A **radial basis function** (**RBF**) is a real-valued function whose value depends only on the distance between the input and some fixed point, either the origin, so that , or some other fixed point , called a *center*, so that . Any function that satisfies the property is a radial function. The distance is usually Euclidean distance, although other metrics are sometimes used. They are often used as a collection which forms a basis for some function space of interest, hence the name.

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 applied to machine learning, in work by David Broomhead and David Lowe in 1988,^{[1]}^{[2]} which stemmed from Michael J. D. Powell's seminal research from 1977.^{[3]}^{[4]}^{[5]}
RBFs are also used as a kernel in support vector classification.^{[6]} The technique has proven effective and flexible enough that radial basis functions are now applied in a variety of engineering applications.^{[7]}^{[8]}

## Definition

A radial function is a function . When paired with a metric on a vector space a function is said to be a radial kernel centered at . A Radial function and the associated radial kernels are said to be radial basis functions if, for any set of nodes

- The kernels are linearly independent (for example in is not a radial basis function)

- The kernels form a basis for a Haar Space, meaning that the interpolation matrix

^{[9]}

^{[10]}

### Examples

Commonly used types of radial basis functions include (writing and using to indicate a *shape parameter* that can be used to scale the input of the radial kernel^{[11]}):

- Infinitely Smooth RBFs

These radial basis functions are from and are strictly positive definite functions^{[12]} that require tuning a shape parameter

- Polyharmonic spline:
**For even-degree polyharmonic splines**, to avoid numerical problems at where , the computational implementation is often written as .*

- Thin plate spline (a special polyharmonic spline):

- Compactly Supported RBFs

These RBFs are compactly supported and thus are non-zero only within a radius of , and thus have sparse differentiation matrices

## Approximation

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

where the approximating function is represented as a sum of radial basis functions, each associated with a different center , and weighted by an appropriate coefficient The weights 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 and 3D reconstruction in computer graphics (for example, hierarchical RBF and Pose Space Deformation).

## RBF Network

The sum

*of radial basis functions is used.*

The approximant is differentiable with respect to the weights *. 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.^{[citation needed]}

## See also

## References

**^**Radial Basis Function networks Archived 2014-04-23 at the Wayback Machine**^**Broomhead, David H.; Lowe, David (1988). "Multivariable Functional Interpolation and Adaptive Networks" (PDF).*Complex Systems*.**2**: 321–355. Archived from the original (PDF) on 2014-07-14.**^**Michael J. D. Powell (1977). "Restart procedures for the conjugate gradient method".*Mathematical Programming*.**12**(1): 241–254. doi:10.1007/bf01593790.**^**Sahin, Ferat (1997).*A Radial Basis Function Approach to a Color Image Classification Problem in a Real Time Industrial Application*(M.Sc.). Virginia Tech. p. 26.Radial basis functions were first introduced by Powell to solve the real multivariate interpolation problem.

**^**Broomhead & Lowe 1988, p. 347: "We would like to thank Professor M.J.D. Powell at the Department of Applied Mathematics and Theoretical Physics at Cambridge University for providing the initial stimulus for this work."**^**VanderPlas, Jake (6 May 2015). "Introduction to Support Vector Machines". [O'Reilly]. Retrieved 14 May 2015.**^**Buhmann, Martin Dietrich (2003).*Radial basis functions : theory and implementations*. Cambridge University Press. ISBN 978-0511040207. OCLC 56352083.**^**Biancolini, Marco Evangelos (2018).*Fast radial basis functions for engineering applications*. Springer International Publishing. ISBN 9783319750118. OCLC 1030746230.**^**Fasshauer, Gregory E. (2007).*Meshfree Approximation Methods with MATLAB*. Singapore: World Scientific Publishing Co. Pte. Ltd. pp. 17–25. ISBN 9789812706331.**^**Wendland, Holger (2005).*Scattered Data Approximation*. Cambridge: Cambridge University Press. pp. 11, 18–23, 64–66. ISBN 0521843359.**^**Fasshauer, Gregory E. (2007).*Meshfree Approximation Methods with MATLAB*. Singapore: World Scientific Publishing Co. Pte. Ltd. p. 37. ISBN 9789812706331.**^**Fasshauer, Gregory E. (2007).*Meshfree Approximation Methods with MATLAB*. Singapore: World Scientific Publishing Co. Pte. Ltd. pp. 37–45. ISBN 9789812706331.

## Further reading

This article includes a list of general references, but it remains largely unverified because it lacks sufficient corresponding inline citations. (June 2013) (Learn how and when to remove this template message) |

- Hardy, R.L. (1971). "Multiquadric equations of topography and other irregular surfaces".
*Journal of Geophysical Research*.**76**(8): 1905–1915. Bibcode:1971JGR....76.1905H. doi:10.1029/jb076i008p01905. - Hardy, R.L. (1990). "Theory and applications of the multiquadric-biharmonic method, 20 years of Discovery, 1968 1988".
*Comp. Math Applic*.**19**(8/9): 163–208. doi:10.1016/0898-1221(90)90272-l. - Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 3.7.1. Radial Basis Function Interpolation",
*Numerical Recipes: The Art of Scientific Computing*(3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8 - Sirayanone, S., 1988, Comparative studies of kriging, multiquadric-biharmonic, and other methods for solving mineral resource problems, PhD. Dissertation, Dept. of Earth Sciences, Iowa State University, Ames, Iowa.
- Sirayanone, S.; Hardy, R.L. (1995). "The Multiquadric-biharmonic Method as Used for Mineral Resources, Meteorological, and Other Applications".
*Journal of Applied Sciences and Computations*.**1**: 437–475.