the least squares method with p...

Smoothing splines are a popular method for performing nonparametric regression. Most important in the implementation of this method is the choice of the smoothing parameter. This article provides a simulation study of several smoothing parameter selection methods, including two so{called risk estimation methods. To the best of the author's knowledge, the empirical performances of these two risk...

Smoothing noisy data with spline functions is well known in approximation theory. Smoothing splines have been used to deal with the problem of numerical differentiation. In this paper, we extend this method to estimate the fractional derivatives of a smooth signal from its discrete noisy data. We begin with finding a smoothing spline by solving the Tikhonov regularization problem. Then, we prop...

Smoothing of noisy sample covariances is an important component in functional data analysis. We propose a novel covariance smoothing method based on penalized splines and associated software. The proposed method is a bivariate spline smoother that is designed for covariance smoothing and can be used for sparse functional or longitudinal data. We propose a fast algorithm for covariance smoothing...

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...

Experimental data are often in need of smoothing without a pre-determined trend line equation. This manuscript demonstrates how spreadsheet software can be programmed to smooth experimental data via cubic splines. Once the trend lines (splines) have been constructed, it is also simple to interpolate values and calculate derivatives and integrals. The formulas to carry out the calculation are li...

This document reproduces the data analyses presented in Bühlmann and Hothorn (2007). For a description of the theory behind applications shown here we refer to the original manuscript. The results differ slightly due to technical changes or bugfixes in mboost that have been implemented after the paper was printed. Most important, gamboost uses penalized B-splines instead of smoothing splines as...