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<span class="vector-toc-numb">4.2.1</span> <span>Generalized cross validation</span> </div> </a> <ul id="toc-Generalized_cross_validation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Constraints" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Constraints"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Constraints</span> </div> </a> <ul id="toc-Constraints-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Pros_and_cons" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Pros_and_cons"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Pros and cons</span> </div> </a> <ul id="toc-Pros_and_cons-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Extensions_and_related_concepts" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" 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vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" 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data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Non-parametric regression technique</div> <p>In <a href="/wiki/Statistics" title="Statistics">statistics</a>, <b>multivariate adaptive regression splines</b> (<b>MARS</b>) is a form of <a href="/wiki/Regression_analysis" title="Regression analysis">regression analysis</a> introduced by <a href="/wiki/Jerome_H._Friedman" title="Jerome H. Friedman">Jerome H. Friedman</a> in 1991.<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> It is a <a href="/wiki/Non-parametric_regression" class="mw-redirect" title="Non-parametric regression">non-parametric regression</a> technique and can be seen as an extension of <a href="/wiki/Linear_model" title="Linear model">linear models</a> that automatically models nonlinearities and interactions between variables. </p><p>The term "MARS" is trademarked and licensed to Salford Systems. In order to avoid trademark infringements, many open-source implementations of MARS are called "Earth".<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><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> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="The_basics">The basics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_adaptive_regression_spline&action=edit&section=1" title="Edit section: The basics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This section introduces MARS using a few examples. We start with a set of data: a matrix of input variables <i>x</i>, and a vector of the observed responses <i>y</i>, with a response for each row in <i>x</i>. For example, the data could be: </p> <table> <tbody><tr> <th><i>x</i></th> <th><i>y</i> </th></tr> <tr> <td>10.5</td> <td>16.4 </td></tr> <tr> <td>10.7</td> <td>18.8 </td></tr> <tr> <td>10.8</td> <td>19.7 </td></tr> <tr> <td>...</td> <td>... </td></tr> <tr> <td>20.6</td> <td>77.0 </td></tr></tbody></table> <p>Here there is only one <a href="/wiki/Dependent_and_independent_variables" title="Dependent and independent variables">independent variable</a>, so the <i>x</i> matrix is just a single column. Given these measurements, we would like to build a model which predicts the expected <i>y</i> for a given <i>x</i>. </p> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/File:Friedmans_mars_linear_model.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/6f/Friedmans_mars_linear_model.png" decoding="async" width="240" height="240" class="mw-file-element" data-file-width="240" data-file-height="240" /></a><figcaption>A linear model</figcaption></figure> <p>A <a href="/wiki/Linear_model" title="Linear model">linear model</a> for the above data is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {y}}=-37+5.1x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mn>37</mn> <mo>+</mo> <mn>5.1</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {y}}=-37+5.1x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3340bb66317fd5fa483dfab59639e03553ad76b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.676ex; height:2.509ex;" alt="{\displaystyle {\widehat {y}}=-37+5.1x}"></span></dd></dl> <p>The hat on the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5716eb05cd3a7b14b6e7d45837c25edb501c4775" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.302ex; height:2.509ex;" alt="{\displaystyle {\widehat {y}}}"></span> indicates that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5716eb05cd3a7b14b6e7d45837c25edb501c4775" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.302ex; height:2.509ex;" alt="{\displaystyle {\widehat {y}}}"></span> is estimated from the data. The figure on the right shows a plot of this function: a line giving the predicted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5716eb05cd3a7b14b6e7d45837c25edb501c4775" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.302ex; height:2.509ex;" alt="{\displaystyle {\widehat {y}}}"></span> versus <i>x</i>, with the original values of <i>y</i> shown as red dots. </p><p>The data at the extremes of <i>x</i> indicates that the relationship between <i>y</i> and <i>x</i> may be non-linear (look at the red dots relative to the regression line at low and high values of <i>x</i>). We thus turn to MARS to automatically build a model taking into account non-linearities. MARS software constructs a model from the given <i>x</i> and <i>y</i> as follows </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\widehat {y}}=&\ 25\\&{}+6.1\max(0,x-13)\\&{}-3.1\max(0,13-x)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo>=</mo> </mtd> <mtd> <mtext> </mtext> <mn>25</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>+</mo> <mn>6.1</mn> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>x</mi> <mo>−</mo> <mn>13</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>−</mo> <mn>3.1</mn> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>13</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\widehat {y}}=&\ 25\\&{}+6.1\max(0,x-13)\\&{}-3.1\max(0,13-x)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da690bde7c606b464a00486dc4733550766ee8a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:25.533ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}{\widehat {y}}=&\ 25\\&{}+6.1\max(0,x-13)\\&{}-3.1\max(0,13-x)\end{aligned}}}"></span></dd></dl> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/File:Friedmans_mars_simple_model.