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<title>Additive model</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Additive model</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about the statistical method. For additive color models, see <a href="Additive_color" title="Additive color">Additive color</a>.</div>

<p>In <a href="Statistics" title="Statistics">statistics</a>, an <b>additive model</b> (<b>AM</b>) is a <a href="Nonparametric_regression" title="Nonparametric regression">nonparametric regression</a> method. It was suggested by <a href="Jerome_H._Friedman" title="Jerome H. Friedman">Jerome H. Friedman</a> and Werner Stuetzle (1981)<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> and is an essential part of the <a href="Alternating_conditional_expectations" title="Alternating conditional expectations">ACE</a> algorithm. The <i>AM</i> uses a one-dimensional <a href="Smoothing" title="Smoothing">smoother</a> to build a restricted class of nonparametric regression models. Because of this, it is less affected by the <a href="Curse_of_dimensionality" title="Curse of dimensionality">curse of dimensionality</a> than a <i>p</i>-dimensional smoother. Furthermore, the <i>AM</i> is more flexible than a <a href="Linear_regression" title="Linear regression">standard linear model</a>, while being more interpretable than a general regression surface at the cost of approximation errors. Problems with <i>AM</i>, like many other machine-learning methods, include <a href="Model_selection" title="Model selection">model selection</a>, <a href="Overfitting" title="Overfitting">overfitting</a>, and <a href="Multicollinearity" title="Multicollinearity">multicollinearity</a>.

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

<p>Given a <a href="Data" title="Data">data</a> set <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 \{y_{i},\,x_{i1},\ldots ,x_{ip}\}_{i=1}^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \{y_{i},\,x_{i1},\ldots ,x_{ip}\}_{i=1}^{n}}</annotation>

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</math></span><img src="./bb65235f66e69d8c663b673c5952ee7a64e9246d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.671ex; height:3.009ex;" alt="{\displaystyle \{y_{i},\,x_{i1},\ldots ,x_{ip}\}_{i=1}^{n}}" loading="lazy"></span> of <i>n</i> <a href="Statistical_unit" title="Statistical unit">statistical units</a>, 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 \{x_{i1},\ldots ,x_{ip}\}_{i=1}^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \{x_{i1},\ldots ,x_{ip}\}_{i=1}^{n}}</annotation>

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</math></span><img src="./6be9d5f9087f4d3694f8304ae3d75e2311db6408.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.311ex; height:3.009ex;" alt="{\displaystyle \{x_{i1},\ldots ,x_{ip}\}_{i=1}^{n}}" loading="lazy"></span> represent predictors 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 y_{i}}">

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

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</math></span><img src="./67d30d30b6c2dbe4d6f150d699de040937ecc95f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.009ex;" alt="{\displaystyle y_{i}}" loading="lazy"></span> is the outcome, the <i>additive model</i> takes the form

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {E} [y_{i}|x_{i1},\ldots ,x_{ip}]=\beta _{0}+\sum _{j=1}^{p}f_{j}(x_{ij})}</annotation>

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</math></span><img src="./81d5cc6bcb970849325c676648504fa8a39a763b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:35.264ex; height:7.343ex;" alt="{\displaystyle \mathrm {E} [y_{i}|x_{i1},\ldots ,x_{ip}]=\beta _{0}+\sum _{j=1}^{p}f_{j}(x_{ij})}" loading="lazy"></span></dd></dl>

<p>or

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<annotation encoding="application/x-tex">{\displaystyle Y=\beta _{0}+\sum _{j=1}^{p}f_{j}(X_{j})+\varepsilon }</annotation>

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</math></span><img src="./8b75dc8895dc8eced4ed4207fc9ecb7404d71b83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:24.44ex; height:7.343ex;" alt="{\displaystyle Y=\beta _{0}+\sum _{j=1}^{p}f_{j}(X_{j})+\varepsilon }" loading="lazy"></span></dd></dl>

<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 \mathrm {E} [\epsilon ]=0}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {E} [\epsilon ]=0}</annotation>

