devel/gam.html



<!doctype html>
<html lang="en" class="no-js">
  <head>
    
      <meta charset="utf-8">
      <meta name="viewport" content="width=device-width,initial-scale=1">
      
      
        <link rel="prev" href="generated/statsmodels.genmod.cov_struct.Nested.update.html">
      
      
        <link rel="next" href="generated/statsmodels.gam.generalized_additive_model.GLMGam.html">
      
      
      <link rel="icon" href="_static/favicon.ico">
    
    
        <title>Generalized Additive Models (GAM) - statsmodels 0.15.0 (+647)</title>
      
    
  <link rel="icon" type="image/png" sizes="32x32" href="_static/icons/favicon-32x32.png">
  <link rel="icon" type="image/png" sizes="16x16" href="_static/icons/favicon-16x16.png">
  <link rel="manifest" href="_static/icons/site.webmanifest">
  <link rel="mask-icon" href="_static/icons/safari-pinned-tab.svg" color="#919191">
  <meta name="msapplication-TileColor" content="#2b5797">
  <meta name="msapplication-config" content="_static/icons/browserconfig.xml">
  <link rel="stylesheet" href="_static/stylesheets/examples.css">
  <link rel="stylesheet" href="_static/stylesheets/deprecation.css">
    
      
        <style>:root{--md-text-font:"Roboto";--md-code-font:"Roboto Mono"}</style>
      
    
        <link rel="stylesheet" type="text/css" href="_static/sphinx_immaterial_theme.96fe8683ff2bd71e9.min.css?v=13adf062" />
        <link rel="stylesheet" type="text/css" href="_static/graphviz.css?v=fd3f3429" />
        <link rel="stylesheet" type="text/css" href="_static/plot_directive.css" />
    <script>__md_scope=new URL(".",location),__md_hash=e=>[...e].reduce(((e,_)=>(e<<5)-e+_.charCodeAt(0)),0),__md_get=(e,_=localStorage,t=__md_scope)=>JSON.parse(_.getItem(t.pathname+"."+e)),__md_set=(e,_,t=localStorage,a=__md_scope)=>{try{t.setItem(a.pathname+"."+e,JSON.stringify(_))}catch(e){}}</script>
    
      
  </head>
  
  
    <body dir="ltr" data-md-color-scheme="default" data-md-color-primary="indigo" data-md-color-accent="blue">
  
    
    <input class="md-toggle" data-md-toggle="drawer" type="checkbox" id="__drawer" autocomplete="off">
    <input class="md-toggle" data-md-toggle="search" type="checkbox" id="__search" autocomplete="off">
    <label class="md-overlay" for="__drawer"></label>
    <div data-md-component="skip">
      
        
        <a href="#examples" class="md-skip">
          Skip to content
        </a>
      
    </div>
    <div data-md-component="announce">
      
    </div>
    
      <div data-md-color-scheme="default" data-md-component="outdated" hidden>
        
      </div>
    
    
<header class="md-header md-header--shadow" data-md-component="header">
  <nav class="md-header__inner md-grid" aria-label="Header">
    <a href="index.html" title="statsmodels 0.15.0 (+647)" class="md-header__button md-logo" aria-label="statsmodels 0.15.0 (+647)" data-md-component="logo">
      <img src="_static/statsmodels-logo-v2-bw.svg" alt="logo">
    </a>
    <label class="md-header__button md-icon" for="__drawer">
      
      <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 6h18v2H3zm0 5h18v2H3zm0 5h18v2H3z"/></svg>
    </label>
    <div class="md-header__title" data-md-component="header-title">
      <div class="md-header__ellipsis">
        <div class="md-header__topic">
          <span class="md-ellipsis">
            statsmodels 0.15.0 (+647)
          </span>
        </div>
        <div class="md-header__topic" data-md-component="header-topic">
          <span class="md-ellipsis">
            
              Generalized Additive Models (GAM)
            
