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<title>Radial basis function</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Radial basis function</span></span>

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<p>In mathematics a <b>radial basis function</b> (<b>RBF</b>) is a <a href="Real-valued_function" title="Real-valued function">real-valued function</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \varphi }">

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<annotation encoding="application/x-tex">{\textstyle \varphi }</annotation>

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</math></span><img src="./99015519246670af1cb5592e439ad64a27fb4830.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\textstyle \varphi }" loading="lazy"></span> whose value depends only on the distance between the input and some <a href="Fixed_point_(mathematics)" title="Fixed point (mathematics)">fixed point</a>, either the <a href="Origin_(mathematics)" title="Origin (mathematics)">origin</a>, so that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \varphi (\mathbf {x} )={\hat {\varphi }}(\left\|\mathbf {x} \right\|)}">

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<annotation encoding="application/x-tex">{\textstyle \varphi (\mathbf {x} )={\hat {\varphi }}(\left\|\mathbf {x} \right\|)}</annotation>

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</math></span><img src="./cff9e5ffd6edd02c07c826bad319fc535b601bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.919ex; height:2.843ex;" alt="{\textstyle \varphi (\mathbf {x} )={\hat {\varphi }}(\left\|\mathbf {x} \right\|)}" loading="lazy"></span>, or some other fixed point <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {c} }">

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<annotation encoding="application/x-tex">{\textstyle \mathbf {c} }</annotation>

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</math></span><img src="./d0d8239c0502ae3b5f33956596b3309fcb61bbc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.188ex; height:1.676ex;" alt="{\textstyle \mathbf {c} }" loading="lazy"></span>, called a <i>center</i>, so that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \varphi (\mathbf {x} )={\hat {\varphi }}(\left\|\mathbf {x} -\mathbf {c} \right\|)}">

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<annotation encoding="application/x-tex">{\textstyle \varphi (\mathbf {x} )={\hat {\varphi }}(\left\|\mathbf {x} -\mathbf {c} \right\|)}</annotation>

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</math></span><img src="./899417ad2db3139b7818bee69f49c8068b4ea001.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.947ex; height:2.843ex;" alt="{\textstyle \varphi (\mathbf {x} )={\hat {\varphi }}(\left\|\mathbf {x} -\mathbf {c} \right\|)}" loading="lazy"></span>. Any function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \varphi }">

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</math></span><img src="./99015519246670af1cb5592e439ad64a27fb4830.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\textstyle \varphi }" loading="lazy"></span> that satisfies the property <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \varphi (\mathbf {x} )={\hat {\varphi }}(\left\|\mathbf {x} \right\|)}">

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<annotation encoding="application/x-tex">{\textstyle \varphi (\mathbf {x} )={\hat {\varphi }}(\left\|\mathbf {x} \right\|)}</annotation>

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</math></span><img src="./cff9e5ffd6edd02c07c826bad319fc535b601bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.919ex; height:2.843ex;" alt="{\textstyle \varphi (\mathbf {x} )={\hat {\varphi }}(\left\|\mathbf {x} \right\|)}" loading="lazy"></span> is a <a href="Radial_function" title="Radial function">radial function</a>. The distance is usually <a href="Euclidean_distance" title="Euclidean distance">Euclidean distance</a>, although other <a href="Distance_function" class="mw-redirect" title="Distance function">metrics</a> are sometimes used. They are often used as a collection <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\varphi _{k}\}_{k}}">

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<annotation encoding="application/x-tex">{\displaystyle \{\varphi _{k}\}_{k}}</annotation>

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</math></span><img src="./d38a867a532fc3c2c3ec0026c5527bc6336d70cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.023ex; height:2.843ex;" alt="{\displaystyle \{\varphi _{k}\}_{k}}" loading="lazy"></span> which forms a <a href="Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> for some <a href="Function_space" title="Function space">function space</a> of interest, hence the name.

