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:::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 of linear least squares.

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Radial basis function network — A radial basis function network is an artificial neural network that uses radial basis functions as activation functions. They are used in function approximation, time series prediction, and control.Network architectureRadial basis function (RBF) …   Wikipedia

  • Basis function — In mathematics, particularly numerical analysis, a basis function is an element of the basis for a function space. The term is a degeneration of the term basis vector for a more general vector space; that is, each function in the function space… …   Wikipedia

  • Function approximation — The need for function approximations arises in many branches of applied mathematics, and computer science in particular. In general, a function approximation problem asks us to select a function among a well defined class that closely matches (… …   Wikipedia

  • Activation function — In computational networks, the activation function of a node defines the output of that node given an input or set of inputs. A standard computer chip circuit can be seen as a digital network of activation functions that can be ON (1) or OFF (0) …   Wikipedia

  • Correlation function — For other uses, see Correlation function (disambiguation). A correlation function is the correlation between random variables at two different points in space or time, usually as a function of the spatial or temporal distance between the points.… …   Wikipedia

  • Bessel function — In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel s differential equation: for an arbitrary real or complex number α (the order of the …   Wikipedia

  • RBFN — radial basis function network …   Medical dictionary

  • RBF — • radial basis function; • regional blood flow; • regional bone mass; • renal blood flow …   Dictionary of medical acronyms & abbreviations

  • RBFN — • radial basis function network …   Dictionary of medical acronyms & abbreviations

  • Polyharmonic spline — In mathematics, polyharmonic splines are used for function approximation and data interpolation.They are very useful for interpolation of scattered datain many dimensions.Polyharmonic splines are a special case of radial basis functions andare… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.