          </span>
        </div>
      </div>
    </div>
    
      
      <label class="md-header__button md-icon" for="__search">
        
        <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M9.5 3A6.5 6.5 0 0 1 16 9.5c0 1.61-.59 3.09-1.56 4.23l.27.27h.79l5 5-1.5 1.5-5-5v-.79l-.27-.27A6.52 6.52 0 0 1 9.5 16 6.5 6.5 0 0 1 3 9.5 6.5 6.5 0 0 1 9.5 3m0 2C7 5 5 7 5 9.5S7 14 9.5 14 14 12 14 9.5 12 5 9.5 5"/></svg>
      </label>
      <div class="md-search" data-md-component="search" role="dialog">
  <label class="md-search__overlay" for="__search"></label>
  <div class="md-search__inner" role="search">
    <form class="md-search__form" name="search">
      <input type="text" class="md-search__input" name="query" aria-label="Search" placeholder="Search" autocapitalize="off" autocorrect="off" autocomplete="off" spellcheck="false" data-md-component="search-query" required>
      <label class="md-search__icon md-icon" for="__search">
        
        <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M9.5 3A6.5 6.5 0 0 1 16 9.5c0 1.61-.59 3.09-1.56 4.23l.27.27h.79l5 5-1.5 1.5-5-5v-.79l-.27-.27A6.52 6.52 0 0 1 9.5 16 6.5 6.5 0 0 1 3 9.5 6.5 6.5 0 0 1 9.5 3m0 2C7 5 5 7 5 9.5S7 14 9.5 14 14 12 14 9.5 12 5 9.5 5"/></svg>
        
        <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M20 11v2H8l5.5 5.5-1.42 1.42L4.16 12l7.92-7.92L13.5 5.5 8 11z"/></svg>
      </label>
      <nav class="md-search__options" aria-label="Search">
        
        <button type="reset" class="md-search__icon md-icon" title="Clear" aria-label="Clear" tabindex="-1">
          
          <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M19 6.41 17.59 5 12 10.59 6.41 5 5 6.41 10.59 12 5 17.59 6.41 19 12 13.41 17.59 19 19 17.59 13.41 12z"/></svg>
        </button>
      </nav>
      
    </form>
    <div class="md-search__output">
      <div class="md-search__scrollwrap" tabindex="0" data-md-scrollfix>
        <div class="md-search-result" data-md-component="search-result">
          <div class="md-search-result__meta">
            Initializing search
          </div>
          <ol class="md-search-result__list" role="presentation"></ol>
        </div>
      </div>
    </div>
  </div>
</div>
    
    
      <div class="md-header__source">
        <a href="https://github.com/statsmodels/statsmodels/" title="Go to repository" class="md-source" data-md-component="source">
  <div class="md-source__icon md-icon">
    
    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 496 512"><!--! Font Awesome Free 6.7.2 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2024 Fonticons, Inc.--><path d="M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6m-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3m44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9M244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8M97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1m-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7m32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1m-11.4-14.7c-1.6 1-1.6 3.6 0 5.9s4.3 3.3 5.6 2.3c1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2"/></svg>
  </div>
  <div class="md-source__repository">
    statsmodels
  </div>
</a>
      </div>
    
  </nav>
  
</header>
    
    <div class="md-container" data-md-component="container">
      
      
      <main class="md-main" data-md-component="main">
        <div class="md-main__inner md-grid">
          
            
              <div class="md-sidebar md-sidebar--primary" data-md-component="sidebar" data-md-type="navigation" >
                <div class="md-sidebar__scrollwrap">
                  <div class="md-sidebar__inner">
                    

<nav class="md-nav md-nav--primary" aria-label="Navigation" data-md-level="0">
  <label class="md-nav__title" for="__drawer">
    <a href="index.html" title="statsmodels 0.15.0 (+647)" class="md-nav__button md-logo" aria-label="statsmodels 0.15.0 (+647)" data-md-component="logo">
      <img src="_static/statsmodels-logo-v2-bw.svg" alt="logo">
    </a>
    statsmodels 0.15.0 (+647)
  </label>
  
