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Convergence rates of "thin plate" smoothing splines wihen the data are noisy

Preliminary report
  • Grace Wahba
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 757)

Abstract

We study the use of "thin plate" smoothing splines for smoothing noisy d dimensional data. The model is
$$z_i = u(t_i ) + \varepsilon _i ,i = 1,2,...,n,$$
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, Eεiεj2, 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 Hm(Ω) of functions with partial derivatives up to order m in L2(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
$$\frac{1}{n}\sum\limits_{i = 1}^n {(\tilde u(t_i ) - z_i ) + \lambda _i } ,\sum\limits_{1,...,i_m = 1_R d}^d {(\frac{{\partial ^m \tilde u}}{{\partial x_{i_1 } \partial x_{i_2 } ...\partial x_{i_m } }})^2 dx_1 ,dx_2 ,...,dx_d }$$
This minimization problem is known to have a solution for λ>0, m>d/2, n≥M=( d m+d−1 ), provided the tl, ..., tn are "unisolvent". We consider the integrated mean square error
$$R(\lambda ) = \frac{1}{{\left| \Omega \right|}}\int\limits_\Omega {(u_{n,m,\lambda } (t) - u(t))^2 dt,} \left| \Omega \right| = \int\limits_\Omega {dt} ,$$
, and ER(λ), as {ti} i=l n 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 Hm(Ω). Then
$$\mathop {\min }\limits_\lambda ER(\lambda ) = 0(n^{{{ - 2m} \mathord{\left/{\vphantom {{ - 2m} {(2m + d)}}} \right.\kern-\nulldelimiterspace} {(2m + d)}}} )$$
.
Suppose uεH2m(Ω) and certain other conditions are satisfied. Then
$$minER(\lambda ) = 0(n^{{{ - 4m} \mathord{\left/{\vphantom {{ - 4m} {(4m + d)}}} \right.\kern-\nulldelimiterspace} {(4m + d)}}} )$$
.

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Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • Grace Wahba
    • 1
  1. 1.Department of StatisticsUniversity of WisconsinMadisonUSA

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