Uses linear combinations of basis functions, such as 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 Radial Basis Function model is augmented with a polynomial function.
$f\left(x\right)=\sum _{i=1}^{n}{\lambda }_{i}\varphi \left(‖x-{x}_{i}‖\right)+\sum _{j=1}^{n}{c}_{j}{p}_{j}\left(x\right)$

where $n$ is the number of sampling points, $x$ is a vector of input variables, ${x}_{i}$ is the ith sampling point, $‖x-{x}_{i}‖$ is the Euclidean norm, $\varphi$ is a basis function, and ${\lambda }_{i}$ is the coefficient for the ith basis function. ${p}_{j}\left(x\right)$ is a low-order (constant or linear) polynomial function; $k$ is the total number of terms in the polynomial, and $cj\left(j=1,2...k\right)$ are the unknown coefficients.

## Usability Characteristics

• Attempts 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 Radial Basis Function Fit, it is suggested that you use a testing matrix.
• Suitable for modeling highly nonlinear output response data that does not contain numerical noise.
• Applicability of HyperKriging and Radial Basis Function 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, Radial Basis Function is suggested for studies with a large number of variables.
Note: As a result, Radial Basis Function Fit are evaluated faster than HyperKriging Fits when used in approaches.

## Settings

In the Specifications step, Settings tab, change method settings.
Note: For most applications the default settings work optimally.
Parameter Default Range Description
Augmented Function Constant
• Constant
• Linear
• Custom
Type of augmented function.
Maximum Points 2000 >=100 Maximum number of points for building Radial Basis Function; 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.