Function approximation: Difference between revisions
Jump to navigation
Jump to search
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 |
imported>Villaida moved the see also section |
||
| Line 1: | Line 1: | ||
{{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}} | ||
[[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]] | |||
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> | ||
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> | |||
==See also== | ==See also== | ||
| Line 23: | Line 15: | ||
*[[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}} | ||
Latest revision as of 18:50, 25 November 2025
Template:Short description Script error: No such module "Distinguish".
In general, a function approximation problem asks us to select a function that closely matches ("approximates") a function in a task-specific way.[1]Template:Better source needed The need for function approximations arises, for example, predicting the growth of microbes in microbiology.[2] Function approximations are used where theoretical models are unavailable or hard to compute.[2]
First, for known target functions approximation theory is the branch of numerical analysis that investigates how certain known functions (for example, special functions) can be approximated by a specific class of functions (for example, polynomials or rational functions) that often have desirable properties (inexpensive computation, continuity, integral and limit values, etc.).[3]
Secondly, for example, if g is an operation on the real numbers, 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 classification problem instead.[4]
See also
- Approximation theory
- Fitness approximation
- Kriging
- Least squares (function approximation)
- Radial basis function network
References
<templatestyles src="Reflist/styles.css" />
Script error: No such module "Check for unknown parameters".