    <div class="md-nav__source">
      <a href="https://github.com/statsmodels/statsmodels/" title="Go to repository" class="md-source" data-md-component="source">
  <div class="md-source__icon md-icon">
    
    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 496 512"><!--! Font Awesome Free 6.7.2 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2024 Fonticons, Inc.--><path d="M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6m-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3m44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9M244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8M97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1m-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7m32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1m-11.4-14.7c-1.6 1-1.6 3.6 0 5.9s4.3 3.3 5.6 2.3c1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2"/></svg>
  </div>
  <div class="md-source__repository">
    statsmodels
  </div>
</a>
    </div>
  
  <ul class="md-nav__list" data-md-scrollfix>
    
      
    <li class="md-nav__item">
      <a href="install.html" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    <span title="/install.rst (reference label)">Installing statsmodels</span>
    
  </span>
  

      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="gettingstarted.html" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    <span title="/gettingstarted.rst (reference label)">Getting started</span>
    
  </span>
  

      </a>
    </li>
  

    <li class="md-nav__item md-nav__item--active md-nav__item--nested">
      
        
        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" checked>
        
          
          <div class="md-nav__link md-nav__container">
            <a href="user-guide.html" class="md-nav__link ">
              
  
  <span class="md-ellipsis">
    <span title="/user-guide.rst (reference label)">User Guide</span>
    
  </span>
  

            </a>
            
              
              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="0">
                <span class="md-nav__icon md-icon"></span>
              </label>
            
          </div>
        
        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="true">
          <label class="md-nav__title" for="__nav_3">
            <span class="md-nav__icon md-icon"></span>
            <span title="/user-guide.rst (reference label)">User Guide</span>
          </label>
          <ul class="md-nav__list" data-md-scrollfix>
            
              
    <li class="md-nav__item">
      <a href="user-guide.html#background" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    <span title="/user-guide.rst#background (reference label)">Background</span>
    
  </span>
  

      </a>
    </li>
  

    <li class="md-nav__item md-nav__item--active md-nav__item--nested">
      
        
        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3_2" checked>
        
          
          <div class="md-nav__link md-nav__container">
            <a href="user-guide.html#regression-and-linear-models" class="md-nav__link ">
              
  
  <span class="md-ellipsis">
    <span title="/user-guide.rst#regression-and-linear-models (reference label)">Regression and Linear Models</span>
    
  </span>
  

            </a>
            
              
              <label class="md-nav__link " for="__nav_3_2" id="__nav_3_2_label" tabindex="0">
                <span class="md-nav__icon md-icon"></span>
              </label>
            
          </div>
        
        <nav class="md-nav" data-md-level="2" aria-labelledby="__nav_3_2_label" aria-expanded="true">
          <label class="md-nav__title" for="__nav_3_2">
            <span class="md-nav__icon md-icon"></span>
            <span title="/user-guide.rst#regression-and-linear-models (reference label)">Regression and Linear Models</span>
          </label>
          <ul class="md-nav__list" data-md-scrollfix>
            
              
    <li class="md-nav__item">
      <a href="regression.html" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    <span title="/regression.rst (reference label)">Linear Regression</span>
    
  </span>
  

      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="glm.html" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    <span title="/glm.rst (reference label)">Generalized Linear Models</span>
    
  </span>
  

      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="gee.html" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    <span title="/gee.rst (reference label)">Generalized Estimating Equations</span>
    
  </span>
  

      </a>
    </li>
  

    <li class="md-nav__item md-nav__item--active md-nav__current-nested md-nav__item--nested">
      
        
        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3_2_4" checked>
        
          
          <div class="md-nav__link md-nav__container">
            <a href="#" class="md-nav__link md-nav__link--active">
              
