Function approximation - Revision history

imported>Villaida: moved the see also section

2025-11-25T18:50:14Z

moved the see also section

← Previous revision		Revision as of 18:50, 25 November 2025
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	{{Short description\|Approximating an arbitrary function with a well-behaved one}}		{{Short description\|Approximating an arbitrary function with a well-behaved one}}
	{{distinguish\|Curve fitting}}		{{distinguish\|Curve fitting}}
	~~{{More citations needed\|date=August 2019}}~~		[[File:Step function approximation.png\|alt=Several approximations of a step function\|thumb\|Several progressively more accurate approximations of the [[step function]]]]
	[[File:Step function approximation.png\|alt=Several approximations of a step function\|thumb\|Several progressively more accurate approximations of the [[step function]].]]		[[File:Regression pic gaussien dissymetrique bruite.svg\|alt=An asymmetrical Gaussian function fit to a noisy curve using regression.\|thumb\|An asymmetrical [[Gaussian function]] fit to a noisy curve using regression]]
	[[File:Regression pic gaussien dissymetrique bruite.svg\|alt=An asymmetrical Gaussian function fit to a noisy curve using regression.\|thumb\|An asymmetrical [[Gaussian function]] fit to a noisy curve using regression.]]		In general, a '''function approximation''' problem asks us to select a [[function (mathematics)\|function]] that closely matches ("approximates") a function in a task-specific way.<ref>{{Cite book\|last1=Lakemeyer\|first1=Gerhard\|url=https://books.google.com/books?id=PW1qCQAAQBAJ&dq=%22function+approximation+is%22&pg=PA49\|title=RoboCup 2006: Robot Soccer World Cup X\|last2=Sklar\|first2=Elizabeth\|last3=Sorrenti\|first3=Domenico G.\|last4=Takahashi\|first4=Tomoichi\|date=2007-09-04\|publisher=Springer\|isbn=978-3-540-74024-7\|language=en}}</ref>{{Better source needed\|reason=Find a source that actually explicitly makes this kind of definition; this one doesn't quite do so\|date=January 2022}} The need for function approximations arises, for example, predicting the growth of microbes in [[microbiology]].<ref name=":0">{{Cite journal\|last1=Basheer\|first1=I.A.\|last2=Hajmeer\|first2=M.\|date=2000\|title=Artificial neural networks: fundamentals, computing, design, and application\|url=https://web.archive.org/web/20230627001502/ethologie.unige.ch/etho5.10/pdf/basheer.hajmeer.2000.fundamentals.design.and.application.of.neural.networks.review.pdf\|journal=Journal of Microbiological Methods\|volume=43\|issue=1\|pages=3–31\|doi=10.1016/S0167-7012(00)00201-3\|pmid=11084225\|s2cid=18267806 }}</ref> Function approximations are used where theoretical models are unavailable or hard to compute.<ref name=":0">{{Cite journal\|last1=Basheer\|first1=I.A.\|last2=Hajmeer\|first2=M.\|date=2000\|title=Artificial neural networks: fundamentals, computing, design, and application\|url=http://ethologie.unige.ch/etho5.10/pdf/basheer.hajmeer.2000.fundamentals.design.and.application.of.neural.networks.review.pdf\|journal=Journal of Microbiological Methods\|volume=43\|issue=1\|pages=3–31\|doi=10.1016/S0167-7012(00)00201-3\|pmid=11084225\|s2cid=18267806 }}</ref>
	In general, a '''function approximation''' problem asks us to select a [[function (mathematics)\|function]] ~~among a {{Citation needed span\|text=well-defined class\|date=January 2022\|reason=This exact phrase is not used in the cited source}}{{Clarify\|date=October 2017}}~~ that closely matches ("approximates") a ~~{{Citation needed span\|text=target~~ function~~\|date=January 2022\|reason=This exact phrase is not used in the cited source.}}~~ in a task-specific way.<ref>{{Cite book\|last1=Lakemeyer\|first1=Gerhard\|url=https://books.google.com/books?id=PW1qCQAAQBAJ&dq=%22function+approximation+is%22&pg=PA49\|title=RoboCup 2006: Robot Soccer World Cup X\|last2=Sklar\|first2=Elizabeth\|last3=Sorrenti\|first3=Domenico G.\|last4=Takahashi\|first4=Tomoichi\|date=2007-09-04\|publisher=Springer\|isbn=978-3-540-74024-7\|language=en}}</ref>{{Better source needed\|reason=Find a source that actually explicitly makes this kind of definition; this one doesn't quite do so\|date=January 2022}} The need for function approximations arises ~~in many branches of [[applied mathematics]]~~, ~~and [[computer science]] in particular {{why\|date=October 2017}}~~,~~{{Citation needed\|date=January 2022}} such as~~ predicting the growth of microbes in [[microbiology]].<ref name=":0">{{Cite journal\|last1=Basheer\|first1=I.A.\|last2=Hajmeer\|first2=M.\|date=2000\|title=Artificial neural networks: fundamentals, computing, design, and application\|url=~~http~~://ethologie.unige.ch/etho5.10/pdf/basheer.hajmeer.2000.fundamentals.design.and.application.of.neural.networks.review.pdf\|journal=Journal of Microbiological Methods\|volume=43\|issue=1\|pages=3–31\|doi=10.1016/S0167-7012(00)00201-3\|pmid=11084225\|s2cid=18267806 }}</ref> Function approximations are used where theoretical models are unavailable or hard to compute.<ref name=":0">{{Cite journal\|last1=Basheer\|first1=I.A.\|last2=Hajmeer\|first2=M.\|date=2000\|title=Artificial neural networks: fundamentals, computing, design, and application\|url=http://ethologie.unige.ch/etho5.10/pdf/basheer.hajmeer.2000.fundamentals.design.and.application.of.neural.networks.review.pdf\|journal=Journal of Microbiological Methods\|volume=43\|issue=1\|pages=3–31\|doi=10.1016/S0167-7012(00)00201-3\|pmid=11084225\|s2cid=18267806 }}</ref>

