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Sums of radial basis functions are typically used to [[function approximation|approximate given functions]]. This approximation process can also be interpreted as a simple kind of [[artificial neural network|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,<ref>[http://www.anc.ed.ac.uk/rbf/intro/node8.html Radial Basis Function networks] {{webarchive|url=https://web.archive.org/web/20140423232029/http://www.anc.ed.ac.uk/rbf/intro/node8.html |date=2014-04-23 }}</ref><ref>{{cite journal |first1 = David H. |last1 = Broomhead |first2 = David |last2 = Lowe |title = Multivariable Functional Interpolation and Adaptive Networks |journal = Complex Systems |volume = 2 |pages = 321–355 |year = 1988 |url = https://www.complex-systems.com/pdf/02-3-5.pdf |archive-url = https://web.archive.org/web/20140714173428/https://www.complex-systems.com/pdf/02-3-5.pdf |archive-date = 2014-07-14}}</ref> which stemmed from [[Michael J. D. Powell]]'s seminal research from 1977.<ref>{{cite journal |title = Restart procedures for the conjugate gradient method |author = Michael J. D. Powell |journal = [[Mathematical Programming]] |volume = 12 |number = 1 |pages = 241–254 |year = 1977 |doi=10.1007/bf01593790|s2cid = 9500591 |author-link = Michael J. D. Powell }}</ref><ref>{{cite thesis |type = M.Sc. |first = Ferat |last = Sahin |title = A Radial Basis Function Approach to a Color Image Classification Problem in a Real Time Industrial Application |publisher = [[Virginia Tech]] |year = 1997 |quote = Radial basis functions were first introduced by Powell to solve the real multivariate interpolation problem. |page = 26 |hdl = 10919/36847 |url = http://hdl.handle.net/10919/36847 }}</ref><ref name="CITEREFBroomheadLowe1988">{{Harvnb|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."</ref><!--this doesn't seem to be working, probably a bug with {{sfn}}: <ref>{{sfn|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."</ref>-->
RBFs are also used as a [[Radial basis function kernel|kernel]] in [[support vector machine|support vector classification]].<ref>{{cite web |url=https://beta.oreilly.com/learning/intro-to-svm |title=Introduction to Support Vector Machines |last=VanderPlas |first=Jake |publisher=[O'Reilly] |date=6 May 2015 |access-date=14 May 2015 |archive-date=5 September 2015 |archive-url=https://web.archive.org/web/20150905100859/https://beta.oreilly.com/learning/intro-to-svm |url-status=dead }}</ref> The technique has proven effective and flexible enough that radial basis functions are now applied in a variety of engineering applications.<ref>{{Cite book|title=Radial basis functions : theory and implementations|first=Martin Dietrich|last=Buhmann|date=2003|publisher=Cambridge University Press|isbn=978-0511040207|oclc=56352083}}</ref><ref>{{Cite book|title=Fast radial basis functions for engineering applications|last=Biancolini|first=Marco Evangelos|date=2018|isbn=9783319750118|publisher=Springer International Publishing|oclc=1030746230}}</ref>
== Definition ==
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[[File:WiKipic1new.svg|thumb|Comparison of RTH, Multiquadric (ε = 2), and Linear RBFs]]
[[File:Bump function shape.png|thumb|Plot of the scaled [[bump function]] with several choices of <math>\varepsilon</math>]]
| [[Multiquadric]]:▼
{{NumBlk||<math display="block">\varphi(r) = \sqrt{1 + (\varepsilon r)^2}, </math>|{{EquationRef|3}}}}▼
| [[Inverse quadratic]]:
{{NumBlk||<math display="block">\varphi(r) = \dfrac{1}{1+(\varepsilon r)^2}, </math>|{{EquationRef|
| [[Inverse multiquadric]]:
{{NumBlk||<math display="block">\varphi(r) = \dfrac{1}{\sqrt{1 + (\varepsilon r)^2}}, </math>|{{EquationRef|
}}
|Other Infinitely Smooth RBFs
{{pb}}
These radial basis functions are also from <math>C^\infty(\mathbb{R})</math> and require tuning a shape parameter <math>\varepsilon</math>, but they are not strictly [[Positive-definite function|positive definite]].
{{bulleted list
▲| [[Multiquadric]]:
▲{{NumBlk||<math display="block">\varphi(r) = \sqrt{1 + (\varepsilon r)^2}, </math>|{{EquationRef|
| RTH:<ref>{{cite journal
| last1 = Heidari
| first1 = Mohammad
| last2 = Mohammadi
| first2 = Maryam
| last3 = De Marchi
| first3 = Stefano
| authorlink3 = Stefano De Marchi
| title = A shape preserving quasi-interpolation operator based on a new transcendental RBF
| journal = Dolomites Research Notes on Approximation
| volume = 14
| issue = 1
| pages = 56–73
| year = 2021
| doi = 10.14658/PUPJ-DRNA-2021-1-6
}}</ref>
{{NumBlk||<math display="block">\varphi(r) = r \tanh(\varepsilon r), </math>|{{EquationRef|6}}}}
}}
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*{{cite journal|last1=Hardy|first1=R.L.|year=1971|title=Multiquadric equations of topography and other irregular surfaces|journal=Journal of Geophysical Research|volume=76|issue=8|pages=1905–1915|doi=10.1029/jb076i008p01905|bibcode=1971JGR....76.1905H}}
* {{cite journal | last1 = Hardy | first1 = R.L. | year = 1990 | title = Theory and applications of the multiquadric-biharmonic method, 20 years of Discovery, 1968 1988 | journal = Comp. Math Applic | volume = 19 | issue = 8/9| pages = 163–208 | doi=10.1016/0898-1221(90)90272-l| doi-access = free }}
* {{Citation |last1 = Press |first1 = WH |last2 = Teukolsky |first2 = SA |last3 = Vetterling |first3 = WT |last4 = Flannery |first4 = BP |year = 2007 |title = Numerical Recipes: The Art of Scientific Computing |edition = 3rd |publisher = Cambridge University Press
* 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.
* {{cite journal | last1 = Sirayanone | first1 = S. | last2 = Hardy | first2 = R.L. | year = 1995 | title = The Multiquadric-biharmonic Method as Used for Mineral Resources, Meteorological, and Other Applications | journal = Journal of Applied Sciences and Computations | volume = 1 | pages = 437–475 }}
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[[Category:Interpolation]]
[[Category:Numerical analysis]]
[[Category:1988 in artificial intelligence]]
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