Abstract
Given data $z_i = g(t_i ) + \varepsilon _i , 1 \leqq i \leqq n$, where g is the unknown function, the $t_i $ are known d-dimensional variables in a domain $\Omega $, and the $\varepsilon _i $ are i.i.d. random errors, the smoothing spline estimate gnu is defined to be the minimizes over h of $n^{ - 1} \Sigma (z_i - h(t_i ))^2 + \lambda J_m (h)$, where $\lambda > 0$ is a smoothing parameter and $J_m (h)$ is the sum of the integrals over $\Omega $ of the squares of all the mth order derivatives of h. Under the assumptions that $\Omega $ is bounded and has a smooth boundary, $\lambda \to 0$ appropriately, and the $t_i $ become dense in $\Omega $ as $n \to \infty $, bounds on the rate of convergence of the expected square of pth order Sobolev norm ($L_2$ norm of pth derivatives) are obtained. These extend known results in the one-dimensional case. The method of proof utilizes an approximation to the smoothing spline based on a Green's function for a linear elliptic boundary value problem. Using eigenvalue approximation techniques, these rate of convergence results are extended to fairly arbitrary domains including $\Omega = \mathbb{R}^d $, but only for the case $p = 0$, i.e. $L_2$ norm.
Keywords
MathematicsSmoothingSobolev spaceCombinatoricsBounded functionSpline (mechanical)Rate of convergenceNorm (philosophy)Uniform normOmegaLambdaMathematical analysis
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Publication Info
- Year
- 1984
- Type
- article
- Volume
- 21
- Issue
- 4
- Pages
- 789-813
- Citations
- 109
- Access
- Closed
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Cite This
Dennis D. Cox
(1984).
Multivariate Smoothing Spline Functions.
SIAM Journal on Numerical Analysis
, 21
(4)
, 789-813.
https://doi.org/10.1137/0721053
Identifiers
- DOI
- 10.1137/0721053