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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
[SerializableAttribute]
public class RadialBasis

The RadialBasis type exposes the following members.

Constructors
  NameDescription
Public methodRadialBasis
Creates a new instance of RadialBasis.
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Methods
  NameDescription
Public methodEquals
Determines whether the specified object is equal to the current object.
(Inherited from Object.)
Public methodEval(Double)
Returns the value of the radial basis at a point.
Public methodEval(Double)
Returns the value of the radial basis approximation at a point.
Protected methodFinalize
Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection.
(Inherited from Object.)
Public methodGetHashCode
Serves as a hash function for a particular type.
(Inherited from Object.)
Public methodGetType
Gets the Type of the current instance.
(Inherited from Object.)
Public methodGradient
Returns the gradient of the radial basis approximation at a point.
Protected methodMemberwiseClone
Creates a shallow copy of the current Object.
(Inherited from Object.)
Public methodToString
Returns a string that represents the current object.
(Inherited from Object.)
Public methodUpdate(Double, Double)
Adds a set of data points, all with weight = 1.
Public methodUpdate(Double, Double)
Adds a data point with weight = 1.
Public methodUpdate(Double, Double, Double)
Adds a set of data points with user-specified weights.
Public methodUpdate(Double, Double, Double)
Adds a data point with a specified weight.
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Properties
  NameDescription
Public propertyANOVA
Returns the ANOVA statistics from the linear regression.
Public propertyRadialFunction
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 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.

See Also

Reference

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