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/a/a7/Friedmans_mars_simple_model.png" decoding="async" width="240" height="240" class="mw-file-element" data-file-width="240" data-file-height="240" /></a><figcaption>A simple MARS model of the same data</figcaption></figure> <p>The figure on the right shows a plot of this function: the predicted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5716eb05cd3a7b14b6e7d45837c25edb501c4775" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.302ex; height:2.509ex;" alt="{\displaystyle {\widehat {y}}}"></span> versus <i>x</i>, with the original values of <i>y</i> once again shown as red dots. The predicted response is now a better fit to the original <i>y</i> values. </p><p>MARS has automatically produced a kink in the predicted <i>y</i> to take into account non-linearity. The kink is produced by <i>hinge functions</i>. The hinge functions are the expressions starting with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">max</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8e49fca3e322708b32d21eaa8b095dc05f09538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.326ex; height:1.676ex;" alt="{\displaystyle \max }"></span> (where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max(a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max(a,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0603f0969a6eb143615dc0339002987aa0dd9daa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.396ex; height:2.843ex;" alt="{\displaystyle \max(a,b)}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a>b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83fc0063781fb9bf4ec7608b2fd11ed6d5b05a13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a>b}"></span>, else <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>). Hinge functions are described in more detail below. </p><p>In this simple example, we can easily see from the plot that <i>y</i> has a non-linear relationship with <i>x</i> (and might perhaps guess that y varies with the square of <i>x</i>). However, in general there will be multiple <a href="/wiki/Dependent_and_independent_variables" title="Dependent and independent variables">independent variables</a>, and the relationship between <i>y</i> and these variables will be unclear and not easily visible by plotting. We can use MARS to discover that non-linear relationship. </p><p>An example MARS expression with multiple variables is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathrm {ozone} =&\ 5.2\\&{}+0.93\max(0,\mathrm {temp} -58)\\&{}-0.64\max(0,\mathrm {temp} -68)\\&{}-0.046\max(0,234-\mathrm {ibt} )\\&{}-0.016\max(0,\mathrm {wind} -7)\max(0,200-\mathrm {vis} )\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">z</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">e</mi> </mrow> <mo>=</mo> </mtd> <mtd> <mtext> </mtext> <mn>5.2</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>+</mo> <mn>0.93</mn> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">p</mi> </mrow> <mo>−</mo> <mn>58</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>−</mo> <mn>0.64</mn> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">p</mi> </mrow> <mo>−</mo> <mn>68</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>−</mo> <mn>0.046</mn> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>234</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">t</mi> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>−</mo> <mn>0.016</mn> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">w</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> <mo>−</mo> <mn>7</mn> <mo stretchy="false">)</mo> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>200</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathrm {ozone} =&\ 5.2\\&{}+0.93\max(0,\mathrm {temp} -58)\\&{}-0.64\max(0,\mathrm {temp} -68)\\&{}-0.046\max(0,234-\mathrm {ibt} )\\&{}-0.016\max(0,\mathrm {wind} -7)\max(0,200-\mathrm {vis} )\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b22575b479c1d8e20b7839a9fce0a7cfc11dd47e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:52.492ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}\mathrm {ozone} =&\ 5.2\\&{}+0.93\max(0,\mathrm {temp} -58)\\&{}-0.64\max(0,\mathrm {temp} -68)\\&{}-0.046\max(0,234-\mathrm {ibt} )\\&{}-0.016\max(0,\mathrm {wind} -7)\max(0,200-\mathrm {vis} )\end{aligned}}}"></span></dd></dl> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/File:Friedmans_mars_ozone_model.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/9/9e/Friedmans_mars_ozone_model.png" decoding="async" width="280" height="280" class="mw-file-element" data-file-width="280" data-file-height="280" /></a><figcaption>Variable interaction in a MARS model</figcaption></figure> <p>This expression models air pollution (the ozone level) as a function of the temperature and a few other variables. Note that the last term in the formula (on the last line) incorporates an interaction between <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {wind} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">w</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {wind} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35d77bc1714fe62b66e93cab1642eba12693b941" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.91ex; height:2.176ex;" alt="{\displaystyle \mathrm {wind} }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {vis} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {vis} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25d2054be22f67d8c27f035f245d3ff0f2ddc2ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.791ex; height:2.176ex;" alt="{\displaystyle \mathrm {vis} }"></span>. </p><p>The figure on the right plots the predicted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {ozone} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">z</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">e</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {ozone} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e22f4da57a37abe9650dc084dc0e9ae80d8b2b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.682ex; height:1.676ex;" alt="{\displaystyle \mathrm {ozone} }"></span> as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {wind} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">w</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {wind} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35d77bc1714fe62b66e93cab1642eba12693b941" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.91ex; height:2.176ex;" alt="{\displaystyle \mathrm {wind} }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {vis} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">s</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {vis} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25d2054be22f67d8c27f035f245d3ff0f2ddc2ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.791ex; height:2.176ex;" alt="{\displaystyle \mathrm {vis} }"></span> vary, with the other variables fixed at their median values. The figure shows that wind does not affect the ozone