RadialBasis Class

Computes a least-squares fit to scattered data.

For a list of all members of this type, see RadialBasis Members.

System.Object
Imsl.Math.RadialBasis

public class RadialBasis

Thread Safety

Public static (Shared in Visual Basic) members of this type are safe for multithreaded operations. Instance members are not guaranteed to be thread-safe.

Remarks

RadialBasis computes a least-squares fit to scattered data in ${\bf R}^d$ , where d is the dimension. More precisely, we are given data points

$x_0,\ldots,x_{n-1} \in {\bf R}^d$

and function values

$f_0,\ldots,f_{n-1} \in {\bf R}^1$

The radial basis fit to the data is a function F which approximates the above data in the sense that it minimizes the sum-of-squares error

$\sum_{i=0}^{n-1} w_i \left( F(x_i)-f_i \right)^2$

where w are the weights. Of course, we must restrict the functional form of F. Here we assume it is a linear combination of radial functions:

$F(x) \equiv \sum_{j=0}^{m-1} \alpha_j \phi(\|x-c_j\|)$

The

are the centers .

A radial function, $\phi(r)$ , maps $[0, \infty)$ into ${\bf R}^1$ . The default radial function is the Hardy multiquadric,

$\phi(r) \equiv \sqrt{r^2+\delta^2}$

with $\delta=1$ . An alternate radial function is the Gaussian, $e^{-ax^2}$ .

By default, the centers are points in a Faure sequence, scaled to cover the box containing the data.

Requirements

Namespace: Imsl.Math

Assembly: ImslCS (in ImslCS.dll)

RadialBasis Class

Thread Safety

Remarks

Requirements

See Also