Moving Least Squares Method (MLSM)

Moving least squares method (MLSM) builds a weighted least squares model where the weights associated with the sampling points do not remain constant. Instead, they are functions of the normalized distance from a sampling point to a point x, where the surrogate model is evaluated. The weight, associated to a sampling point, decays as the evaluation point moves away from it. The decay is defined through a decay function. For each point x it reconstructs a continuous function biased towards the region around that point.

Usability Characteristics

•	Suggested to be used for nonlinear and noisy output responses.

•	Residuals and diagnostics should be used to gain an understanding of the quality of the Fit.

•	Use a Validation matrix in addition to an Input matrix for better diagnostics.

•	Quality of an MLSM Fit is a function of the number of runs, order of the polynomial and the behavior of the application.

•	If the residuals and diagnostics are not good for an MLSM Fit, than you can increase the order of the Fit provided you have enough runs to fit that specific order.

•	Because the weights are not constant in MLSM, there is no analytical form and an equation can not be provided.

Settings

In the Specifications step, you can change the settings of MLSM from the Settings tab.

Note:

For most applications the default settings work optimally, and you may only need to change the Order to improve the Fit quality.

In the Settings tab, you can access the following settings:

Parameter

Default

Range

Description

Fit Parameter

(dFitParameter)

5.0

>= 0.0

<= 10.0

The parameter controls the effect of screening out noise; the larger value, the less effect.

Minimum Weight

(dWMin)

0.001

> 0.0

Minimum weight.

Number of Excess Points

(NExcessPts)

>=0

Number of excessive points to build MLSM.

Order

nOrder)

1 + interactions

2 + interactions

3 + interaction

The order of polynomial function.

•	1 (Linear model)

•	2 (Quadratic model)

•	3 (Cubic model)

•	1 + interactions (Linear + interactions terms)

•	2 + interactions (Quadratic + interactions terms)

•	3 + interaction (Cubic + interactions terms)

Weighting Function

(nWeightingFunc)

Gaussian

Cubic

Fourth Order

Fifth Order

Seventh Order

The type of weighting function.

where is the normalized distance from the i-th sampling point to a current point. The parameter defines the "closeness of fit", the case =0 is equivalent to the traditional least squares regression. When the parameter is large, it is possible to obtain a very close fit through the sampling points, if desired. The following figures illustrate the change of the weight over the interval [0,1] where the sampling point is at r = 0:

mslm_0_1

mslm_010

mslm_0100

where is the normalized radius of the sphere of influence:

mslm_cubic_poly

The normalized radius of the sphere of influence inversely relates to the closeness of fit parameter, for example the smaller the value of , the closer fit is obtained.

mslm_4th

mslm_5th

mslm_7th