	~~One can distinguish{{Citation needed\|date=January 2022}} two major classes of function approximation problems:~~

	First, for known target functions [[approximation theory]] is the branch of [[numerical analysis]] that investigates how certain known functions (for example, [[special function]]s) can be approximated by a specific class of functions (for example, [[polynomial]]s or [[rational function]]s) that often have desirable properties (inexpensive computation, continuity, integral and limit values, etc.).<ref>{{Cite book\|last1=Mhaskar\|first1=Hrushikesh Narhar\|url=https://books.google.com/books?id=643OA9qwXLgC&dq=%22approximation+theory%22&pg=PA1\|title=Fundamentals of Approximation Theory\|last2=Pai\|first2=Devidas V.\|date=2000\|publisher=CRC Press\|isbn=978-0-8493-0939-7\|language=en}}</ref>		First, for known target functions [[approximation theory]] is the branch of [[numerical analysis]] that investigates how certain known functions (for example, [[special function]]s) can be approximated by a specific class of functions (for example, [[polynomial]]s or [[rational function]]s) that often have desirable properties (inexpensive computation, continuity, integral and limit values, etc.).<ref>{{Cite book\|last1=Mhaskar\|first1=Hrushikesh Narhar\|url=https://books.google.com/books?id=643OA9qwXLgC&dq=%22approximation+theory%22&pg=PA1\|title=Fundamentals of Approximation Theory\|last2=Pai\|first2=Devidas V.\|date=2000\|publisher=CRC Press\|isbn=978-0-8493-0939-7\|language=en}}</ref>

