A radial basis function (RBF) is a realvalued 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, although other distance functions are also possible. For example by using LukaszykKarmowski metric it is for some radial functions possible^{[1]} to avoid problems with ill conditioning of the matrix solved to determine coefficients w_{i} (see below), since the is always greater than zero.
Sums of radial basis functions are typically used to approximate given functions. This approximation process can also be interpreted as a simple kind of neural network.
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Commonly used types of radial basis functions include :
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}. The weights w_{i} can be estimated using the matrix methods of linear least squares, because the approximating function is linear in the weights.
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 example, hierarchical RBF).
The sum
can also be interpreted as a rather simple singlelayer 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 is used.
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.
