Smoothing noisy data with spline functions
 Peter Craven,
 Grace Wahba
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Smoothing splines are well known to provide nice curves which smooth discrete, noisy data. We obtain a practical, effective method for estimating the optimum amount of smoothing from the data. Derivatives can be estimated from the data by differentiating the resulting (nearly) optimally smoothed spline.
We consider the modely _{ i }(t _{ i })+ε_{ i },i=1, 2, ...,n,t _{i}∈[0, 1], whereg∈W _{2} ^{(m)} ={f:f,f′, ...,f ^{(m−1)} abs. cont.,f ^{(m)}∈ℒ_{2}[0,1]}, and the {ε_{ i }} are random errors withEε_{ i }=0,Eε_{ i }ε_{ j }=σ^{2}δ_{ ij }. The error variance σ^{2} may be unknown. As an estimate ofg we take the solutiong _{n, λ} to the problem: Findf∈W _{2} ^{(m)} to minimize \(\frac{1}{n}\sum\limits_{j = 1}^n {(f(t_j )  y_j )^2 + \lambda \int\limits_0^1 {(f^{(m)} (u))^2 du} }\) . The functiong _{n, λ} is a smoothing polynomial spline of degree 2m−1. The parameter λ controls the tradeoff between the “roughness” of the solution, as measured by \(\int\limits_0^1 {[f^{(m)} (u)]^2 du}\) , and the infidelity to the data as measured by \(\frac{1}{n}\sum\limits_{j = 1}^n {(f(t_j )  y_j )^2 }\) , and so governs the average square errorR(λ; g)=R(λ) defined by $$R(\lambda ) = \frac{1}{n}\sum\limits_{j = 1}^n {(g_{n,\lambda } (t_j )  g(t_j ))^2 }$$ . We provide an estimate \(\hat \lambda\) , called the generalized crossvalidation estimate, for the minimizer ofR(λ). The estimate \(\hat \lambda\) is the minimizer ofV(λ) defined by \(V(\lambda ) = \frac{1}{n}\parallel (I  A(\lambda ))y\parallel ^2 /\left[ {\frac{1}{n}{\text{Trace(}}I  A(\lambda ))} \right]^2\) , wherey=(y _{1}, ...,y _{n})^{t} andA(λ) is then×n matrix satisfying(g _{n, λ} (t _{1}), ...,g _{n, λ} (t _{n}))^{t}=A (λ) y. We prove that there exist a sequence of minimizers \(\tilde \lambda = \tilde \lambda (n)\) ofEV(λ), such that as the (regular) mesh{t _{i}} _{i=1} ^{n} becomes finer, \(\mathop {\lim }\limits_{n \to \infty } ER(\tilde \lambda )/\mathop {\min }\limits_\lambda ER(\lambda ) \downarrow 1\) . A Monte Carlo experiment with several smoothg's was tried withm=2,n=50 and several values of σ^{2}, and typical values of \(R(\hat \lambda )/\mathop {\min }\limits_\lambda R(\lambda )\) were found to be in the range 1.01–1.4. The derivativeg′ ofg can be estimated by \(g'_{n,\hat \lambda } (t)\) . In the Monte Carlo examples tried, the minimizer of \(R_D (\lambda ) = \frac{1}{n}\sum\limits_{j = 1}^n {(g'_{n,\lambda } (t_j )  } g'(t_j ))\) tended to be close to the minimizer ofR(λ), so that \(\hat \lambda\) was also a good value of the smoothing parameter for estimating the derivative.
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 Title
 Smoothing noisy data with spline functions
 Journal

Numerische Mathematik
Volume 31, Issue 4 , pp 377403
 Cover Date
 19781201
 DOI
 10.1007/BF01404567
 Print ISSN
 0029599X
 Online ISSN
 09453245
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 MOS:65D10
 CR:5.17
 MOS:65D25
 Industry Sectors
 Authors

 Peter Craven ^{(1)}
 Grace Wahba ^{(2)}
 Author Affiliations

 1. The Computer Laboratory, The University of Liverpool, Liverpool, England
 2. Department of Statistics, University of Wisconsin, 53706, Madison, WI, USA