</p><p>Sums of radial basis functions are typically used to <a href="Function_approximation" title="Function approximation">approximate given functions</a>. This approximation process can also be interpreted as a simple kind of <a href="Artificial_neural_network" class="mw-redirect" title="Artificial neural network">neural network</a>; this was the context in which they were originally applied to <a href="Machine_learning" title="Machine learning">machine learning</a>, in work by <a href="David_Broomhead" title="David Broomhead">David Broomhead</a> and David Lowe in 1988,<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><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> which stemmed from <a href="Michael_J._D._Powell" title="Michael J. D. Powell">Michael J. D. Powell</a>'s seminal research from 1977.<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><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><sup id="cite_ref-CITEREFBroomheadLowe1988_5-0" class="reference"><a href="#cite_note-CITEREFBroomheadLowe1988-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

RBFs are also used as a <a href="Radial_basis_function_kernel" title="Radial basis function kernel">kernel</a> in <a href="Support_vector_machine" title="Support vector machine">support vector classification</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The technique has proven effective and flexible enough that radial basis functions are now applied in a variety of engineering applications.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>A radial function is a function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \varphi :[0,\infty )\to \mathbb {R} }">

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<annotation encoding="application/x-tex">{\textstyle \varphi :[0,\infty )\to \mathbb {R} }</annotation>

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</math></span><img src="./f370fe84a9237623971e8e8182fb46d0fbe186a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.821ex; height:2.843ex;" alt="{\textstyle \varphi :[0,\infty )\to \mathbb {R} }" loading="lazy"></span>. When paired with a <a href="Norm_(mathematics)" title="Norm (mathematics)">norm</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \|\cdot \|:V\to [0,\infty )}">

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<annotation encoding="application/x-tex">{\textstyle \|\cdot \|:V\to [0,\infty )}</annotation>

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</math></span><img src="./92056932f853e837c535192621e2e5cadecabc42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.414ex; height:2.843ex;" alt="{\textstyle \|\cdot \|:V\to [0,\infty )}" loading="lazy"></span> on a vector space, a function of the form <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \varphi _{\mathbf {c} }=\varphi (\|\mathbf {x} -\mathbf {c} \|)}">

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<annotation encoding="application/x-tex">{\textstyle \varphi _{\mathbf {c} }=\varphi (\|\mathbf {x} -\mathbf {c} \|)}</annotation>

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</math></span><img src="./f038c25db3c5b3b0bedbab33f5939fc5b0e350ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.785ex; height:2.843ex;" alt="{\textstyle \varphi _{\mathbf {c} }=\varphi (\|\mathbf {x} -\mathbf {c} \|)}" loading="lazy"></span> is said to be a <b>radial kernel</b> centered at <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {c} \in V}">

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<annotation encoding="application/x-tex">{\textstyle \mathbf {c} \in V}</annotation>

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</math></span><img src="./f5085c80dfa9c8b05911692e5828a350fe57645e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.816ex; height:2.176ex;" alt="{\textstyle \mathbf {c} \in V}" loading="lazy"></span>. A radial function and the associated radial kernels are said to be radial basis functions if, for any <a href="Finite_set" title="Finite set">finite set</a> of nodes <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\mathbf {x} _{k}\}_{k=1}^{n}\subseteq V}">

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<annotation encoding="application/x-tex">{\displaystyle \{\mathbf {x} _{k}\}_{k=1}^{n}\subseteq V}</annotation>

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</math></span><img src="./a1a0250d892408f323223a26ae8ecd1d6f99d0e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.9ex; height:3.009ex;" alt="{\displaystyle \{\mathbf {x} _{k}\}_{k=1}^{n}\subseteq V}" loading="lazy"></span>, all of the following conditions are true:

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<div><ul><li>The kernels <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{\mathbf {x} _{1}},\varphi _{\mathbf {x} _{2}},\dots ,\varphi _{\mathbf {x} _{n}}}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{\mathbf {x} _{1}},\varphi _{\mathbf {x} _{2}},\dots ,\varphi _{\mathbf {x} _{n}}}</annotation>

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</math></span><img src="./723cf47a40626e329cfb651edaafd92d98e086d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.09ex; height:2.343ex;" alt="{\displaystyle \varphi _{\mathbf {x} _{1}},\varphi _{\mathbf {x} _{2}},\dots ,\varphi _{\mathbf {x} _{n}}}" loading="lazy"></span> are linearly independent (for example <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (r)=r^{2}}">