	~~Second~~, the target function, call it ''g'', may be unknown; instead of an explicit formula, only a set of points of the form (''x'', ''g''(''x'')) is provided.{{Citation needed\|date=January 2022}} Depending on the structure of the [[domain of a function\|domain]] and [[codomain]] of ''g'', several techniques for ~~approximating ''g'' may be applicable. For~~ example, if ''g'' is an operation on the [[real number]]s, techniques of [[interpolation]], [[extrapolation]], [[regression analysis]], and [[curve fitting]] can be used. If the [[codomain]] (range or target set) of ''g'' is a finite set, one is dealing with a [[statistical classification\|classification]] problem instead.<ref>{{Cite journal\|last1=Charte\|first1=David\|last2=Charte\|first2=Francisco\|last3=García\|first3=Salvador\|last4=Herrera\|first4=Francisco\|date=2019-04-01\|title=A snapshot on nonstandard supervised learning problems: taxonomy, relationships, problem transformations and algorithm adaptations\|url=https://doi.org/10.1007/s13748-018-00167-7\|journal=Progress in Artificial Intelligence\|language=en\|volume=8\|issue=1\|pages=1–14\|doi=10.1007/s13748-018-00167-7\|arxiv=1811.12044\|s2cid=53715158\|issn=2192-6360}}</ref>		Secondly, for example, if ''g'' is an operation on the [[real number]]s, techniques of [[interpolation]], [[extrapolation]], [[regression analysis]], and [[curve fitting]] can be used. If the [[codomain]] (range or target set) of ''g'' is a finite set, one is dealing with a [[statistical classification\|classification]] problem instead.<ref>{{Cite journal\|last1=Charte\|first1=David\|last2=Charte\|first2=Francisco\|last3=García\|first3=Salvador\|last4=Herrera\|first4=Francisco\|date=2019-04-01\|title=A snapshot on nonstandard supervised learning problems: taxonomy, relationships, problem transformations and algorithm adaptations\|url=https://doi.org/10.1007/s13748-018-00167-7\|journal=Progress in Artificial Intelligence\|language=en\|volume=8\|issue=1\|pages=1–14\|doi=10.1007/s13748-018-00167-7\|arxiv=1811.12044\|s2cid=53715158\|issn=2192-6360}}</ref>

	To some extent, the different problems (regression, classification, [[fitness approximation]]) have received a unified treatment in [[statistical learning theory]], where they are viewed as [[supervised learning]] problems.{{Citation needed\|date=January 2022}}

	~~== References ==~~
	~~{{Reflist}}~~

	==See also==		==See also==
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	*[[Least squares (function approximation)]]		*[[Least squares (function approximation)]]
	*[[Radial basis function network]]		*[[Radial basis function network]]

			{{DEFAULTSORT:Function Approximation}}
			[[Category:Regression analysis]]
			[[Category:Statistical approximations]]
			== References ==
			{{Reflist}}

	{{DEFAULTSORT:Function Approximation}}		{{DEFAULTSORT:Function Approximation}}

imported>Jacobolus: rv. taking an article that is nearly a stub and cramming it with detailed information about a single recent paper written by the editor without commensurately expanding the rest to properly survey the field puts undue weight

2024-07-16T21:08:24Z

rv. taking an article that is nearly a stub and cramming it with detailed information about a single recent paper written by the editor without commensurately expanding the rest to properly survey the field puts undue weight