level unless visibility is low. We see that MARS can build quite flexible regression surfaces by combining hinge functions. </p><p>To obtain the above expression, the MARS model building procedure automatically selects which variables to use (some variables are important, others not), the positions of the kinks in the hinge functions, and how the hinge functions are combined. </p> <div class="mw-heading mw-heading2"><h2 id="The_MARS_model">The MARS model</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_adaptive_regression_spline&action=edit&section=2" title="Edit section: The MARS model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>MARS builds models of the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {f}}(x)=\sum _{i=1}^{k}c_{i}B_{i}(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo>^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {f}}(x)=\sum _{i=1}^{k}c_{i}B_{i}(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed68af49e29e98c540c751847c4bdf82fe77f2d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.907ex; height:7.343ex;" alt="{\displaystyle {\widehat {f}}(x)=\sum _{i=1}^{k}c_{i}B_{i}(x).}"></span></dd></dl> <p>The model is a weighted sum of basis functions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{i}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{i}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2afbb9ab6dd4f6a42fd07d63629584caf06afc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.703ex; height:2.843ex;" alt="{\displaystyle B_{i}(x)}"></span>. Each <span class="mwe-math-element"><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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01acb7953ba52c2aa44264b5d0f8fd223aa178a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.807ex; height:2.009ex;" alt="{\displaystyle c_{i}}"></span> is a constant coefficient. For example, each line in the formula for ozone above is one basis function multiplied by its coefficient. </p><p>Each <a href="/wiki/Basis_function" title="Basis function">basis function</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{i}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{i}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2afbb9ab6dd4f6a42fd07d63629584caf06afc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.703ex; height:2.843ex;" alt="{\displaystyle B_{i}(x)}"></span> takes one of the following three forms: </p><p>1) a constant 1. There is just one such term, the intercept. In the ozone formula above, the intercept term is 5.2. </p><p>2) a <i>hinge</i> function. A hinge function has the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max(0,x-{\text{constant}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>constant</mtext> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max(0,x-{\text{constant}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74bf24f738aaa7b59734b178074a0527157a972c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.169ex; height:2.843ex;" alt="{\displaystyle \max(0,x-{\text{constant}})}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max(0,{\text{constant}}-x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>constant</mtext> </mrow> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max(0,{\text{constant}}-x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e05d39fa65bd6e8b3aee32bcc8edfc00fde22aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.169ex; height:2.843ex;" alt="{\displaystyle \max(0,{\text{constant}}-x)}"></span>. MARS automatically selects variables and values of those variables for knots of the hinge functions. Examples of such basis functions can be seen in the middle three lines of the ozone formula. </p><p>3) a product of two or more hinge functions. These basis functions can model interaction between two or more variables. An example is the last line of the ozone formula. </p> <div class="mw-heading mw-heading2"><h2 id="Hinge_functions">Hinge functions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_adaptive_regression_spline&action=edit&section=3" title="Edit section: Hinge functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/File:Friedmans_mars_hinge_functions.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/7/74/Friedmans_mars_hinge_functions.png" decoding="async" width="280" height="220" class="mw-file-element" data-file-width="280" data-file-height="220" /></a><figcaption>A mirrored pair of hinge functions with a knot at x=3.1</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1236090951">.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}}</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Hinge_function" class="mw-redirect" title="Hinge function">Hinge function</a></div> <p>A key part of MARS models are <i>hinge functions</i> taking the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max(0,x-c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>x</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max(0,x-c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/388baa5bdf1f2f1c722aa348131f9784cf0f0741" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.508ex; height:2.843ex;" alt="{\displaystyle \max(0,x-c)}"></span></dd></dl> <p>or </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max(0,c-x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>c</mi> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max(0,c-x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf73bdf8a9febadd3fef9a7a3120a6817489e5fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.508ex; height:2.843ex;" alt="{\displaystyle \max(0,c-x)}"></span></dd></dl> <p>where <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> is a constant, called the <i>knot</i>. The figure on the right shows a mirrored pair of hinge functions with a knot at 3.1. </p><p>A hinge function is zero for part of its range, so can be used to partition the data into disjoint regions, each of which can be treated independently. Thus for example a mirrored pair of hinge functions in the expression </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6.1\max(0,x-13)-3.1\max(0,13-x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6.1</mn> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>x</mi> <mo>−</mo> <mn>13</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>3.1</mn> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>13</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6.1\max(0,x-13)-3.1\max(0,13-x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a82eb9d2aba94e2a3dd49af110633b9cc3662ff3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.211ex; height:2.843ex;" alt="{\displaystyle 6.1\max(0,x-13)-3.1\max(0,13-x)}"></span></dd></dl> <p>creates