Smoothing splines using compactly supported, positive definite, radial basis functions

In this paper, we develop a fast algorithm for a smoothing spline estimator in multivariate regression. To accomplish this, we employ general concepts associated with roughness penalty methods in conjunction with the theory of radial basis functions and reproducing kernel Hilbert spaces. It is shown that through the use of compactly supported radial basis functions it becomes possible to recover the band structured matrix feature of univariate spline smoothing and thereby obtain a fast computational algorithm. Given n data points in R <Superscript>2</Superscript>, the new algorithm has complexity O(n <Superscript>2</Superscript>) compared to O(n <Superscript>3</Superscript>), the order for the thin plate multivariate smoothing splines. Copyright Springer-Verlag 2012