# Radial basis function

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**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 usuallyEuclidean distance .Radial basis functions are typically used to build up

function approximation s 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 intime series prediction and control ofnonlinear systems exhibiting sufficiently simple chaotic behaviour.The sum can also be interpreted as a rather simple single-layer type of

artificial neural network called aradial 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.**RBF types**Commonly used types of radial basis functions include$(r\; =\; ||mathbf\{x\}\; -\; mathbf\{c\}\_i||)$:

* Gaussian:::$phi(r)\; =\; exp(-eta\; r^2),$ for some $eta\; 0$

*Multiquadric : ::$phi(r)\; =\; sqrt\{r^2\; +\; eta^2\},$ for some $eta\; 0$

*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 oflinear least squares .

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