the <a href="/wiki/Piecewise" class="mw-redirect" title="Piecewise">piecewise</a> linear graph shown for the simple MARS model in the previous section. </p><p>One might assume that only piecewise linear functions can be formed from hinge functions, but hinge functions can be multiplied together to form non-linear functions. </p><p>Hinge functions are also called <a href="/wiki/Ramp_function" title="Ramp function">ramp</a>, <a href="/wiki/Ice_hockey_stick" title="Ice hockey stick">hockey stick</a>, or <a href="/wiki/Rectifier_(neural_networks)" title="Rectifier (neural networks)">rectifier</a> functions. Instead of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">max</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8e49fca3e322708b32d21eaa8b095dc05f09538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.326ex; height:1.676ex;" alt="{\displaystyle \max }"></span> notation used in this article, hinge functions are often represented by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\pm (x_{i}-c)]_{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo>±</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\pm (x_{i}-c)]_{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4164761539b8c42b6abe93e151683852d9ec043" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.399ex; height:2.843ex;" alt="{\displaystyle [\pm (x_{i}-c)]_{+}}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [\cdot ]_{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo>⋅</mo> <msub> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\cdot ]_{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0196f79f9c7c8c6a7777ab193018b24c6f4b1a92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.451ex; height:2.843ex;" alt="{\displaystyle [\cdot ]_{+}}"></span> means take the positive part. </p> <div class="mw-heading mw-heading2"><h2 id="The_model_building_process">The model building process</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_adaptive_regression_spline&action=edit&section=4" title="Edit section: The model building process"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Stepwise_regression" title="Stepwise regression">Stepwise regression</a></div> <p>MARS builds a model in two phases: the forward and the backward pass. This two-stage approach is the same as that used by <a href="/wiki/Recursive_partitioning" title="Recursive partitioning">recursive partitioning</a> trees. </p> <div class="mw-heading mw-heading3"><h3 id="The_forward_pass">The forward pass</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_adaptive_regression_spline&action=edit&section=5" title="Edit section: The forward pass"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>MARS starts with a model which consists of just the intercept term (which is the mean of the response values). </p><p>MARS then repeatedly adds basis function in pairs to the model. At each step it finds the pair of basis functions that gives the maximum reduction in sum-of-squares <a href="/wiki/Errors_and_residuals_in_statistics" class="mw-redirect" title="Errors and residuals in statistics">residual</a> error (it is a <a href="/wiki/Greedy_algorithm" title="Greedy algorithm">greedy algorithm</a>). The two basis functions in the pair are identical except that a different side of a mirrored hinge function is used for each function. Each new basis function consists of a term already in the model (which could perhaps be the intercept term) multiplied by a new hinge function. A hinge function is defined by a variable and a knot, so to add a new basis function, MARS must search over all combinations of the following: </p><p>1) existing terms (called <i>parent terms</i> in this context) </p><p>2) all variables (to select one for the new basis function) </p><p>3) all values of each variable (for the knot of the new hinge function). </p><p>To calculate the coefficient of each term, MARS applies a linear regression over the terms. </p><p>This process of adding terms continues until the change in residual error is too small to continue or until the maximum number of terms is reached. The maximum number of terms is specified by the user before model building starts. </p><p>The search at each step is usually done in a <a href="/wiki/Brute-force_search" title="Brute-force search">brute-force</a> fashion, but a key aspect of MARS is that because of the nature of hinge functions, the search can be done quickly using a fast least-squares update technique. Brute-force search can be sped up by using a <a href="/wiki/Heuristics" class="mw-redirect" title="Heuristics">heuristic</a> that reduces the number of parent terms considered at each step ("Fast MARS"<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> <div class="mw-heading mw-heading3"><h3 id="The_backward_pass">The backward pass</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_adaptive_regression_spline&action=edit&section=6" title="Edit section: The backward pass"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The forward pass usually <a href="/wiki/Overfit" class="mw-redirect" title="Overfit">overfits</a> the model. To build a model with better generalization ability, the backward pass prunes the model, deleting the least effective term at each step until it finds the best submodel. Model subsets are compared using the Generalized cross validation (GCV) criterion described below. </p><p>The backward pass has an advantage over the forward pass: at any step it can choose any term to delete, whereas the forward pass at each step can only see the next pair of terms. </p><p>The forward pass adds terms in pairs, but the backward pass typically discards one side of the pair and so terms are often not seen in pairs in the final model. A paired hinge can be seen in the equation for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {y}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5716eb05cd3a7b14b6e7d45837c25edb501c4775" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.302ex; height:2.509ex;" alt="{\displaystyle {\widehat {y}}}"></span> in the first MARS example above; there are no complete pairs retained in the ozone example. </p> <div class="mw-heading mw-heading4"><h4 id="Generalized_cross_validation">Generalized cross validation</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_adaptive_regression_spline&action=edit&section=7" title="Edit section: Generalized cross validation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross-validation (statistics)</a>, <a href="/wiki/Model_selection" title="Model selection">Model selection</a>, and <a href="/wiki/Akaike_information_criterion" title="Akaike information