RadialBasis computes a least-squares fit to scattered data in {\bf R}^d, where d is the dimension.

Namespace: Imsl.Math
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.0.0

Syntax

C#
[SerializableAttribute]
public class RadialBasis
Visual Basic (Declaration)
<SerializableAttribute> _
Public Class RadialBasis
Visual C++
[SerializableAttribute]
public ref class RadialBasis

Remarks

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 c_j 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.

Two Update methods allow the user to specify weights for each data point in the approximation scheme. In this way the user can influence the fit of the radial basis function. For example, if weights are in the range [0,1] then 0-weighted points are effectively removed from computations and 1-weighted points will have more influence than any others. When the number of centers equals the number of data points, the RBF fit will be "exact", otherwise it will be an approximation (useful for large or noisy data sets). Provided the ratios of the weights are not too extreme, weights will not appreciably change the accuracy of the fit to the data, but they will affect the shape of the approximating function away from the data: Greater weights result in greater influence at greater distances.

Inheritance Hierarchy

System..::.Object
Imsl.Math..::.RadialBasis

See Also