Smoothing noisy data with spline functions

@article{Craven1978SmoothingND,
  title={Smoothing noisy data with spline functions},
  author={Peter Craven and Grace Wahba},
  journal={Numerische Mathematik},
  year={1978},
  volume={31},
  pages={377-403}
}
SummarySmoothing 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 modelyi(ti)+εi,i=1, 2, ...,n,ti∈[0, 1], whereg∈W2(m)={f:f,f′, ...,f(m−1) abs. cont.,f(m)∈ℒ2[0,1]}, and the {εi} are random errors withEεi=0,Eεiεj=σ2δij. The error variance… 

AnL1 smoothing spline algorithm with cross validation

TLDR
The combined LAD-CV smoothing spline algorithm is a continuation scheme in λ↘0 taken on the above SQPs parametrized inλ, with the optimal smoothing parameter taken to be that value ofλ at which theCV(λ) score first begins to increase.

LOWLAD: a locally weightedL1 smoothing spline algorithm with cross validated choice of smoothing parameters

TLDR
A locally weighted, LAD smoothing spline based smoother is suggested, and preliminary results will be discussed, and an algorithm for its computation is proposed.

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

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

Adaptive Spline Estimates for Nonparametric Regression Models

where the are independent standart Gaussian random variables, while the regressors x are deterministic and equally spaced, i.e., x (2i-1)/(2n). We suppose that the unknown function f(.) is

Two constrained deconvolution methods using spline functions

  • D. Verotta
  • Mathematics
    Journal of Pharmacokinetics and Biopharmaceutics
  • 2005
TLDR
Two new methods to solve the following estimation problem of the response of a system to a knowninput and an unknowninput, recasting the problem in terms of inequality-constrained linear regression.

Fitting noisy data using cross-validated cubic smoothing splines

An algorithm is described for approximating an unknown function f(x), given many function values containing random noise. The approximation constructed is a cubic spline g(x) with sufficient basis

The Discrete k-Functional and Spline Smoothing of Noisy Data

TLDR
A discrete analog of Peetre's K-functional is defined and it is shown how to use $k_m $ and its connection to the mth order modulus of continuity to assess the smoothness of f and to choose a good smoothing spline approximation to f and some of its derivatives.

Smoothing splines approximation using Hilbert curve basis selection

TLDR
An ef-ficient algorithm that is adaptive to the unknown probability density function of the predictors and has the same convergence rate as the full-basis estimator when q is roughly at the order of O.

Approximate Smoothing Spline Methods for Large DataSets in the Binary

TLDR
A randomized version of the GACV function is proposed, which is numerically stable and uses a clustering algorithm to choose a set of basis functions with which to approximate the exact additive smoothing spline estimate, which has a basis function for every data point.
...

References

SHOWING 1-10 OF 30 REFERENCES

Improper Priors, Spline Smoothing and the Problem of Guarding Against Model Errors in Regression

SUMMARY Spline and generalized spline smoothing is shown to be equivalent to Bayesian estimation with a partially improper prior. This result supports the idea that spline smoothing is a natural

Error bounds for polynomial spline interpolation

New upper and lower bounds for the L2 and L- norms of derivatives of the error in polynomial spline interpolation are derived. These results improve corresponding results of Ahlberg, Nilson, and

Practical Approximate Solutions to Linear Operator Equations When the Data are Noisy

  • G. Wahba
  • Mathematics, Computer Science
  • 1977
TLDR
It is shown that the weighted cross-validation estimate of $\hat \lambda $ estimates the value of $\lambda $ which minimizes $({1 / n) E\sum\nolimits_{j = 1}^n {[(\mathcal{K}f_{n,\lambda } )(t_j ) - (\mathcal(K)f)(t-j )]} ^2 $ .

A Note on the Convergence of Interpolatory Cubic Splines

It is shown that if $x \in C^4 [a,b]$ is approximated by a natural cubic spline, then the error is $O(h^4 )$ in a closed interval which is asymptotic to $[a,b]$ as h, the maximum interval length,

Convergence Properties of the Method of Regularization for Noisy Linear Operation Equations.

Abstract : Convergence properties of the method of regulation for finding approximate solutions to the linear operator equation g = Kf are found when g is contaminated by noise. If f belongs to (H

Numerical differentiation procedures for non-exact data

AbstractThe numerical differentiation of data divides naturally into two distinct problems:(i)the differentiation of exact data, and(ii)the differentiation of non-exact (experimental) data. In this

Spline Functions in Data Analysis.

Abstract : This paper discusses the approximation of non-exact data by smooth functions. It is shown that optimal approximations for a large class of criteria are spline functions, and that a

A Time Series Approach To Numerical Differentiation

TLDR
A parametric family of models is introduced, and estimation of the parameters is discussed; the theory of time series analysis provides useful tools for discussing such a model.

SPLINE FUNCTIONS AND THE PROBLEM OF GRADUATION.

  • I. J. Schoenberg
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1964
TLDR
The aim of this note is to extend some of the recent work on spline interpolation so as to include also a solution of the problem of graduation of data and the qualitative aspects of the new method are described.