Radial basis function

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 "φ"("x")="φ"(||"x"||) is a radial function. The norm is usually Euclidean distance.

Radial basis functions are typically used to build up function approximations of the form :$y(mathbf\{x\})\; =\; sum\_\{i=1\}^N\; w\_i\; ,\; phi(||mathbf\{x\}\; -\; mathbf\{c\}\_i||),$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". Approximation schemes of this kind have been particularly used in time series prediction and control of nonlinear systems exhibiting sufficiently simple chaotic behaviour.

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.

₁=0.75 and "c"₂=3.25.

RBF types

Commonly used types of radial basis functions include$(r\; =\; ||mathbf\{x\}\; -\; mathbf\{c\}\_i||)$:
* Gaussian::: for some
* Multiquadric: :: for some
* Polyharmonic spline:::$phi(r)\; =\; r^k,;\; k=1,3,5,...$::$phi(r)\; =\; r^k\; ln(r),;\; k=2,4,6,...$
* Thin plate spline (a special polyharmonic spline):::$phi(r)\; =\; r^2\; ln(r);$

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.