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</math></span><img src="./1454b5bea2ccf7dd54a7a2eee207e338e0ba6323.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.579ex; height:3.176ex;" alt="{\displaystyle \varphi (r)=r^{2}}" loading="lazy"></span> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=\mathbb {R} }">

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</math></span><img src="./63d897bc226139b01f3467c554249c117a944f4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.564ex; height:2.176ex;" alt="{\displaystyle V=\mathbb {R} }" loading="lazy"></span> is not a radial basis function)</li><li>The kernels <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{\mathbf {x} _{1}},\varphi _{\mathbf {x} _{2}},\dots ,\varphi _{\mathbf {x} _{n}}}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{\mathbf {x} _{1}},\varphi _{\mathbf {x} _{2}},\dots ,\varphi _{\mathbf {x} _{n}}}</annotation>

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</math></span><img src="./723cf47a40626e329cfb651edaafd92d98e086d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.09ex; height:2.343ex;" alt="{\displaystyle \varphi _{\mathbf {x} _{1}},\varphi _{\mathbf {x} _{2}},\dots ,\varphi _{\mathbf {x} _{n}}}" loading="lazy"></span> form a basis for a <a href="Haar_space" title="Haar space">Haar Space</a>, meaning that the <a href="Radial_basis_function_interpolation" title="Radial basis function interpolation">interpolation matrix</a> (given below) is non-<a href="Singular_matrix" title="Singular matrix">singular</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-wendland2005_10-0" class="reference"><a href="#cite_note-wendland2005-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></li></ul></div>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\varphi (\|\mathbf {x} _{1}-\mathbf {x} _{1}\|)&\varphi (\|\mathbf {x} _{2}-\mathbf {x} _{1}\|)&\dots &\varphi (\|\mathbf {x} _{n}-\mathbf {x} _{1}\|)\\\varphi (\|\mathbf {x} _{1}-\mathbf {x} _{2}\|)&\varphi (\|\mathbf {x} _{2}-\mathbf {x} _{2}\|)&\dots &\varphi (\|\mathbf {x} _{n}-\mathbf {x} _{2}\|)\\\vdots &\vdots &\ddots &\vdots \\\varphi (\|\mathbf {x} _{1}-\mathbf {x} _{n}\|)&\varphi (\|\mathbf {x} _{2}-\mathbf {x} _{n}\|)&\dots &\varphi (\|\mathbf {x} _{n}-\mathbf {x} _{n}\|)\\\end{bmatrix}}}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span></td></tr></tbody></table>

<p><br>

</p>

<div class="mw-heading mw-heading3"><h3 id="Examples">Examples</h3></div>

<p>Commonly used types of radial basis functions include (writing <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle r=\left\|\mathbf {x} -\mathbf {x} _{i}\right\|}">

<mrow>

<mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle r=\left\|\mathbf {x} -\mathbf {x} _{i}\right\|}</annotation>

</semantics>

</math></span><img src="./e81323236a9f30fba864c97c9a5664c762cc7276.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.934ex; height:2.843ex;" alt="{\textstyle r=\left\|\mathbf {x} -\mathbf {x} _{i}\right\|}" loading="lazy"></span> and using <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \varepsilon }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \varepsilon }</annotation>

</semantics>

</math></span><img src="./2193f5fc8a6dd05f24e01d0789b78ec2b57515ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\textstyle \varepsilon }" loading="lazy"></span> to indicate a <a href="Shape_parameter" title="Shape parameter">shape parameter</a> that can be used to scale the input of the radial kernel<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>):

</p>

<div><ul><li>Infinitely Smooth RBFs

<p>These radial basis functions are from <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }(\mathbb {R} )}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C^{\infty }(\mathbb {R} )}</annotation>

</semantics>

</math></span><img src="./c913ced284ad05bd42c8f004a3a83dee6323cb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.161ex; height:2.843ex;" alt="{\displaystyle C^{\infty }(\mathbb {R} )}" loading="lazy"></span> and are strictly <a href="Positive-definite_function" title="Positive-definite function">positive definite functions</a><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> that require tuning a shape parameter <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation>

</semantics>

</math></span><img src="./a30c89172e5b88edbd45d3e2772c7f5e562e5173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" loading="lazy"></span>

</p>

<div><ul><li><a href="Gaussian_function" title="Gaussian function">Gaussian</a>:

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (r)=e^{-(\varepsilon r)^{2}},}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span></td></tr></tbody></table>

</li><li>Inverse quadratic:

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (r)={\dfrac {1}{1+(\varepsilon r)^{2}}},}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_3" class="reference nourlexpansion" style="font-weight:bold;">3</span></td></tr></tbody></table></li><li>Inverse multiquadric:

<mfrac>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (r)={\dfrac {1}{\sqrt {1+(\varepsilon r)^{2}}}},}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_4" class="reference nourlexpansion" style="font-weight:bold;">4</span></td></tr></tbody></table></li></ul></div></li><li>Other Infinitely Smooth RBFs

<p>These radial basis functions are also from <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }(\mathbb {R} )}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C^{\infty }(\mathbb {R} )}</annotation>

</semantics>

</math></span><img src="./c913ced284ad05bd42c8f004a3a83dee6323cb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.161ex; height:2.843ex;" alt="{\displaystyle C^{\infty }(\mathbb {R} )}" loading="lazy"></span> and require tuning a shape parameter <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation>

</semantics>

</math></span><img src="./a30c89172e5b88edbd45d3e2772c7f5e562e5173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" loading="lazy"></span>, but they are not strictly <a href="Positive-definite_function" title="Positive-definite function">positive definite</a>.

</p>

<div><ul><li>Multiquadric:

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (r)={\sqrt {1+(\varepsilon r)^{2}}},}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_5" class="reference nourlexpansion" style="font-weight:bold;">5</span></td></tr></tbody></table></li><li>RTH:<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (r)=r\tanh(\varepsilon r),}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_6" class="reference nourlexpansion" style="font-weight:bold;">6</span></td></tr></tbody></table></li></ul></div></li><li><a href="Polyharmonic_spline" title="Polyharmonic spline">Polyharmonic spline</a>:

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\varphi (r)&=r^{k},&k&=1,3,5,\dotsc \\\varphi (r)&=r^{k}\ln(r),&k&=2,4,6,\dotsc \end{aligned}}}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_7" class="reference nourlexpansion" style="font-weight:bold;">7</span></td></tr></tbody></table>

<i>*For even-degree polyharmonic splines</i> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k=2,4,6,\dotsc )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (k=2,4,6,\dotsc )}</annotation>

</semantics>

</math></span><img src="./a03fc4595ea17fbf4d2ee406bd3c66bab7b6fbd3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.431ex; height:2.843ex;" alt="{\displaystyle (k=2,4,6,\dotsc )}" loading="lazy"></span><i>, to avoid numerical problems at <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r=0}</annotation>

</semantics>

</math></span><img src="./894a83e863728b4ee2e12f3a999a09f5f2bf1c89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r=0}" loading="lazy"></span> where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(0)=-\infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ln(0)=-\infty }</annotation>

</semantics>

</math></span><img src="./6ce280837cfb2a00cfbd338d2ca1a68ffb60177e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.141ex; height:2.843ex;" alt="{\displaystyle \ln(0)=-\infty }" loading="lazy"></span>, the computational implementation is often written as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (r)=r^{k-1}\ln(r^{r})}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (r)=r^{k-1}\ln(r^{r})}</annotation>

</semantics>

</math></span><img src="./92133dcad1a62a83d70d961661ab8624fb4659bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.873ex; height:3.176ex;" alt="{\displaystyle \varphi (r)=r^{k-1}\ln(r^{r})}" loading="lazy"></span>.</i></li><li><a href="Thin_plate_spline" title="Thin plate spline">Thin plate spline</a> (a special polyharmonic spline):

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (r)=r^{2}\ln(r),}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_8" class="reference nourlexpansion" style="font-weight:bold;">8</span></td></tr></tbody></table></li><li>Compactly <a href="Support_(mathematics)" title="Support (mathematics)">Supported</a> RBFs

<p>These RBFs are compactly supported and thus are non-zero only within a radius of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/\varepsilon }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1/\varepsilon }</annotation>

</semantics>

</math></span><img src="./c8a61384fb438a8f4019445c369caa1e9ae12e0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.408ex; height:2.843ex;" alt="{\displaystyle 1/\varepsilon }" loading="lazy"></span>, and thus have sparse differentiation matrices

