Radial basis function

In mathematics a radial basis function (RBF) is a real-valued function ${\textstyle \varphi }$ whose value depends only on the distance between the input and some fixed point, either the origin, so that ${\textstyle \varphi (\mathbf {x} )={\hat {\varphi }}(\left\|\mathbf {x} \right\|)}$, or some other fixed point ${\textstyle \mathbf {c} }$, called a center, so that ${\textstyle \varphi (\mathbf {x} )={\hat {\varphi }}(\left\|\mathbf {x} -\mathbf {c} \right\|)}$. Any function ${\textstyle \varphi }$ that satisfies the property ${\textstyle \varphi (\mathbf {x} )={\hat {\varphi }}(\left\|\mathbf {x} \right\|)}$ is a

are sometimes used. They are often used as a collection ${\displaystyle \{\varphi _{k}\}_{k}}$ which forms a basis for some function space of interest, hence the name.

Sums of radial basis functions are typically used to

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 ${\textstyle \varphi :[0,\infty )\to \mathbb {R} }$. When paired with a metric on a vector space ${\textstyle \|\cdot \|:V\to [0,\infty )}$ a function ${\textstyle \varphi _{\mathbf {c} }=\varphi (\|\mathbf {x} -\mathbf {c} \|)}$ is said to be a radial kernel centered at ${\textstyle \mathbf {c} }$. A Radial function and the associated radial kernels are said to be radial basis functions if, for any set of nodes ${\displaystyle \{\mathbf {x} _{k}\}_{k=1}^{n}}$

Examples

Commonly used types of radial basis functions include (writing ${\textstyle r=\left\|\mathbf {x} -\mathbf {x} _{i}\right\|}$ and using ${\textstyle \varepsilon }$ to indicate a shape parameter that can be used to scale the input of the radial kernel^[11]):

Approximation

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

$${\displaystyle y(\mathbf {x} )=\sum _{i=1}^{N}w_{i}\,\varphi (\left\|\mathbf {x} -\mathbf {x} _{i}\right\|),}$$

(9)

where the approximating function ${\textstyle y(\mathbf {x} )}$ is represented as a sum of ${\displaystyle N}$ radial basis functions, each associated with a different center ${\textstyle \mathbf {x} _{i}}$, and weighted by an appropriate coefficient ${\textstyle w_{i}.}$ The weights ${\textstyle w_{i}}$ can be estimated using the matrix methods of linear least squares, because the approximating function is linear in the weights ${\textstyle w_{i}}$.

Approximation schemes of this kind have been particularly used^[

).

RBF Network

Main article: radial basis function network

The sum

$${\displaystyle y(\mathbf {x} )=\sum _{i=1}^{N}w_{i}\,\varphi (\left\|\mathbf {x} -\mathbf {x} _{i}\right\|),}$$

(10)

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 ${\textstyle N}$ of radial basis functions is used.

The approximant ${\textstyle y(\mathbf {x} )}$ is differentiable with respect to the weights ${\textstyle w_{i}}$. 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]

RBFs for PDEs

Main article: Kansa method

Radial basis functions are used to approximate functions and so can be used to discretize and numerically solve Partial Differential Equations (PDEs). This was first done in 1990 by E. J. Kansa who developed the first RBF based numerical method. It is called the

advection-diffusion equation

. The function values at points ${\displaystyle \mathbf {x} }$ in the domain are approximated by the linear combination of RBFs:

$${\displaystyle u(\mathbf {x} )=\sum _{i=1}^{N}\lambda _{i}\,\varphi (\left\|\mathbf {x} -\mathbf {x} _{i}\right\|),\quad \mathbf {x} \in \mathbb {R} ^{d}}$$

(11)

The derivatives are approximated as such:

$${\displaystyle {\frac {\partial ^{n}u({\textbf {x}})}{\partial x^{n}}}=\sum _{i=1}^{N}\lambda _{i}\,{\frac {\partial ^{n}}{\partial x^{n}}}\varphi (\left\|\mathbf {x} -\mathbf {x} _{i}\right\|),\quad \mathbf {x} \in \mathbb {R} ^{d}}$$

(12)

where ${\displaystyle N}$ are the number of points in the discretized domain, ${\displaystyle d}$ the dimension of the domain and ${\displaystyle \lambda }$ the scalar coefficients that are unchanged by the differential operator.^[13]

Different numerical methods based on Radial Basis Functions were developed thereafter. Some methods are the RBF-FD method,^[14][15] the RBF-QR method^[16] and the RBF-PUM method.^[17]

See also

• Matérn covariance function
• Radial basis function interpolation
• Kansa method

References

1. ^ Radial Basis Function networks Archived 2014-04-23 at the Wayback Machine
2. ^ 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.
3. S2CID 9500591
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4. hdl:10919/36847
   . Radial basis functions were first introduced by Powell to solve the real multivariate interpolation problem.
5. ^ 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."
6. ^ VanderPlas, Jake (6 May 2015). "Introduction to Support Vector Machines". [O'Reilly]. Retrieved 14 May 2015.
7. OCLC 56352083
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8. OCLC 1030746230
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9. ISBN 9789812706331
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10. ISBN 0521843359
    .
11. ISBN 9789812706331
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12. ISBN 9789812706331
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13. ISSN 0898-1221
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14. S2CID 121511032
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15. ISSN 0045-7825
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16. ISSN 1064-8275
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17. S2CID 254691757
    .

Further reading

This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise 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.
  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.