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 |firstfirst1 = David H. |lastlast1 = 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}}</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>