Moving Least Squares surrogate models

Specification

Alias: none

Argument(s): none

Required/Optional

Description of Group

Dakota Keyword

Dakota Keyword Description

Optional

poly_order

Polynomial order for the MLS bases

Optional

weight_function

Selects the weight function for the MLS model

Optional

export_model_file

Export surrogate to Surfpack model file

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:

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