Sometimes, the relationship between an outcome (dependent) variable and the explanatory (independent) variable(s) is not linear. Restricted cubic splines are a way of testing the hypothesis that the relationship is not linear or summarizing a relationship that is too non-linear to be usefully summarized by a linear relationship. Restricted cubic splines are just a transformation of an independe...

It is an important fact that general families of Chebyshev and L-splines can be locally represented, i.e. there exists a basis of B-splines which spans the entire space. We develop a special technique to calculate with 4 order Chebyshev splines of minimum deficiency on nonuniform meshes, which leads to a numerically stable algorithm, at least in case one special Hermite interpolant can be const...

The asymptotic behavior of penalised spline estimators is studied in the univariate case. B-splines are used and a penalty is placed on mth order differences of the coefficients. The number of knots is assumed to converge to ∞ as the sample size increases. We show that penalised splines behave similarly to Nadaraya-Watson kernel estimators with an “equivalent” kernels depending upon m. The equi...

In this paper, local cubic quasi-interpolating splines on non-uniform grids are described. The splines are computed by fast computational algorithms that utilize the relation between splines and cubic interpolation polynomials. These splines provide an efficient tool for real-time signal processing. As an input, they use either clean or noised arbitrarily-spaced samples. Exact estimations of th...

Hex-splines are a novel family of bivariate splines, which are well suited to handle hexagonally sampled data. Similar to classical 1D B-splines, the spline coefficients need to be computed by a prefilter. Unfortunately, the elegant implementation of this prefilter by causal and anti-causal recursive filtering is not applicable for the (non-separable) hex-splines. Therefore, in this paper we in...