</p>

<div><ul><li><a href="Bump_function" title="Bump function">Bump function</a>:

<mrow>

<mtr>

<mtd>

<mrow>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mtd>

<mtd>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mtext> otherwise</mtext>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (r)={\begin{cases}\exp \left(-{\frac {1}{1-(\varepsilon r)^{2}}}\right)&{\text{ for }}r<{\frac {1}{\varepsilon }}\\0&{\text{ otherwise}}\end{cases}},}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_9" class="reference nourlexpansion" style="font-weight:bold;">9</span></td></tr></tbody></table></li></ul></div></li></ul></div>

<div class="mw-heading mw-heading2"><h2 id="Approximation">Approximation</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Kernel_smoothing" class="mw-redirect" title="Kernel smoothing">Kernel smoothing</a></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Radial_basis_function_interpolation" title="Radial basis function interpolation">Radial basis function interpolation</a></div>

<p>Radial basis functions are typically used to build up <a href="Function_approximation" title="Function approximation">function approximations</a> of the form

</p>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y(\mathbf {x} )=\sum _{i=1}^{N}w_{i}\,\varphi (\left\|\mathbf {x} -\mathbf {x} _{i}\right\|),}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_10" class="reference nourlexpansion" style="font-weight:bold;">10</span></td></tr></tbody></table>

<p>where the approximating function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle y(\mathbf {x} )}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle y(\mathbf {x} )}</annotation>

</semantics>

</math></span><img src="./376ab167e65c6eb75b3f6eb5e317fb3c72925e42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.376ex; height:2.843ex;" alt="{\textstyle y(\mathbf {x} )}" loading="lazy"></span> is represented as a sum of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> radial basis functions, each associated with a different center <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {x} _{i}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mathbf {x} _{i}}</annotation>

</semantics>

</math></span><img src="./ab4ab075c8df670c8449ece2d4fce70d20844b2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.211ex; height:2.009ex;" alt="{\textstyle \mathbf {x} _{i}}" loading="lazy"></span>, and weighted by an appropriate coefficient <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle w_{i}.}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle w_{i}.}</annotation>

</semantics>

</math></span><img src="./f954a6e415e55b47dc4659d00200a19a036097ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.111ex; height:2.009ex;" alt="{\textstyle w_{i}.}" loading="lazy"></span> The weights <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle w_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle w_{i}}</annotation>

</semantics>

</math></span><img src="./586a2dd08d3247e8956143e61d66ba129940bb10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.464ex; height:2.009ex;" alt="{\textstyle w_{i}}" loading="lazy"></span> can be estimated using the matrix methods of <a href="Weighted_least_squares" title="Weighted least squares">linear least squares</a>, because the approximating function is <i>linear</i> in the weights <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle w_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle w_{i}}</annotation>

</semantics>

</math></span><img src="./586a2dd08d3247e8956143e61d66ba129940bb10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.464ex; height:2.009ex;" alt="{\textstyle w_{i}}" loading="lazy"></span>.

</p><p>Approximation schemes of this kind have been particularly used in <a href="Time_series_prediction" class="mw-redirect" title="Time series prediction">time series prediction</a> and <a href="Control_theory" title="Control theory">control</a> of <a href="Nonlinear_systems" class="mw-redirect" title="Nonlinear systems">nonlinear systems</a> exhibiting sufficiently simple <a href="Chaos_theory" title="Chaos theory">chaotic</a> behaviour and 3D reconstruction in <a href="Computer_graphics" title="Computer graphics">computer graphics</a> (for example, <a href="Hierarchical_RBF" title="Hierarchical RBF">hierarchical RBF</a> and <a href="Pose_Space_Deformation" class="mw-redirect" title="Pose Space Deformation">Pose Space Deformation</a>).