  
  <span class="md-ellipsis">
    <span title="/gam.rst (reference label)">Generalized Additive Models <wbr>(GAM)</span>
    
  </span>
  

            </a>
            
              
              <label class="md-nav__link md-nav__link--active" for="__nav_3_2_4" id="__nav_3_2_4_label" tabindex="0">
                <span class="md-nav__icon md-icon"></span>
              </label>
            
          </div>
        
        <nav class="md-nav" data-md-level="3" aria-labelledby="__nav_3_2_4_label" aria-expanded="true">
          <label class="md-nav__title" for="__nav_3_2_4">
            <span class="md-nav__icon md-icon"></span>
            <span title="/gam.rst (reference label)">Generalized Additive Models <wbr>(GAM)</span>
          </label>
          <ul class="md-nav__list" data-md-scrollfix>
            
              
    <li class="md-nav__item md-nav__item--nested">
      
        
        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3_2_4_1" >
        
          
          <div class="md-nav__link md-nav__container">
            <a href="#module-statsmodels.gam.generalized_additive_model" class="md-nav__link ">
              
  
  <span class="md-ellipsis">
    <span title="statsmodels.gam.generalized_additive_model (Python module)">Module Reference</span>
    
  </span>
  

            </a>
            
              
              <label class="md-nav__link " for="__nav_3_2_4_1" id="__nav_3_2_4_1_label" tabindex="0">
                <span class="md-nav__icon md-icon"></span>
              </label>
            
          </div>
        
        <nav class="md-nav" data-md-level="4" aria-labelledby="__nav_3_2_4_1_label" aria-expanded="false">
          <label class="md-nav__title" for="__nav_3_2_4_1">
            <span class="md-nav__icon md-icon"></span>
            <span title="statsmodels.gam.generalized_additive_model (Python module)">Module Reference</span>
          </label>
          <ul class="md-nav__list" data-md-scrollfix>
            
              
    <li class="md-nav__item md-nav__item--nested">
      
        
        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3_2_4_1_1" >
        
          
          <div class="md-nav__link md-nav__container">
            <a href="#model-class" class="md-nav__link ">
              
  
  <span class="md-ellipsis">
    <span title="/gam.rst#model-class (reference label)">Model Class</span>
    
  </span>
  

            </a>
            
              
              <label class="md-nav__link " for="__nav_3_2_4_1_1" id="__nav_3_2_4_1_1_label" tabindex="0">
                <span class="md-nav__icon md-icon"></span>
              </label>
            
          </div>
        
        <nav class="md-nav" data-md-level="5" aria-labelledby="__nav_3_2_4_1_1_label" aria-expanded="false">
          <label class="md-nav__title" for="__nav_3_2_4_1_1">
            <span class="md-nav__icon md-icon"></span>
            <span title="/gam.rst#model-class (reference label)">Model Class</span>
          </label>
          <ul class="md-nav__list" data-md-scrollfix>
            
              
    <li class="md-nav__item">
      <a href="generated/statsmodels.gam.generalized_additive_model.GLMGam.html" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    <span title="/generated/statsmodels.gam.generalized_additive_model.glmgam.rst (reference label)">statsmodels.<wbr>gam.<wbr>generalized_<wbr>additive_<wbr>model.<wbr>GLMGam</span>
    
  </span>
  

      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="generated/statsmodels.gam.generalized_additive_model.LogitGam.html" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    <span title="/generated/statsmodels.gam.generalized_additive_model.logitgam.rst (reference label)">statsmodels.<wbr>gam.<wbr>generalized_<wbr>additive_<wbr>model.<wbr>Logit<wbr>Gam</span>
    
  </span>
  

      </a>
    </li>
  

          </ul>
        </nav>
      
    </li>
  
  
    <li class="md-nav__item md-nav__item--nested">
      
        
        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3_2_4_1_2" >
        
          
          <div class="md-nav__link md-nav__container">
            <a href="#results-classes" class="md-nav__link ">
              
