HyperStudy

Radial Basis Function

Radial Basis Function

 Previous topic Next topic No expanding text in this topic  

Radial Basis Function

 Previous topic Next topic JavaScript is required for expanding text JavaScript is required for the print function  

Radial basis function method is a Fit method that uses linear combinations of basis functions. Typical basis functions are linear, cubic, thin-plate spline, Gaussian, multiquadric, and inverse-multiquadric. These basis functions are observed to be accurate for highly nonlinear output responses but not for linear output responses. To remedy this deficiency, in HyperStudy, a RBF model is augmented with a polynomial function:

rbf2

where n is the number of sampling points, x is a vector of input variables, xi3 is the ith sampling point, xxi is the Euclidean norm, basisfunction is a basis function, and coefficient is the coefficient for the ith basis function. pjx is a low-order (constant or linear) polynomial function; k is the total number of terms in the polynomial, and cj are the unknown coefficients.

 

Usability Characteristics

•RBF tries to go through the exact sampling points, and in general, the residuals are small, if not zero. As a result, diagnostic measures using only the complete input matrix do not produce meaningful values. Cross-validation results provide some diagnostics using a special scheme using only the input points. To get detailed diagnostics on the quality of a RBF fit, it is suggested that you use a validation matrix.
•Suitable for modeling highly nonlinear output response data that does not contain numerical noise. 
•Applicability of HyperKriging and Radial Basis Function (RBF) methods are similar in terms of physics (they both are suggested for highly nonlinear output responses with no noise). It is suggested that you use HyperKriging for large studies that contain a large number of sampling points, whereas, RBF is suggested for studies with a large number of variables. 
Note:As a result, RBF Fits are evaluated faster than HyperKriging Fits when used in approaches.

 

Settings

In the Specifications step, you can change the the following RBF settings from the Settings tab.

Note:For most applications the default settings work optimally.

Parameter

Default

Range

Description

Augmented Function

Constant

Constant or Linear

Type of augmented function.

Maximum Points

2000

>= 100

The maximum number of points for building RBF; if number of building points is larger than maxnpt, then the point reduction algorithm is activated and a warning message is shown; the purpose of introducing maxnpt is to reduce computational effort for large scale problems.

RBF Type

CS21

Multiquadric

CS21 (formally knows as Wu's Compactly Supported (2,1))

Gaussian

Type of RBF.

Relaxation Parameter

1.0

>= 0.0

Relaxation parameter d used in RBF; if RBF is CS21 or Gaussian, and d is set to 0.0 by users, then RBF will automatically set d = 1.0e-6.