Abstract
We study the use of "thin plate" smoothing splines for smoothing noisy d dimensional data. The model is
where u is a real valued function on a closed, bounded subset Ω of Euclidean d-space and the εi are random variables satisfying Eεi=0, Eεiεj=σ2, i=j, =0, i≠j, tiεΩ. The zi are observed. It is desired to estimate u, given zl, ..., zn. u is only assumed to be "smooth", more precisely we assume that u is in the Sobolev space H m(Ω) of functions with partial derivatives up to order m in L 2(Ω), with m>d/2. u is estimated by un,m,λ, the restriction to Ω of ũn,m,λ, where ũn,m,λ is the solution to: Find ũ (in an appropriate space of functions on Rd) to minimize
This minimization problem is known to have a solution for λ>0, m>d/2, n≥M=( m+d−1d ), provided the tl, ..., tn are "unisolvent". We consider the integrated mean square error
, and ER(λ), as {ti} ni=l become dense in Ω. An estimate of λ which asymptotically minimizes ER(λ) can be obtained by the method of generalized cross-validation. In this paper we give plausible arguments and numerical evidence supporting the following conjectures:
Suppose u ε H m(Ω). Then
.
Suppose uεH 2m(Ω) and certain other conditions are satisfied. Then
.
Research supported by the Office of Naval Research under Grant No. N00014-77-C-0675.
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Wahba, G. (1979). Convergence rates of "thin plate" smoothing splines wihen the data are noisy. In: Gasser, T., Rosenblatt, M. (eds) Smoothing Techniques for Curve Estimation. Lecture Notes in Mathematics, vol 757. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0098499
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DOI: https://doi.org/10.1007/BFb0098499
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