</p>

<div class="mw-heading mw-heading2"><h2 id="RBF_Network">RBF Network</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Radial_basis_function_network" title="Radial basis function network">radial basis function network</a></div>

<p>The sum

</p>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y(\mathbf {x} )=\sum _{i=1}^{N}w_{i}\,\varphi (\left\|\mathbf {x} -\mathbf {x} _{i}\right\|),}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_11" class="reference nourlexpansion" style="font-weight:bold;">11</span></td></tr></tbody></table>

<p>can also be interpreted as a rather simple single-layer type of <a href="Artificial_neural_network" class="mw-redirect" title="Artificial neural network">artificial neural network</a> called a <a href="Radial_basis_function_network" title="Radial basis function network">radial basis function network</a>, with the radial basis functions taking on the role of the activation functions of the network. It can be shown that any <a href="Continuous_function" title="Continuous function">continuous function</a> on a <a href="Compact_space" title="Compact space">compact</a> interval can in principle be interpolated with arbitrary accuracy by a sum of this form, if a sufficiently large number <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle N}</annotation>

</semantics>

</math></span><img src="./0d21d55fc102ec49600d3d5522a59ae4561acc22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\textstyle N}" loading="lazy"></span> of radial basis functions is used.

</p><p>The approximant <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle y(\mathbf {x} )}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle y(\mathbf {x} )}</annotation>

</semantics>

</math></span><img src="./376ab167e65c6eb75b3f6eb5e317fb3c72925e42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.376ex; height:2.843ex;" alt="{\textstyle y(\mathbf {x} )}" loading="lazy"></span> is differentiable with respect to the weights <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle w_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle w_{i}}</annotation>

</semantics>

</math></span><img src="./586a2dd08d3247e8956143e61d66ba129940bb10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.464ex; height:2.009ex;" alt="{\textstyle w_{i}}" loading="lazy"></span>. The weights could thus be learned using any of the standard iterative methods for neural networks.

</p><p>Using radial basis functions in this manner yields a reasonable interpolation approach provided that the fitting set has been chosen such that it covers the entire range systematically (equidistant data points are ideal). However, without a polynomial term that is orthogonal to the radial basis functions, estimates outside the fitting set tend to perform poorly.

</p>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Kansa_method" title="Kansa method">Kansa method</a></div>

<p>Radial basis functions are used to approximate functions and so can be used to discretize and numerically solve Partial Differential Equations (PDEs). This was first done in 1990 by E. J. Kansa who developed the first RBF based numerical method. It is called the <a href="Kansa_method" title="Kansa method">Kansa method</a> and was used to solve the elliptic <a href="Poisson's_equation" title="Poisson's equation">Poisson equation</a> and the linear <a href="Advection-diffusion_equation" class="mw-redirect" title="Advection-diffusion equation">advection-diffusion equation</a>. The function values at points <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation>

</semantics>

</math></span><img src="./32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }" loading="lazy"></span> in the domain are approximated by the linear combination of RBFs:

</p>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u(\mathbf {x} )=\sum _{i=1}^{N}\lambda _{i}\,\varphi (\left\|\mathbf {x} -\mathbf {x} _{i}\right\|),\quad \mathbf {x} \in \mathbb {R} ^{d}}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_12" class="reference nourlexpansion" style="font-weight:bold;">12</span></td></tr></tbody></table>

<p>The derivatives are approximated as such:

</p>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{n}u({\textbf {x}})}{\partial x^{n}}}=\sum _{i=1}^{N}\lambda _{i}\,{\frac {\partial ^{n}}{\partial x^{n}}}\varphi (\left\|\mathbf {x} -\mathbf {x} _{i}\right\|),\quad \mathbf {x} \in \mathbb {R} ^{d}}</annotation>

</semantics>

</math></span></span></td> <td></td> <td class="nowrap"><span id="math_13" class="reference nourlexpansion" style="font-weight:bold;">13</span></td></tr></tbody></table>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math></span><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> are the number of points in the discretized domain, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d}</annotation>

</semantics>

</math></span><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"></span> the dimension of the domain and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math></span><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"></span> the scalar coefficients that are unchanged by the differential operator.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup>

</p><p>Different numerical methods based on Radial Basis Functions were developed thereafter. Some methods are the RBF-FD method,<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> the RBF-QR method<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> and the RBF-PUM method.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup>

</p>

<ul><li><a href="Mat%C3%A9rn_covariance_function" title="Matérn covariance function">Matérn covariance function</a></li>

<li><a href="Radial_basis_function_interpolation" title="Radial basis function interpolation">Radial basis function interpolation</a></li>

