A radial basis function (RBF) is a real-valued function whose value depends only on the distance from the origin, so that $\phi(\mathbf{x}) = \phi(\|\mathbf{x}\|)$; or alternatively on the distance from some other point c, called a center, so that $\phi(\mathbf{x}, \mathbf{c}) = \phi(\|\mathbf{x}-\mathbf{c}\|)$. Any function $\phi$ that satisfies the property $\phi(\mathbf{x}) = \phi(\|\mathbf{x}\|)$ is a radial function. The norm is usually Euclidean distance, although other distance functions are also possible. For example, using Łukaszyk–Karmowski metric, it is possible for some radial functions to avoid problems with ill conditioning of the matrix solved to determine coefficients wi (see below), since the $\|\mathbf{x}\|$ is always greater than zero.[1]

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; this was the context in which they were originally invented, by David Broomhead and David Lowe in 1988.[2][3] RBFs are also used as a kernel in support vector classification.

## Types

Commonly used types of radial basis functions include (writing $r = \|\mathbf{x} - \mathbf{x}_i\|\;$):

The first term, that is used for normalisation of the Gaussian, is missing, because in our sum every Gaussian has a weight, so the normalisation is not necessary.

$\phi(r) = e^{-(\varepsilon r)^2}\,$
$\phi(r) = \sqrt{1 + (\varepsilon r)^2}$
$\phi(r) = \frac{1}{1+(\varepsilon r)^2}$
$\phi(r) = \frac{1}{\sqrt{1 + (\varepsilon r)^2}}$
$\phi(r) = r^k,\; k=1,3,5,\dots$
$\phi(r) = r^k \ln(r),\; k=2,4,6,\dots$
$\phi(r) = r^2 \ln(r)\;$

## Approximation

Main article: Kernel smoothing

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

$y(\mathbf{x}) = \sum_{i=1}^N w_i \, \phi(\|\mathbf{x} - \mathbf{x}_i\|),$

where the approximating function y(x) is represented as a sum of N radial basis functions, each associated with a different center xi, 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.

Approximation schemes of this kind have been particularly used[citation needed] in time series prediction and control of nonlinear systems exhibiting sufficiently simple chaotic behaviour, 3D reconstruction in computer graphics (for example, hierarchical RBF and Pose Space Deformation).

## RBF Network

Two unnormalized Gaussian radial basis functions in one input dimension. The basis function centers are located at x1=0.75 and x2=3.25.

The sum

$y(\mathbf{x}) = \sum_{i=1}^N w_i \, \phi(\|\mathbf{x} - \mathbf{x}_i\|),$

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 is used.

The approximant y(x) is differentiable with respect to the weights wi. The weights could thus be learned using any of the standard iterative methods for neural networks.

Using radial basis functions in this manner yields a reasonable interpolation approach provided that the fitting set has been chosen such that it covers the entire range systematically (equidistant data points are ideal). However, without a polynomial term that is orthogonal to the radial basis functions, estimates outside the fitting set tend to perform poorly.

## References

1. ^ Łukaszyk, S. (2004) A new concept of probability metric and its applications in approximation of scattered data sets. Computational Mechanics, 33, 299-3004. limited access
2. ^ Radial Basis Function networks
3. ^ Broomhead, D H, Lowe, D, Multivariable Functional Interpolation and Adaptive Networks Complex Systems vol 2 321-355 (1988)