  
  <span class="md-ellipsis">
    <span title="/gam.rst#results-classes (reference label)">Results Classes</span>
    
  </span>
  

            </a>
            
              
              <label class="md-nav__link " for="__nav_3_2_4_1_2" id="__nav_3_2_4_1_2_label" tabindex="0">
                <span class="md-nav__icon md-icon"></span>
              </label>
            
          </div>
        
        <nav class="md-nav" data-md-level="5" aria-labelledby="__nav_3_2_4_1_2_label" aria-expanded="false">
          <label class="md-nav__title" for="__nav_3_2_4_1_2">
            <span class="md-nav__icon md-icon"></span>
            <span title="/gam.rst#results-classes (reference label)">Results Classes</span>
          </label>
          <ul class="md-nav__list" data-md-scrollfix>
            
              
    <li class="md-nav__item">
      <a href="generated/statsmodels.gam.generalized_additive_model.GLMGamResults.html" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    <span title="/generated/statsmodels.gam.generalized_additive_model.glmgamresults.rst (reference label)">statsmodels.<wbr>gam.<wbr>generalized_<wbr>additive_<wbr>model.<wbr>GLMGam<wbr>Results</span>
    
  </span>
  

      </a>
    </li>
  

          </ul>
        </nav>
      
    </li>
  
  
    <li class="md-nav__item md-nav__item--nested">
      
        
        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3_2_4_1_3" >
        
          
          <div class="md-nav__link md-nav__container">
            <a href="#module-statsmodels.gam.smooth_basis" class="md-nav__link ">
              
  
  <span class="md-ellipsis">
    <span title="statsmodels.gam.smooth_basis (Python module)">Smooth Basis Functions</span>
    
  </span>
  

            </a>
            
              
              <label class="md-nav__link " for="__nav_3_2_4_1_3" id="__nav_3_2_4_1_3_label" tabindex="0">
                <span class="md-nav__icon md-icon"></span>
              </label>
            
          </div>
        
        <nav class="md-nav" data-md-level="5" aria-labelledby="__nav_3_2_4_1_3_label" aria-expanded="false">
          <label class="md-nav__title" for="__nav_3_2_4_1_3">
            <span class="md-nav__icon md-icon"></span>
            <span title="statsmodels.gam.smooth_basis (Python module)">Smooth Basis Functions</span>
          </label>
          <ul class="md-nav__list" data-md-scrollfix>
            
              
    <li class="md-nav__item">
      <a href="generated/statsmodels.gam.smooth_basis.BSplines.html" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    <span title="/generated/statsmodels.gam.smooth_basis.bsplines.rst (reference label)">statsmodels.<wbr>gam.<wbr>smooth_<wbr>basis.<wbr>BSplines</span>
    
  </span>
  

      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="generated/statsmodels.gam.smooth_basis.CyclicCubicSplines.html" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    <span title="/generated/statsmodels.gam.smooth_basis.cycliccubicsplines.rst (reference label)">statsmodels.<wbr>gam.<wbr>smooth_<wbr>basis.<wbr>Cyclic<wbr>Cubic<wbr>Splines</span>
    
  </span>
  

      </a>
    </li>
  

          </ul>
        </nav>
      
    </li>
  
  
          </ul>
        </nav>
      
    </li>
  
  
          </ul>
        </nav>
      
    </li>
  
  
    <li class="md-nav__item md-nav__item--active md-nav__current-toc">
      
      
      <input class="md-nav__toggle md-toggle" type="checkbox" id="__toc">
      
        <label class="md-nav__link md-nav__link--active" for="__toc">
          
  
  <span class="md-ellipsis">
    <span title="/gam.rst (reference label)">Generalized Additive Models <wbr>(GAM)</span>
    
  </span>
  

          <span class="md-nav__icon md-icon"></span>
        </label>
      
      <a href="#" class="md-nav__link md-nav__link--active">
        
  
  <span class="md-ellipsis">
    <span title="/gam.rst (reference label)">Generalized Additive Models <wbr>(GAM)</span>
    