criterion">Akaike information criterion</a></div> <p>The backward pass compares the performance of different models using Generalized Cross-Validation (GCV), a minor variant on the <a href="/wiki/Akaike_information_criterion" title="Akaike information criterion">Akaike information criterion</a> that approximates the <a href="/wiki/Leave-one-out_cross-validation" class="mw-redirect" title="Leave-one-out cross-validation">leave-one-out cross-validation</a> score in the special case where errors are Gaussian, or where the squared error loss function is used. GCV was introduced by Craven and <a href="/wiki/Grace_Wahba" title="Grace Wahba">Wahba</a> and extended by Friedman for MARS; lower values of GCV indicate better models. The formula for the GCV is </p> <dl><dd>GCV = RSS / (<i>N</i> · (1 − (effective number of parameters) / <i>N</i>)<sup>2</sup>)</dd></dl> <p>where RSS is the residual sum-of-squares measured on the training data and <i>N</i> is the number of observations (the number of rows in the <b>x</b> matrix). </p><p>The effective number of parameters is defined as </p> <dl><dd>(effective number of parameters) = (number of mars terms) + (penalty) · ((number of Mars terms) − 1 ) / 2</dd></dl> <p>where <b>penalty</b> is typically 2 (giving results equivalent to the <a href="/wiki/Akaike_information_criterion" title="Akaike information criterion">Akaike information criterion</a>) but can be increased by the user if they so desire. </p><p>Note that </p> <dl><dd>(number of Mars terms − 1 ) / 2</dd></dl> <p>is the number of hinge-function knots, so the formula penalizes the addition of knots. Thus the GCV formula adjusts (i.e. increases) the training RSS to penalize more complex models. We penalize flexibility because models that are too flexible will model the specific realization of noise in the data instead of just the systematic structure of the data. </p> <div class="mw-heading mw-heading3"><h3 id="Constraints">Constraints</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_adaptive_regression_spline&action=edit&section=8" title="Edit section: Constraints"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One constraint has already been mentioned: the user can specify the maximum number of terms in the forward pass. </p><p>A further constraint can be placed on the forward pass by specifying a maximum allowable degree of interaction. Typically only one or two degrees of interaction are allowed, but higher degrees can be used when the data warrants it. The maximum degree of interaction in the first MARS example above is one (i.e. no interactions or an <i>additive model</i>); in the ozone example it is two. </p><p>Other constraints on the forward pass are possible. For example, the user can specify that interactions are allowed only for certain input variables. Such constraints could make sense because of knowledge of the process that generated the data. </p> <div class="mw-heading mw-heading2"><h2 id="Pros_and_cons">Pros and cons</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_adaptive_regression_spline&action=edit&section=9" title="Edit section: Pros and cons"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Original_research plainlinks metadata ambox ambox-content ambox-Original_research" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/40px-Ambox_important.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/60px-Ambox_important.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/80px-Ambox_important.svg.png 2x" data-file-width="40" data-file-height="40" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>possibly contains <a href="/wiki/Wikipedia:No_original_research" title="Wikipedia:No original research">original research</a></b>.<span class="hide-when-compact"> Please <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Multivariate_adaptive_regression_spline&action=edit">improve it</a> by <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verifying</a> the claims made and adding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a>. Statements consisting only of original research should be removed.</span> <span class="date-container"><i>(<span class="date">October 2016</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>No regression modeling technique is best for all situations. The guidelines below are intended to give an idea of the pros and cons of MARS, but there will be exceptions to the guidelines. It is useful to compare MARS to <a href="/wiki/Recursive_partitioning" title="Recursive partitioning">recursive partitioning</a> and this is done below. (Recursive partitioning is also commonly called <i>regression trees</i>, <i>decision trees</i>, or <a href="/wiki/Predictive_analytics#Classification_and_regression_trees" title="Predictive analytics">CART</a>; see the <a href="/wiki/Decision_tree_learning" title="Decision tree learning">recursive partitioning</a> article for details). </p> <ul><li>MARS models are more flexible than <a href="/wiki/Linear_regression" title="Linear regression">linear regression</a> models.</li> <li>MARS models are simple to understand and interpret.<sup id="cite_ref-:0_5-0" class="reference"><a href="#cite_note-:0-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Compare the equation for ozone concentration above to, say, the innards of a trained <a href="/wiki/Artificial_neural_network" class="mw-redirect" title="Artificial neural network">neural network</a> or a <a href="/wiki/Random_forest" title="Random forest">random forest</a>.</li> <li>MARS can handle both continuous and <a href="/wiki/Categorical_data" class="mw-redirect" title="Categorical data">categorical data</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><sup id="cite_ref-Friedman_1991_7-0" class="reference"><a href="#cite_note-Friedman_1991-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> MARS tends to be better than recursive partitioning for numeric data because hinges are more appropriate for numeric variables than the piecewise constant segmentation used by recursive partitioning.</li> <li>Building MARS models often requires little or no data preparation.<sup id="cite_ref-:0_5-1" class="reference"><a href="#cite_note-:0-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The hinge functions automatically partition the input data, so the effect of outliers is contained. In this respect MARS is similar to <a href="/wiki/Recursive_partitioning" title="Recursive partitioning">recursive partitioning</a> which also partitions the data into disjoint regions, although using a different method.