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</math></span><img src="./f7494dee92fe5ac5a8d8a61fa72e73482ad679d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.082ex; height:2.843ex;" alt="{\displaystyle \mathrm {E} [\epsilon ]=0}" loading="lazy"></span>, <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 \mathrm {Var} (\epsilon )=\sigma ^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {Var} (\epsilon )=\sigma ^{2}}</annotation>

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</math></span><img src="./83cf6c76399264ee62920a70c1f871a3220750ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.054ex; height:3.176ex;" alt="{\displaystyle \mathrm {Var} (\epsilon )=\sigma ^{2}}" loading="lazy"></span> 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 \mathrm {E} [f_{j}(X_{j})]=0}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {E} [f_{j}(X_{j})]=0}</annotation>

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</math></span><img src="./ebc45cdd662236cf2546a2a70c1c03383d50ac3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.83ex; height:3.009ex;" alt="{\displaystyle \mathrm {E} [f_{j}(X_{j})]=0}" loading="lazy"></span>. The functions <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 f_{j}(x_{ij})}">

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<annotation encoding="application/x-tex">{\displaystyle f_{j}(x_{ij})}</annotation>

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</math></span><img src="./225d58d77baff64752a0b0e45d477d78057254ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.665ex; height:3.009ex;" alt="{\displaystyle f_{j}(x_{ij})}" loading="lazy"></span> are unknown <a href="Smooth_function" class="mw-redirect" title="Smooth function">smooth functions</a> fit from the data. Fitting the <i>AM</i> (i.e. the functions <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 f_{j}(x_{ij})}">

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<annotation encoding="application/x-tex">{\displaystyle f_{j}(x_{ij})}</annotation>

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</math></span><img src="./225d58d77baff64752a0b0e45d477d78057254ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.665ex; height:3.009ex;" alt="{\displaystyle f_{j}(x_{ij})}" loading="lazy"></span>) can be done using the <a href="Backfitting_algorithm" title="Backfitting algorithm">backfitting algorithm</a> proposed by Andreas Buja, <a href="Trevor_Hastie" title="Trevor Hastie">Trevor Hastie</a> and <a href="Robert_Tibshirani" title="Robert Tibshirani">Robert Tibshirani</a> (1989).<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>

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<ul><li><a href="Generalized_additive_model" title="Generalized additive model">Generalized additive model</a></li>

<li><a href="Backfitting_algorithm" title="Backfitting algorithm">Backfitting algorithm</a></li>

<li><a href="Projection_pursuit_regression" title="Projection pursuit regression">Projection pursuit regression</a></li>

<li><a href="Generalized_additive_model_for_location%2C_scale%2C_and_shape" class="mw-redirect" title="Generalized additive model for location, scale, and shape">Generalized additive model for location, scale, and shape</a> (GAMLSS)</li>

<li><a href="Median_polish" title="Median polish">Median polish</a></li>

<li><a href="Projection_pursuit" title="Projection pursuit">Projection pursuit</a></li></ul>

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

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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="Friedman%2C_J.H." class="mw-redirect" title="Friedman, J.H.">Friedman, J.H.</a> and Stuetzle, W. (1981). "Projection Pursuit Regression", <i>Journal of the American Statistical Association</i> 76:817–823. <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F01621459.1981.10477729">10.1080/01621459.1981.10477729</a></span>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Buja, A., Hastie, T., and Tibshirani, R. (1989). "Linear Smoothers and Additive Models", <i>The Annals of Statistics</i> 17(2):453–555. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2241560">2241560</a></span>

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

<ul><li>Breiman, L. and <a href="Friedman%2C_J.H." class="mw-redirect" title="Friedman, J.H.">Friedman, J.H.</a> (1985). "Estimating Optimal Transformations for Multiple Regression and Correlation", <i><a href="Journal_of_the_American_Statistical_Association" title="Journal of the American Statistical Association">Journal of the American Statistical Association</a></i> 80:580–598. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F01621459.1985.10478157">10.1080/01621459.1985.10478157</a></li></ul></div><div><div class="zim-footer">

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