RadialBasis Class

RadialBasis Class

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

Inheritance Hierarchy

SystemObject
Imsl.MathRadialBasis

Namespace: Imsl.Math
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0

Syntax

C++

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[SerializableAttribute]
public class RadialBasis

<SerializableAttribute>
Public Class RadialBasis

[SerializableAttribute]
public ref class RadialBasis

[<SerializableAttribute>]
type RadialBasis =  class end

The RadialBasis type exposes the following members.

Constructors

	Name	Description
	RadialBasis	Creates a new instance of RadialBasis.

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Methods

	Name	Description
	Equals	Determines whether the specified object is equal to the current object. (Inherited from Object.)
	Eval(Double)	Returns the value of the radial basis at a point.
	Eval(Double)	Returns the value of the radial basis approximation at a point.
	Finalize	Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.)
	GetHashCode	Serves as a hash function for a particular type. (Inherited from Object.)
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	Gradient	Returns the gradient of the radial basis approximation at a point.
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	ToString	Returns a string that represents the current object. (Inherited from Object.)
	Update(Double, Double)	Adds a set of data points, all with weight = 1.
	Update(Double, Double)	Adds a data point with weight = 1.
	Update(Double, Double, Double)	Adds a set of data points with user-specified weights.
	Update(Double, Double, Double)	Adds a data point with a specified weight.

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Properties

	Name	Description
	ANOVA	Returns the ANOVA statistics from the linear regression.
	RadialFunction	The radial function.

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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

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.

Reference

Imsl.Math Namespace

Other Resources

Example