</li> <li>MARS (like recursive partitioning) does automatic <a href="/wiki/Feature_selection" title="Feature selection">variable selection</a> (meaning it includes important variables in the model and excludes unimportant ones). However, there can be some arbitrariness in the selection, especially when there are correlated predictors, and this can affect interpretability.<sup id="cite_ref-:0_5-2" class="reference"><a href="#cite_note-:0-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li> <li>MARS models tend to have a good bias-variance trade-off. The models are flexible enough to model non-linearity and variable interactions (thus MARS models have fairly low bias), yet the constrained form of MARS basis functions prevents too much flexibility (thus MARS models have fairly low variance).</li> <li>MARS is suitable for handling large datasets, and implementations run very quickly. However, recursive partitioning can be faster than MARS<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (March 2019)">citation needed</span></a></i>]</sup>.</li> <li>With MARS models, as with any non-parametric regression, parameter confidence intervals and other checks on the model cannot be calculated directly (unlike <a href="/wiki/Linear_regression" title="Linear regression">linear regression</a> models). <a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">Cross-validation</a> and related techniques must be used for validating the model instead.</li> <li>The <code>earth</code>, <code>mda</code>, and <code>polspline</code> implementations do not allow missing values in predictors, but free implementations of regression trees (such as <code>rpart</code> and <code>party</code>) do allow missing values using a technique called surrogate splits.</li> <li>MARS models can make predictions very quickly, as they only require evaluating a linear function of the predictors.</li> <li>The resulting fitted function is continuous, unlike recursive partitioning, which can give a more realistic model in some situations. (However, the model is not smooth or differentiable).</li></ul> <div class="mw-heading mw-heading2"><h2 id="Extensions_and_related_concepts">Extensions and related concepts</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_adaptive_regression_spline&action=edit&section=10" title="Edit section: Extensions and related concepts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Generalized_linear_model" title="Generalized linear model">Generalized linear models</a> (GLMs) can be incorporated into MARS models by applying a link function after the MARS model is built. Thus, for example, MARS models can incorporate <a href="/wiki/Logistic_regression" title="Logistic regression">logistic regression</a> to predict probabilities.</li> <li><a href="/wiki/Nonlinear_regression" title="Nonlinear regression">Non-linear regression</a> is used when the underlying form of the function is known and regression is used only to estimate the parameters of that function. MARS, on the other hand, estimates the functions themselves, albeit with severe constraints on the nature of the functions. (These constraints are necessary because discovering a model from the data is an <a href="/wiki/Inverse_problem" title="Inverse problem">inverse problem</a> that is not <a href="/wiki/Well-posed_problem" title="Well-posed problem">well-posed</a> without constraints on the model.)</li> <li><a href="/wiki/Recursive_partitioning" title="Recursive partitioning">Recursive partitioning</a> (commonly called CART). MARS can be seen as a generalization of recursive partitioning that allows for continuous models, which can provide a better fit for numerical data.</li> <li><a href="/wiki/Generalized_additive_model" title="Generalized additive model">Generalized additive models</a>. Unlike MARS, GAMs fit smooth <a href="/wiki/Local_regression" title="Local regression">loess</a> or polynomial <a href="/wiki/Spline_(mathematics)" title="Spline (mathematics)">splines</a> rather than hinge functions, and they do not automatically model variable interactions. The smoother fit and lack of regression terms reduces variance when compared to MARS, but ignoring variable interactions can worsen the bias.</li> <li><a href="/w/index.php?title=TSMARS&action=edit&redlink=1" class="new" title="TSMARS (page does not exist)">TSMARS</a>. Time Series Mars is the term used when MARS models are applied in a time series context. Typically in this set up the predictors are the lagged time series values resulting in autoregressive spline models. These models and extensions to include moving average spline models are described in "Univariate Time Series Modelling and Forecasting using TSMARS: A study of threshold time series autoregressive, seasonal and moving average models using TSMARS".</li> <li><a href="/w/index.php?title=Bayesian_MARS&action=edit&redlink=1" class="new" title="Bayesian MARS (page does not exist)">Bayesian MARS</a> (BMARS) uses the same model form, but builds the model using a Bayesian approach. It may arrive at different optimal MARS models because the model building approach is different. The result of BMARS is typically an ensemble of posterior samples of MARS models, which allows for probabilistic prediction.<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></li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_adaptive_regression_spline&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Linear_regression" title="Linear regression">Linear regression</a></li> <li><a href="/wiki/Local_regression" title="Local regression">Local regression</a></li> <li><a href="/wiki/Rational_function_modeling" class="mw-redirect" title="Rational function modeling">Rational function modeling</a></li> <li><a href="/wiki/Segmented_regression" title="Segmented regression">Segmented regression</a></li> <li><a href="/wiki/Spline_interpolation" title="Spline interpolation">Spline interpolation</a></li> <li><a href="/wiki/Spline_regression" class="mw-redirect" title="Spline regression">Spline regression</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_adaptive_regression_spline&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFFriedman1991" class="citation journal cs1">Friedman, J. H. (1991). "Multivariate Adaptive Regression Splines". <i>The Annals of Statistics</i>. <b>19</b> (1): <span class="nowrap">1–</span>67. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.382.970">10.1.1.382.970</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176347963">10.1214/aos/1176347963</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2241837">2241837</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1091842">1091842</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0765.62064">0765.62064</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Annals+of+Statistics&rft.atitle=Multivariate+Adaptive+Regression+Splines&rft.volume=19&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E67&rft.date=1991&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0765.62064%23id-name%3DZbl&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2241837%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1214%2Faos%2F1176347963&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.382.970%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1091842%23id-name%3DMR&rft.aulast=Friedman&rft.aufirst=J.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultivariate+adaptive+regression+spline" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://cran.r-project.org/web/packages/earth/index.html">CRAN Package earth</a></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://orange.biolab.si/blog/2011/12/20/earth-multivariate-adaptive-regression-splines/">Earth – Multivariate adaptive regression splines in Orange (Python machine learning library)</a></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="/wiki/Friedman,_J._H." class="mw-redirect" title="Friedman, J. H.">Friedman, J. H.</a> (1993) <i>Fast MARS</i>, Stanford University Department of Statistics, Technical Report 110</span> </li> <li id="cite_note-:0-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_5-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_5-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKuhnJohnson2013" class="citation book cs1">Kuhn, Max; Johnson, Kjell (2013). <i>Applied Predictive Modeling</i>. New York, NY: Springer New York. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4614-6849-3">10.1007/978-1-4614-6849-3</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781461468486" title="Special:BookSources/9781461468486"><bdi>9781461468486</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+Predictive+Modeling&rft.place=New+York%2C+NY&rft.pub=Springer+New+York&rft.date=2013&rft_id=info%3Adoi%2F10.1007%2F978-1-4614-6849-3&rft.isbn=9781461468486&rft.aulast=Kuhn&rft.aufirst=Max&rft.au=Johnson%2C+Kjell&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultivariate+adaptive+regression+spline" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFriedman1993" class="citation book cs1"><a href="/wiki/Friedman,_J._H." class="mw-redirect" title="Friedman, J. H.">Friedman, Jerome H.</a> (1993). "Estimating Functions of Mixed Ordinal and Categorical Variables Using Adaptive Splines". In Stephan Morgenthaler; Elvezio Ronchetti; Werner Stahel (eds.). <i>New Directions in Statistical Data Analysis and Robustness</i>. Birkhauser.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Estimating+Functions+of+Mixed+Ordinal+and+Categorical+Variables+Using+Adaptive+Splines&rft.btitle=New+Directions+in+Statistical+Data+Analysis+and+Robustness&rft.pub=Birkhauser&rft.date=1993&rft.aulast=Friedman&rft.aufirst=Jerome+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultivariate+adaptive+regression+spline" class="Z3988"></span></span> </li> <li id="cite_note-Friedman_1991-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Friedman_1991_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFriedman1991" class="citation journal cs1">Friedman, Jerome H. (1991-06-01). <a rel="nofollow" class="external text" href="https://apps.dtic.mil/sti/citations/ADA590939">"Estimating Functions of Mixed Ordinal and Categorical Variables Using Adaptive Splines"</a>. <i>DTIC</i>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220411085148/https://apps.dtic.mil/sti/citations/ADA590939">Archived</a> from the original on April 11, 2022<span class="reference-accessdate">. Retrieved <span class="nowrap">2022-04-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=DTIC&rft.atitle=Estimating+Functions+of+Mixed+Ordinal+and+Categorical+Variables+Using+Adaptive+Splines&rft.date=1991-06-01&rft.aulast=Friedman&rft.aufirst=Jerome+H.&rft_id=https%3A%2F%2Fapps.dtic.mil%2Fsti%2Fcitations%2FADA590939&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultivariate+adaptive+regression+spline" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDenisonMallickSmith1998" class="citation journal cs1">Denison, D. G. T.; Mallick, B. K.; Smith, A. F. M. (1 December 1998). <a rel="nofollow" class="external text" href="https://link.springer.com/content/pdf/10.1023/A:1008824606259.pdf">"Bayesian MARS"</a> <span class="cs1-format">(PDF)</span>. <i>Statistics and Computing</i>. <b>8</b> (4): <span class="nowrap">337–</span>346. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1008824606259">10.1023/A:1008824606259</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1573-1375">1573-1375</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:12570055">12570055</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Statistics+and+Computing&rft.atitle=Bayesian+MARS&rft.volume=8&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E337-%3C%2Fspan%3E346&rft.date=1998-12-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A12570055%23id-name%3DS2CID&rft.issn=1573-1375&rft_id=info%3Adoi%2F10.1023%2FA%3A1008824606259&rft.aulast=Denison&rft.aufirst=D.+G.+T.&rft.au=Mallick%2C+B.+K.&rft.au=Smith%2C+A.+F.+M.&rft_id=https%3A%2F%2Flink.springer.com%2Fcontent%2Fpdf%2F10.1023%2FA%3A1008824606259.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultivariate+adaptive+regression+spline" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_adaptive_regression_spline&action=edit&section=13" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Hastie T., Tibshirani R., and Friedman J.H. (2009) <a rel="nofollow" class="external text" href="http://www-stat.stanford.edu/~tibs/ElemStatLearn"><i>The Elements of Statistical Learning</i></a>, 2nd edition. Springer, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-84857-0" title="Special:BookSources/978-0-387-84857-0">978-0-387-84857-0</a> (has a section on MARS)</li> <li>Faraway J. (2005) <a rel="nofollow" class="external text" href="http://www.maths.bath.ac.uk/~jjf23"><i>Extending the Linear Model with R</i></a>, CRC, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-58488-424-8" title="Special:BookSources/978-1-58488-424-8">978-1-58488-424-8</a> (has an example using MARS with R)</li> <li>Heping Zhang and Burton H. Singer (2010) <a rel="nofollow" class="external text" href="https://www.amazon.com/Recursive-Partitioning-Applications-Springer-Statistics/dp/1441968237"><i>Recursive Partitioning and Applications</i></a>, 2nd edition. Springer, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4419-6823-4" title="Special:BookSources/978-1-4419-6823-4">978-1-4419-6823-4</a> (has a chapter on MARS and discusses some tweaks to the