<li><a href="Kansa_method" title="Kansa method">Kansa method</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><cite id="CITEREFShuDingYeo2003" class="citation journal cs1">Shu, C; Ding, H; Yeo, K. S (2003-02-14). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/pii/S0045782502006187">"Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier–Stokes equations"</a></span>. <i>Computer Methods in Applied Mechanics and Engineering</i>. <b>192</b> (7): <span class="nowrap">941–</span>954. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2003CMAME.192..941S">2003CMAME.192..941S</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0045-7825%2802%2900618-7">10.1016/S0045-7825(02)00618-7</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/0045-7825">0045-7825</a>.</cite></span>

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<li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><cite id="CITEREFFornbergLarssonFlyer2011" class="citation journal cs1">Fornberg, Bengt; Larsson, Elisabeth; Flyer, Natasha (2011-01-01). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://epubs.siam.org/doi/10.1137/09076756X">"Stable Computations with Gaussian Radial Basis Functions"</a></span>. <i>SIAM Journal on Scientific Computing</i>. <b>33</b> (2): <span class="nowrap">869–</span>892. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2011SJSC...33..869F">2011SJSC...33..869F</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F09076756X">10.1137/09076756X</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/1064-8275">1064-8275</a>.</cite></span>

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<li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><cite id="CITEREFSafdari-VaighaniHeryudonoLarsson2015" class="citation journal cs1">Safdari-Vaighani, Ali; Heryudono, Alfa; Larsson, Elisabeth (2015-08-01). <a rel="nofollow" class="external text" href="https://link.springer.com/article/10.1007/s10915-014-9935-9">"A Radial Basis Function Partition of Unity Collocation Method for Convection–Diffusion Equations Arising in Financial Applications"</a>. <i>Journal of Scientific Computing</i>. <b>64</b> (2): <span class="nowrap">341–</span>367. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10915-014-9935-9">10.1007/s10915-014-9935-9</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/1573-7691">1573-7691</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:254691757">254691757</a>.</cite></span>

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<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

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<ul><li><cite id="CITEREFHardy1971" class="citation journal cs1">Hardy, R.L. (1971). "Multiquadric equations of topography and other irregular surfaces". <i>Journal of Geophysical Research</i>. <b>76</b> (8): <span class="nowrap">1905–</span>1915. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1971JGR....76.1905H">1971JGR....76.1905H</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1029%2Fjb076i008p01905">10.1029/jb076i008p01905</a>.</cite></li>

<li><cite id="CITEREFHardy1990" class="citation journal cs1">Hardy, R.L. (1990). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0898-1221%2890%2990272-l">"Theory and applications of the multiquadric-biharmonic method, 20 years of Discovery, 1968 1988"</a>. <i>Comp. Math Applic</i>. <b>19</b> (8/9): <span class="nowrap">163–</span>208. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0898-1221%2890%2990272-l">10.1016/0898-1221(90)90272-l</a></span>.</cite></li>

<li><cite id="CITEREFPressTeukolskyVetterlingFlannery2007" class="citation cs2">Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170311163415/http://apps.nrbook.com/empanel/index.html?pg=139">"Section 3.7.1. Radial Basis Function Interpolation"</a>, <i>Numerical Recipes: The Art of Scientific Computing</i> (3rd ed.), New York: Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-88068-8</bdi>, archived from <a rel="nofollow" class="external text" href="http://apps.nrbook.com/empanel/index.html?pg=139">the original</a> on 2017-03-11<span class="reference-accessdate">, retrieved <span class="nowrap">2011-08-08</span></span></cite></li>

<li>Sirayanone, S., 1988, Comparative studies of kriging, multiquadric-biharmonic, and other methods for solving mineral resource problems, PhD. Dissertation, Dept. of Earth Sciences, Iowa State University, Ames, Iowa.</li>

<li><cite id="CITEREFSirayanoneHardy1995" class="citation journal cs1">Sirayanone, S.; Hardy, R.L. (1995). "The Multiquadric-biharmonic Method as Used for Mineral Resources, Meteorological, and Other Applications". <i>Journal of Applied Sciences and Computations</i>. <b>1</b>: <span class="nowrap">437–</span>475.</cite></li></ul></div><div><div class="zim-footer">

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