# Function approximation

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{{Short description|Approximating an arbitrary function with a well-behaved one}}
{{distinguish|Curve fitting}}
[[File:Step function approximation.png|alt=Several approximations of a step function|thumb|Several progressively more accurate approximations of the [step function](/source/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](/source/Gaussian_function) fit to a noisy curve using regression]]
In general, a '''function approximation''' problem asks us to select a [function](/source/function_(mathematics)) 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](/source/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"/>

First, for known target functions [approximation theory](/source/approximation_theory) is the branch of [numerical analysis](/source/numerical_analysis) that investigates how certain known functions (for example, [special function](/source/special_function)s) can be approximated by a specific class of functions (for example, [polynomial](/source/polynomial)s or [rational function](/source/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>

Secondly, for example, if ''g'' is an operation on the [real number](/source/real_number)s, techniques of [interpolation](/source/interpolation), [extrapolation](/source/extrapolation), [regression analysis](/source/regression_analysis), and [curve fitting](/source/curve_fitting) can be used. If the [codomain](/source/codomain) (range or target set) of ''g'' is a finite set, one is dealing with a [classification](/source/statistical_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>

==See also==
*[Approximation theory](/source/Approximation_theory)
*[Fitness approximation](/source/Fitness_approximation)
*[Kriging](/source/Kriging)
*[Least squares (function approximation)](/source/Least_squares_(function_approximation))
*[Radial basis function network](/source/Radial_basis_function_network)

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

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

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Adapted from the Wikipedia article [Function approximation](https://en.wikipedia.org/wiki/Function_approximation) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Function_approximation?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.