New page

{{Short description|Approximating an arbitrary function with a well-behaved one}}
{{distinguish|Curve fitting}}
{{More citations needed|date=August 2019}}
[[File:Step function approximation.png|alt=Several approximations of a step function|thumb|Several progressively more accurate approximations of the [[step function]].]]
[[File:Regression pic gaussien dissymetrique bruite.svg|alt=An asymmetrical Gaussian function fit to a noisy curve using regression.|thumb|An asymmetrical [[Gaussian function]] fit to a noisy curve using regression.]]
In general, a '''function approximation''' problem asks us to select a [[function (mathematics)|function]] among a {{Citation needed span|text=well-defined class|date=January 2022|reason=This exact phrase is not used in the cited source}}{{Clarify|date=October 2017}} that closely matches ("approximates") a {{Citation needed span|text=target function|date=January 2022|reason=This exact phrase is not used in the cited source.}} in a task-specific way.<ref>{{Cite book|last1=Lakemeyer|first1=Gerhard|url=https://books.google.com/books?id=PW1qCQAAQBAJ&dq=%22function+approximation+is%22&pg=PA49|title=RoboCup 2006: Robot Soccer World Cup X|last2=Sklar|first2=Elizabeth|last3=Sorrenti|first3=Domenico G.|last4=Takahashi|first4=Tomoichi|date=2007-09-04|publisher=Springer|isbn=978-3-540-74024-7|language=en}}</ref>{{Better source needed|reason=Find a source that actually explicitly makes this kind of definition; this one doesn't quite do so|date=January 2022}} The need for function approximations arises in many branches of [[applied mathematics]], and [[computer science]] in particular {{why|date=October 2017}},{{Citation needed|date=January 2022}} such as predicting the growth of microbes in [[microbiology]].<ref name=":0">{{Cite journal|last1=Basheer|first1=I.A.|last2=Hajmeer|first2=M.|date=2000|title=Artificial neural networks: fundamentals, computing, design, and application|url=http://ethologie.unige.ch/etho5.10/pdf/basheer.hajmeer.2000.fundamentals.design.and.application.of.neural.networks.review.pdf|journal=Journal of Microbiological Methods|volume=43|issue=1|pages=3–31|doi=10.1016/S0167-7012(00)00201-3|pmid=11084225|s2cid=18267806 }}</ref> Function approximations are used where theoretical models are unavailable or hard to compute.<ref name=":0">{{Cite journal|last1=Basheer|first1=I.A.|last2=Hajmeer|first2=M.|date=2000|title=Artificial neural networks: fundamentals, computing, design, and application|url=http://ethologie.unige.ch/etho5.10/pdf/basheer.hajmeer.2000.fundamentals.design.and.application.of.neural.networks.review.pdf|journal=Journal of Microbiological Methods|volume=43|issue=1|pages=3–31|doi=10.1016/S0167-7012(00)00201-3|pmid=11084225|s2cid=18267806 }}</ref>

One can distinguish{{Citation needed|date=January 2022}} two major classes of function approximation problems:

First, for known target functions [[approximation theory]] is the branch of [[numerical analysis]] that investigates how certain known functions (for example, [[special function]]s) can be approximated by a specific class of functions (for example, [[polynomial]]s or [[rational function]]s) that often have desirable properties (inexpensive computation, continuity, integral and limit values, etc.).<ref>{{Cite book|last1=Mhaskar|first1=Hrushikesh Narhar|url=https://books.google.com/books?id=643OA9qwXLgC&dq=%22approximation+theory%22&pg=PA1|title=Fundamentals of Approximation Theory|last2=Pai|first2=Devidas V.|date=2000|publisher=CRC Press|isbn=978-0-8493-0939-7|language=en}}</ref>

Second, the target function, call it ''g'', may be unknown; instead of an explicit formula, only a set of points of the form (''x'', ''g''(''x'')) is provided.{{Citation needed|date=January 2022}} Depending on the structure of the [[domain of a function|domain]] and [[codomain]] of ''g'', several techniques for approximating ''g'' may be applicable. For example, if ''g'' is an operation on the [[real number]]s, techniques of [[interpolation]], [[extrapolation]], [[regression analysis]], and [[curve fitting]] can be used. If the [[codomain]] (range or target set) of ''g'' is a finite set, one is dealing with a [[statistical classification|classification]] problem instead.<ref>{{Cite journal|last1=Charte|first1=David|last2=Charte|first2=Francisco|last3=García|first3=Salvador|last4=Herrera|first4=Francisco|date=2019-04-01|title=A snapshot on nonstandard supervised learning problems: taxonomy, relationships, problem transformations and algorithm adaptations|url=https://doi.org/10.1007/s13748-018-00167-7|journal=Progress in Artificial Intelligence|language=en|volume=8|issue=1|pages=1–14|doi=10.1007/s13748-018-00167-7|arxiv=1811.12044|s2cid=53715158|issn=2192-6360}}</ref>

To some extent, the different problems (regression, classification, [[fitness approximation]]) have received a unified treatment in [[statistical learning theory]], where they are viewed as [[supervised learning]] problems.{{Citation needed|date=January 2022}}

== References ==
{{Reflist}}

==See also==
*[[Approximation theory]]
*[[Fitness approximation]]
*[[Kriging]]
*[[Least squares (function approximation)]]
*[[Radial basis function network]]

{{DEFAULTSORT:Function Approximation}}
[[Category:Regression analysis]]
[[Category:Statistical approximations]]

{{mathanalysis-stub}}
{{statistics-stub}}