Moving least squares

Short description: Method for reconstructing continuous functions

Moving least squares is a method of reconstructing continuous functions from a set of unorganized point samples via the calculation of a weighted least squares measure biased towards the region around the point at which the reconstructed value is requested.

In computer graphics, the moving least squares method is useful for reconstructing a surface from a set of points. Often it is used to create a 3D surface from a point cloud through either downsampling or upsampling.

In numerical analysis to handle contributions of geometry where it is difficult to obtain discretizations, the moving least squares methods have also been used and generalized to solve PDEs on curved surfaces and other geometries.[1][2][3] This includes numerical methods developed for curved surfaces for solving scalar parabolic PDEs[1][3] and vector-valued hydrodynamic PDEs.[2]

In machine learning, moving least squares methods have also been used to develop model classes and learning methods. This includes function regression methods[4] and neural network function and operator regression approaches, such as GMLS-Nets.[5]

## Contents

* 1 Definition
* 2 See also
* 3 References
* 4 External links

## Definition

Here is a 1D example. The circles are the sample points and the polygon is a linear interpolation. The blue curve is a smooth approximation of order 3.

Consider a function f:ℝn->ℝ and a set of sample points S={(xi,fi)|f(xi)=fi}. Then, the moving least square approximation of degree m at the point x is p~(x) where p~ minimizes the weighted least-square error

∑i∈I(p(xi)−fi)2θ(‖x−xi‖)

over all polynomials p of degree m in ℝn. θ(s) is the weight and it tends to zero as s->∞.

In the example θ(s)=e−s2. The smooth interpolator of "order 3" is a quadratic interpolator.

## See also

* Local regression
* Diffuse element method
* Moving average

## References

1. ↑ 1.0 1.1 Liang, Jian; Zhao, Hongkai (January 2013). "Solving Partial Differential Equations on Point Clouds". SIAM Journal on Scientific Computing 35 (3): A1461–A1486. doi:10.1137/120869730. Bibcode: 2013SJSC...35A1461L.
2. ↑ 2.0 2.1 Gross, B. J.; Trask, N.; Kuberry, P.; Atzberger, P. J. (15 May 2020). "Meshfree methods on manifolds for hydrodynamic flows on curved surfaces: A Generalized Moving Least-Squares (GMLS) approach". Journal of Computational Physics 409. doi:10.1016/j.jcp.2020.109340. Bibcode: 2020JCoPh.40909340G.
3. ↑ 3.0 3.1 Gross, B. J.; Kuberry, P.; Atzberger, P. J. (15 March 2022). "First-passage time statistics on surfaces of general shape: Surface PDE solvers using Generalized Moving Least Squares (GMLS)" (in en). Journal of Computational Physics 453. doi:10.1016/j.jcp.2021.110932. ISSN 0021-9991. Bibcode: 2022JCoPh.45310932G. https://doi.org/10.1016/j.jcp.2021.110932.
4. ↑ Wang, Hong-Yan; Xiang, Dao-Hong; Zhou, Ding-Xuan (1 March 2010). "Moving least-square method in learning theory" (in en). Journal of Approximation Theory 162 (3): 599–614. doi:10.1016/j.jat.2009.12.002. ISSN 0021-9045.
5. ↑ Trask, Nathaniel; Patel, Ravi G.; Gross, Ben J.; Atzberger, Paul J. (13 September 2019). "GMLS-Nets: A framework for learning from unstructured data". arXiv:1909.05371 [cs.LG].

* The approximation power of moving least squares David Levin, Mathematics of Computation, Volume 67, 1517-1531, 1998 [1]
* 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

## External links

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

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Least squares and regression analysis

Computational statistics|

* Least squares
* Linear least squares
* Non-linear least squares
* Iteratively reweighted least squares

Correlation and dependence|

* Pearson product-moment correlation
* Rank correlation (Spearman's rho
* Kendall's tau)
* Partial correlation
* Confounding variable

Regression analysis|

* Ordinary least squares
* Partial least squares
* Total least squares
* Ridge regression

Regression as a
statistical model|

| Linear regression|

* Simple linear regression
* Ordinary least squares
* Generalized least squares
* Weighted least squares
* General linear model

|

Predictor structure|

* Polynomial regression
* Growth curve (statistics)
* Segmented regression
* Local regression

Non-standard|

* Nonlinear regression
* Nonparametric
* Semiparametric
* Robust
* Quantile
* Isotonic

Non-normal errors|

* Generalized linear model
* Binomial
* Poisson
* Logistic

Decomposition of variance|

* Analysis of variance
* Analysis of covariance
* Multivariate AOV

Model exploration|

* Stepwise regression
* Model selection
* Mallows's Cp
* AIC
* BIC
* Model specification
* Regression validation

Background|

* Mean and predicted response
* Gauss–Markov theorem
* Errors and residuals
* Goodness of fit
* Studentized residual
* Minimum mean-square error

Design of experiments|

* Response surface methodology
* Optimal design
* Bayesian design

Numerical approximation|

* Numerical analysis
* Approximation theory
* Numerical integration
* Gaussian quadrature
* Orthogonal polynomials
* Chebyshev polynomials
* Chebyshev nodes

Applications|

* Curve fitting
* Calibration curve
* Numerical smoothing and differentiation
* System identification
* Moving least squares

* Regression analysis category
* Statistics category
*   * Statistics outline
* Statistics topics

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