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 ; or alternatively on the distance from some other point "c", called a "center", so that . Any function that satisfies the property "φ"("x")="φ"(||"x"||) is a radial function. The norm is usually
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 predictionand control of nonlinear systemsexhibiting sufficiently simple chaotic behaviour.
The sum can also be interpreted as a rather simple single-layer type of
artificial neural networkcalled 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.
Commonly used types of radial basis functions include:
* Gaussian::: for some
Multiquadric: :: for some
Thin plate spline(a special polyharmonic spline):::
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.
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