Radial basis function: Difference between revisions

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A '''radial basis function''' ('''RBF''') is a [[real-valued function]] <math display="inline">\varphi</math> whose value depends only on the distance between the input and some fixed point, either the [[Origin (mathematics)|origin]], so that <math display="inline">\varphi(\mathbf{x}) = \varphi(\left\|\mathbf{x}\right\|)</math>, or some other fixed point <math display="inline">\mathbf{c}</math>, called a ''center'', so that <math display="inline">\varphi(\mathbf{x}) = \varphi(\left\|\mathbf{x}-\mathbf{c}\right\|)</math>. Any function <math display="inline">\varphi</math> that satisfies the property <math display="inline">\varphi(\mathbf{x}) = \varphi(\left\|\mathbf{x}\right\|)</math> is a [[radial function]]. The distance is usually [[Euclidean distance]], although other [[distance function|metric]]s are sometimes used. They are often used as a collection <math>\{ \varphi_k \}_k</math>which forms a [[Basis (linear algebra)|basis]] for some [[function space]] of interest, hence the name.

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 |first = David H. |last = 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 |archiveurlarchive-url = https://web.archive.org/web/20140714173428/https://www.complex-systems.com/pdf/02-3-5.pdf |archivedatearchive-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|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 |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 |website= |access-date=14 May 2015}}</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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== Further reading ==

{{more footnotes|date=June 2013}}

*{{cite journal|last1=Hardy|first1=R.L.|year=1971|title=Multiquadric equations of topography and other irregular surfaces|url=|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 | url = | journal = Comp. Math Applic | volume = 19 | issue = 8/9| pages = 163–208 | doi=10.1016/0898-1221(90)90272-l}}

* {{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| ___location=New York |isbn = 978-0-521-88068-8 |chapter = Section 3.7.1. Radial Basis Function Interpolation|chapter-url=http://apps.nrbook.com/empanel/index.html?pg=139 }}

* 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 | url = | journal = Journal of Applied Sciences and Computations | volume = 1 | issue = | pages = 437–475 }}

[[Category:Artificial neural networks]]