algorithm)</li> <li>Denison D.G.T., Holmes C.C., Mallick B.K., and Smith A.F.M. (2004) <a rel="nofollow" class="external text" href="http://www.stat.tamu.edu/~bmallick/wileybook/book_code.html"><i>Bayesian Methods for Nonlinear Classification and Regression</i></a>, Wiley, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-49036-4" title="Special:BookSources/978-0-471-49036-4">978-0-471-49036-4</a></li> <li>Berk R.A. (2008) <i>Statistical learning from a regression perspective</i>, Springer, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-77500-5" title="Special:BookSources/978-0-387-77500-5">978-0-387-77500-5</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Multivariate_adaptive_regression_spline&action=edit&section=14" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-External_links plainlinks metadata ambox ambox-style ambox-external_links" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article's <b>use of <a href="/wiki/Wikipedia:External_links" title="Wikipedia:External links">external links</a> may not follow Wikipedia's policies or guidelines</b>.<span class="hide-when-compact"> Please <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Multivariate_adaptive_regression_spline&action=edit">improve this article</a> by removing <a href="/wiki/Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_mirror_or_a_repository_of_links,_images,_or_media_files" title="Wikipedia:What Wikipedia is not">excessive</a> or <a href="/wiki/Wikipedia:External_links" title="Wikipedia:External links">inappropriate</a> external links, and converting useful links where appropriate into <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">footnote references</a>.</span> <span class="date-container"><i>(<span class="date">October 2016</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>Several free and commercial software packages are available for fitting MARS-type models. </p> <dl><dt>Free software</dt> <dd></dd></dl> <ul><li><a href="/wiki/R_(programming_language)" title="R (programming language)">R</a> packages: <ul><li><code>earth</code> function in the <code><a rel="nofollow" class="external text" href="https://cran.r-project.org/web/packages/earth/index.html">earth</a></code> package</li> <li><code>mars</code> function in the <code><a rel="nofollow" class="external text" href="https://cran.r-project.org/web/packages/mda/index.html">mda</a></code> package</li> <li><code>polymars</code> function in the <code><a rel="nofollow" class="external text" href="https://cran.r-project.org/web/packages/polspline/index.html">polspline</a></code> package. Not Friedman's MARS.</li> <li><code>bass</code> function in the <code><a rel="nofollow" class="external text" href="https://cran.r-project.org/web/packages/BASS/index.html">BASS</a></code> package for Bayesian MARS.</li></ul></li> <li>Matlab code: <ul><li><a rel="nofollow" class="external text" href="http://www.cs.rtu.lv/jekabsons/regression.html">ARESLab: Adaptive Regression Splines toolbox for Matlab</a></li> <li><a rel="nofollow" class="external text" href="https://web.stat.tamu.edu/~bmallick/wileybook/book_code.html">Code</a> from the book <i>Bayesian Methods for Nonlinear Classification and Regression</i><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> for Bayesian MARS.</li></ul></li> <li>Python <ul><li><a rel="nofollow" class="external text" href="http://orange.biolab.si/blog/2011/12/20/earth-multivariate-adaptive-regression-splines/">Earth – Multivariate adaptive regression splines</a></li> <li><a rel="nofollow" class="external text" href="https://github.com/jcrudy/py-earth/">py-earth</a></li> <li><a rel="nofollow" class="external text" href="https://github.com/lanl/pyBASS">pyBASS</a> for Bayesian MARS.</li></ul></li></ul> <dl><dt>Commercial software</dt> <dd></dd></dl> <ul><li><a rel="nofollow" class="external text" href="http://www.salford-systems.com/mars.php">MARS</a> from Salford Systems. Based on Friedman's implementation.</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20101203023609/http://www.statsoft.com/products/data-mining-solutions/">STATISTICA Data Miner</a> from StatSoft</li> <li><a rel="nofollow" class="external text" href="http://support.sas.com/documentation/cdl/en/statug/65328/HTML/default/viewer.htm#statug_adaptivereg_overview.htm">ADAPTIVEREG</a> from SAS.</li></ul> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDenisonHolmesMallickSmith2002" class="citation book cs1">Denison, D. G. T.; Holmes, C. C.; Mallick, B. K.; Smith, A. F. M. (2002). <i>Bayesian methods for nonlinear classification and regression</i>. Chichester, England: Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-49036-4" title="Special:BookSources/978-0-471-49036-4"><bdi>978-0-471-49036-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Bayesian+methods+for+nonlinear+classification+and+regression&rft.place=Chichester%2C+England&rft.pub=Wiley&rft.date=2002&rft.isbn=978-0-471-49036-4&rft.aulast=Denison&rft.aufirst=D.+G.+T.&rft.au=Holmes%2C+C.+C.&rft.au=Mallick%2C+B.+K.&rft.au=Smith%2C+A.+F.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultivariate+adaptive+regression+spline" class="Z3988"></span></span> </li> </ol></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Multivariate_adaptive_regression_spline&oldid=1180139083">https://en.wikipedia.org/w/index.php?title=Multivariate_adaptive_regression_spline&oldid=1180139083</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Nonparametric_regression" title="Category:Nonparametric regression">Nonparametric regression</a></li><li><a href="/wiki/Category:Machine_learning" title="Category:Machine learning">Machine learning</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Articles_that_may_contain_original_research_from_October_2016" title="Category:Articles that may contain original research from October 2016">Articles that may contain original research from October 2016</a></li><li><a href="/wiki/Category:All_articles_that_may_contain_original_research" title="Category:All articles that may contain original research">All articles that may contain original research</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_March_2019" title="Category:Articles with unsourced statements from March 2019">Articles with unsourced statements from March 2019</a></li><li><a href="/wiki/Category:Wikipedia_external_links_cleanup_from_October_2016" title="Category:Wikipedia external links cleanup from October 2016">Wikipedia external links cleanup from October 2016</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 14 October 2023, at 19:29<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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