Function approximation

{{Short description|Approximating an arbitrary function with a well-behaved one}}

{{distinguish|Curve fitting}}

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File:Step function approximation.png.]]

File:Regression pic gaussien dissymetrique bruite.svg fit to a noisy curve using regression.]]

In general, a function approximation problem asks us to select a 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.{{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}}{{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.{{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 }} Function approximations are used where theoretical models are unavailable or hard to compute.{{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 }}

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 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.).{{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}}

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 and codomain of g, several techniques for approximating g may be applicable. 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.{{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}}

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}}