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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Function approximation</span></span>
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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Curve_fitting" title="Curve fitting">Curve fitting</a>.</div>
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<p>In general, a <b>function approximation</b> problem asks us to select a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> among a <span class="citation-needed-content" style="padding-left:0.1em; padding-right:0.1em; color:var(--color-subtle, #54595d); border:1px solid var(--border-color-subtle, #c8ccd1);">well-defined class</span> that closely matches ("approximates") a <span class="citation-needed-content" style="padding-left:0.1em; padding-right:0.1em; color:var(--color-subtle, #54595d); border:1px solid var(--border-color-subtle, #c8ccd1);">target function</span> in a task-specific way.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The need for function approximations arises in many branches of <a href="Applied_mathematics" title="Applied mathematics">applied mathematics</a>, and <a href="Computer_science" title="Computer science">computer science</a> in particular , such as predicting the growth of microbes in <a href="Microbiology" title="Microbiology">microbiology</a>.<sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Function approximations are used where theoretical models are unavailable or hard to compute.<sup id="cite_ref-:0_2-1" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p><p>One can distinguish two major classes of function approximation problems:
</p><p>First, for known target functions <a href="Approximation_theory" title="Approximation theory">approximation theory</a> is the branch of <a href="Numerical_analysis" title="Numerical analysis">numerical analysis</a> that investigates how certain known functions (for example, <a href="Special_function" class="mw-redirect" title="Special function">special functions</a>) can be approximated by a specific class of functions (for example, <a href="Polynomial" title="Polynomial">polynomials</a> or <a href="Rational_function" title="Rational function">rational functions</a>) that often have desirable properties (inexpensive computation, continuity, integral and limit values, etc.).<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>
</p><p>Second, the target function, call it <i>g</i>, may be unknown; instead of an explicit formula, only a set of points of the form (<i>x</i>, <i>g</i>(<i>x</i>)) is provided. Depending on the structure of the <a href="Domain_of_a_function" title="Domain of a function">domain</a> and <a href="Codomain" title="Codomain">codomain</a> of <i>g</i>, several techniques for approximating <i>g</i> may be applicable. For example, if <i>g</i> is an operation on the <a href="Real_number" title="Real number">real numbers</a>, techniques of <a href="Interpolation" title="Interpolation">interpolation</a>, <a href="Extrapolation" title="Extrapolation">extrapolation</a>, <a href="Regression_analysis" title="Regression analysis">regression analysis</a>, and <a href="Curve_fitting" title="Curve fitting">curve fitting</a> can be used. If the <a href="Codomain" title="Codomain">codomain</a> (range or target set) of <i>g</i> is a finite set, one is dealing with a <a href="Statistical_classification" title="Statistical classification">classification</a> problem instead.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>
</p><p>To some extent, the different problems (regression, classification, <a href="Fitness_approximation" title="Fitness approximation">fitness approximation</a>) have received a unified treatment in <a href="Statistical_learning_theory" title="Statistical learning theory">statistical learning theory</a>, where they are viewed as <a href="Supervised_learning" title="Supervised learning">supervised learning</a> problems.
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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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</style><cite id="CITEREFLakemeyerSklarSorrentiTakahashi2007" class="citation book cs1">Lakemeyer, Gerhard; Sklar, Elizabeth; Sorrenti, Domenico G.; Takahashi, Tomoichi (2007-09-04). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PW1qCQAAQBAJ&dq=%22function+approximation+is%22&pg=PA49"><i>RoboCup 2006: Robot Soccer World Cup X</i></a>. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-74024-7</bdi>.</cite></span>
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<li id="cite_note-:0-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFBasheerHajmeer2000" class="citation journal cs1">Basheer, I.A.; Hajmeer, M. (2000). <a rel="nofollow" class="external text" href="http://ethologie.unige.ch/etho5.10/pdf/basheer.hajmeer.2000.fundamentals.design.and.application.of.neural.networks.review.pdf">"Artificial neural networks: fundamentals, computing, design, and application"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of Microbiological Methods</i>. <b>43</b> (1): <span class="nowrap">3–</span>31. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0167-7012%2800%2900201-3">10.1016/S0167-7012(00)00201-3</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/11084225">11084225</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:18267806">18267806</a>.</cite></span>
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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFMhaskarPai2000" class="citation book cs1">Mhaskar, Hrushikesh Narhar; Pai, Devidas V. (2000). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=643OA9qwXLgC&dq=%22approximation+theory%22&pg=PA1"><i>Fundamentals of Approximation Theory</i></a>. CRC Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8493-0939-7</bdi>.</cite></span>
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<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFCharteCharteGarcíaHerrera2019" class="citation journal cs1">Charte, David; Charte, Francisco; García, Salvador; Herrera, Francisco (2019-04-01). <a rel="nofollow" class="external text" href="https://doi.org/10.1007/s13748-018-00167-7">"A snapshot on nonstandard supervised learning problems: taxonomy, relationships, problem transformations and algorithm adaptations"</a>. <i>Progress in Artificial Intelligence</i>. <b>8</b> (1): <span class="nowrap">1–</span>14. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1811.12044">1811.12044</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs13748-018-00167-7">10.1007/s13748-018-00167-7</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2192-6360">2192-6360</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:53715158">53715158</a>.</cite></span>
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<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Approximation_theory" title="Approximation theory">Approximation theory</a></li>
<li><a href="Fitness_approximation" title="Fitness approximation">Fitness approximation</a></li>
<li><a href="Kriging" title="Kriging">Kriging</a></li>
<li><a href="Least_squares_(function_approximation)" class="mw-redirect" title="Least squares (function approximation)">Least squares (function approximation)</a></li>
<li><a href="Radial_basis_function_network" title="Radial basis function network">Radial basis function network</a></li></ul>
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