Moving least squares

Here is a 2D example. The circles are the samples and the polygon is a linear interpolation. The blue curve is a smooth approximation of order 3.

Consider a function ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }$ and a set of sample points ${\displaystyle S=\{(x_{i},f_{i})|f(x_{i})=f_{i}\}}$. Then, the moving least square approximation of degree ${\displaystyle m}$ at the point ${\displaystyle x}$ is ${\displaystyle {\tilde {p}}(x)}$ where ${\displaystyle {\tilde {p}}}$ minimizes the weighted least-square error

${\displaystyle \sum _{i\in I}(p(x_{i})-f_{i})^{2}\theta (\|x-x_{i}\|)}$

over all polynomials ${\displaystyle p}$ of degree ${\displaystyle m}$ in ${\displaystyle \mathbb {R} ^{n}}$. ${\displaystyle \theta (s)}$ is the weight and it tends to zero as ${\displaystyle s\to \infty }$.

In the example ${\displaystyle \theta (s)=e^{-s^{2}}}$. The smooth interpolator of "order 3" is a quadratic interpolator.

• Local regression
• Diffuse element method
• Moving average

• The approximation power of moving least squares David Levin, Mathematics of Computation, Volume 67, 1517-1531, 1998
• Moving least squares response surface approximation: Formulation and metal forming applications Piotr Breitkopf; Hakim Naceur; Alain Rassineux; Pierre Villon, Computers and Structures, Volume 83, 17-18, 2005.
• Generalizing the finite element method: diffuse approximation and diffuse elements, B Nayroles, G Touzot. Pierre Villon, P, Computational Mechanics Volume 10, pp 307-318, 1992

• An As-Short-As-Possible Introduction to the Least Squares, Weighted Least Squares and Moving Least Squares Methods for Scattered Data Approximation and Interpolation