Radial basis function

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A radial basis function (RBF) is a real-valued function whose value depends only on the distance from the origin, so that ; or alternatively on the distance from some other point c, called a center, so that . Any function φ that satisfies the property is a radial function. The norm is usually Euclidean distance.

Radial basis functions are typically used to build up function approximations of the form

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. Approximation schemes of this kind have been particularly used in time series prediction and control of nonlinear systems exhibiting sufficiently simple chaotic behaviour, 3D reconstruction in computer graphics (for ex. hierarchical RBF).

The sum can also be interpreted as a rather simple single-layer type of artificial neural network called a radial basis function network, with the radial basis functions taking on the role of the activation functions of the network. It can be shown that any continuous function on a compact interval can in principle be interpolated with arbitrary accuracy by a sum of this form, if a sufficiently large number N of radial basis functions are used.

Two unnormalized Gaussian radial basis functions in one input dimension. The basis function centers are located at c₁=0.75 and c₂=3.25.

[edit] RBF types

Commonly used types of radial basis functions include :

• Gaussian:

for some β > 0

• Multiquadric:

for some β > 0

• Polyharmonic spline:

• Thin plate spline (a special polyharmonic spline):

[edit] Estimating the weights

The approximant y(x) is differentiable with respect to the weights w_i. The weights could thus be learned using any of the standard iterative methods for neural networks. But such iterative schemes are not in fact necessary: because the approximating function is linear in the weights w_i, the w_i can simply be estimated directly, using the matrix methods of linear least squares.