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

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 ε_i are random variables satisfying Eε_i=0, Eε_iε_j=σ², i=j, =0, i≠j, t_iεΩ. The z_i are observed. It is desired to estimate u, given z_l, ..., z_n. 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 u_n,m,λ, the restriction to Ω of ũ_n,m,λ, where ũ_n,m,λ is the solution to: Find ũ (in an appropriate space of functions on R^d) 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=( ^m+d−1_d ), provided the t_l, ..., t_n 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 {t_i} ⁿ_i=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

$$\mathop {\min }\limits_\lambda ER(\lambda ) = 0(n^{{{ - 2m} \mathord{\left/{\vphantom {{ - 2m} {(2m + d)}}} \right.\kern-\nulldelimiterspace} {(2m + d)}}} )$$

.

Suppose uεH ^2m(Ω) and certain other conditions are satisfied. Then

$$minER(\lambda ) = 0(n^{{{ - 4m} \mathord{\left/{\vphantom {{ - 4m} {(4m + d)}}} \right.\kern-\nulldelimiterspace} {(4m + d)}}} )$$

.

Research supported by the Office of Naval Research under Grant No. N00014-77-C-0675.