Dakota Reference Manual  Version 6.4
Large-Scale Engineering Optimization and Uncertainty Analysis
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moving_least_squares


Moving Least Squares surrogate models

Specification

Alias: none

Argument(s): none

Required/Optional Description of Group Dakota Keyword Dakota Keyword Description
Optional basis_order

Polynomial order for the MLS bases

Optional weight_function

Selects the weight function for the MLS model

Optional export_model

Exports surrogate model in user-selected format

Description

Moving least squares is a further generalization of weighted least squares where the weighting is "moved" or recalculated for every new point where a prediction is desired[64].

The implementation of moving least squares is still under development. It tends to work well in trust region optimization methods where the surrogate model is constructed in a constrained region over a few points. The present implementation may not work as well globally.

Theory

Moving Least Squares can be considered a more specialized version of linear regression models. In linear regression, one usually attempts to minimize the sum of the squared residuals, where the residual is defined as the difference between the surrogate model and the true model at a fixed number of points.

In weighted least squares, the residual terms are weighted so the determination of the optimal coefficients governing the polynomial regression function, denoted by $\hat{f}({\bf x})$, are obtained by minimizing the weighted sum of squares at N data points:

\[ \sum_{n=1}^{N}w_{n}({\parallel \hat{f}({\bf x_{n}})-f({\bf x_{n}})\parallel}) \]