Package smile.math.rbf
package smile.math.rbf
Radial basis functions. A radial basis function is a real-valued function
whose value depends only on the distance from the origin, so that
φ(x)=φ(||x||); or alternatively on the distance from some other
point c, called a center, so that φ(x,c)=φ(||x-c||). Any function
φ that satisfies the property is a radial function. The norm is usually
Euclidean distance, although other distance functions are also possible.
For example by using probability metric it is for some radial functions
possible to avoid problems with ill conditioning of the matrix solved to
determine coefficients wi (see below), since the ||x|| is always
greater than zero.
Sums of radial basis functions are typically used to approximate given functions:
y(x) = Σ wi φ(||x-ci||)
where the approximating function y(x) is represented as a sum of N radial basis functions, each associated with a different center ci, and weighted by an appropriate coefficient wi. The weights wi can be estimated using the matrix methods of linear least squares, because the approximating function is linear in the weights.
This approximation process can also be interpreted as a simple kind of neural network and has been particularly used in time series prediction and control of nonlinear systems exhibiting sufficiently simple chaotic behavior, 3D reconstruction in computer graphics (for example, hierarchical RBF).
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ClassDescriptionGaussian RBF.Inverse multiquadric RBF.Multiquadric RBF.A radial basis function (RBF) is a real-valued function whose value depends only on the distance from the origin, so that φ(x)=φ(||x||); or alternatively on the distance from some other point c, called a center, so that φ(x,c)=φ(||x-c||).Thin plate RBF.