Package smile.math.rbf

Radial basis functions.

See: Description

Interface Summary

Interface	Description
RadialBasisFunction	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||).

Class Summary

Class	Description
GaussianRadialBasis	Gaussian RBF.
InverseMultiquadricRadialBasis	Inverse multiquadric RBF.
MultiquadricRadialBasis	Multiquadric RBF.
ThinPlateRadialBasis	Thin plate RBF.

Package smile.math.rbf Description

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 w_i (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 w_i φ(||x-c_i||)

where the approximating function y(x) is represented as a sum of N radial basis functions, each associated with a different center c_i, and weighted by an appropriate coefficient w_i. The weights w_i 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).