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Types of Radial Basis Functions

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How to Choose a Kernel Gaussian Thin-Plate Spline Logistic Basis Function Wendland's Compactly Supported Function Multiquadrics Reciprocal Multiquadrics Linear Cubic

How to Choose a Kernel

Within the Model Setup dialog box, you can choose which RBF kernel to use. Kernels are the types of RBF (multiquadric, gaussian, thinplate, and so on). These types are described in the following sections.

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Gaussian

This is the radial basis function most commonly used in the neural network community. Its profile function is

This leads to the radial basis function

In this case, the width parameter is the same as the standard deviation of the gaussian function.

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Thin-Plate Spline

This radial basis function is an example of a smoothing spline, as popularized by Grace Wahba (http://www.stat.wisc.edu/~wahba/). They are usually supplemented by low-order polynomial terms. Its profile function is

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Logistic Basis Function

These radial basis functions are mentioned in Hassoun (see References). They have the profile function

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Wendland's Compactly Supported Function

These form a family of radial basis functions that have a piecewise polynomial profile function and compact support [Wendland, see References]. The member of the family to choose depends on the dimension of the space (n) from which the data is drawn and the desired amount of continuity of the polynomials.

Dimension	Continuity	Profile
n=1	0
	2
	4
n=3	0
	2
	4
n=5	0
	2
	4

We have used the notation for the positive part of a.

When n is even, the radial basis function corresponding to dimension n+1 is used.

Note that each of the radial basis functions is nonzero when r is in [0,1]. It is possible to change the support to be by replacing r by in the preceding formula. The parameter is still referred to as the width of the radial basis function.

Similar formulas for the profile functions exist for n>5, and for even continuity > 4. Wendland's functions are available up to an even continuity of 6, and in any space dimension n.

Notes on Use

• Better approximation properties are usually associated with higher continuity.

• For a given data set the width parameter for Wendland's functions should be larger than the width chosen for the Gaussian.

  

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Multiquadrics

These are a popular tool for scattered data fitting. They have the profile function

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Reciprocal Multiquadrics

These have the profile function

Note that a width of zero is invalid.

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Linear

These have the profile function .

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Cubic

These have the profile function .

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Guide to Radial Basis Functions for Model Building

Fitting Routines

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