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BIT Numerical Mathematics

, Volume 55, Issue 3, pp 781–796 | Cite as

Between moving least-squares and moving least-\(\ell _1\)

  • David LevinEmail author
Article

Abstract

Given function values at scattered points in \({\mathbb {R}}^d\), possibly with noise, one of the ways of generating approximation to the function is by the method of moving least-squares (MLS). The method consists of computing local polynomials which approximate the data in a locally weighted least-squares sense. The resulting approximation is smooth, and is well approximating if the underlying function is smooth. Yet, as any least-squares based method it is quite sensitive to outliers in the data. It is well known that least-\(\ell _1\) approximations are not sensitive to outliers. However, due to the nature of the \(\ell _1\) norm, using it in the framework of a “moving” approximation will not give a smooth, or even a continuous approximation. This paper suggests an error measure which is between the \(\ell _1\) and the \(\ell _2\) norms, with the advantages of both. Namely, yielding smooth approximations which are not too sensitive to outliers. A fast iterative method for computing the approximation is demonstrated and analyzed. It is shown that for a scattered data taken from a smooth function, with few outliers, the new approximation gives an \(O(h)\) approximation error to the function.

Keywords

Moving least-squares Outliers Multivariate approximation 

Mathematics Subject Classification

65D15 33F05 65D10 65D07 

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.School of Mathematical SciencesTel-Aviv UniversityTel AvivIsrael

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