  </span>
  

      </a>
      
        
<nav class="md-nav md-nav--secondary">
  
  
    <label class="md-nav__title" for="__toc">
      <span class="md-nav__icon md-icon"></span>
      Contents
    </label>
    <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
      
        <li class="md-nav__item">
  <a href="#examples" class="md-nav__link">
    
    <span class="md-ellipsis">
      <span title="/gam.rst#examples (reference label)">Examples</span>
    </span>
  </a>
  
    <nav class="md-nav" aria-label="Examples">
      <ul class="md-nav__list">
        
          <li class="md-nav__item">
  <a href="#references" class="md-nav__link">
    
    <span class="md-ellipsis">
      <span title="/gam.rst#references (reference label)">References</span>
    </span>
  </a>
  
</li>
        
      </ul>
    </nav>
  
</li>
      
        <li class="md-nav__item">
  <a href="#module-statsmodels.gam.generalized_additive_model" class="md-nav__link">
    
    <span class="md-ellipsis">
      <span title="statsmodels.gam.generalized_additive_model (Python module)">Module Reference</span>
    </span>
  </a>
  
    <nav class="md-nav" aria-label="Module Reference">
      <ul class="md-nav__list">
        
          <li class="md-nav__item">
  <a href="#model-class" class="md-nav__link">
    
    <span class="md-ellipsis">
      <span title="/gam.rst#model-class (reference label)">Model Class</span>
    </span>
  </a>
  
    <nav class="md-nav" aria-label="Model Class">
      <ul class="md-nav__list">
        
          <li class="md-nav__item">
  <a href="generated/statsmodels.gam.generalized_additive_model.GLMGam.html" class="md-nav__link">
    
    <span class="md-ellipsis">
      <span title="/generated/statsmodels.gam.generalized_additive_model.glmgam.rst (reference label)">statsmodels.<wbr>gam.<wbr>generalized_<wbr>additive_<wbr>model.<wbr>GLMGam</span>
    </span>
  </a>
  
</li>
        
          <li class="md-nav__item">
  <a href="generated/statsmodels.gam.generalized_additive_model.LogitGam.html" class="md-nav__link">
    
    <span class="md-ellipsis">
      <span title="/generated/statsmodels.gam.generalized_additive_model.logitgam.rst (reference label)">statsmodels.<wbr>gam.<wbr>generalized_<wbr>additive_<wbr>model.<wbr>Logit<wbr>Gam</span>
    </span>
  </a>
  
</li>
        
      </ul>
    </nav>
  
</li>
        
          <li class="md-nav__item">
  <a href="#results-classes" class="md-nav__link">
    
    <span class="md-ellipsis">
      <span title="/gam.rst#results-classes (reference label)">Results Classes</span>
    </span>
  </a>
  
    <nav class="md-nav" aria-label="Results Classes">
      <ul class="md-nav__list">
        
          <li class="md-nav__item">
  <a href="generated/statsmodels.gam.generalized_additive_model.GLMGamResults.html" class="md-nav__link">
    
    <span class="md-ellipsis">
      <span title="/generated/statsmodels.gam.generalized_additive_model.glmgamresults.rst (reference label)">statsmodels.<wbr>gam.<wbr>generalized_<wbr>additive_<wbr>model.<wbr>GLMGam<wbr>Results</span>
    </span>
  </a>
  
</li>
        
      </ul>
    </nav>
  
</li>
        
          <li class="md-nav__item">
  <a href="#module-statsmodels.gam.smooth_basis" class="md-nav__link">
    
    <span class="md-ellipsis">
      <span title="statsmodels.gam.smooth_basis (Python module)">Smooth Basis Functions</span>
    </span>
  </a>
  
    <nav class="md-nav" aria-label="Smooth Basis Functions">
      <ul class="md-nav__list">
        
          <li class="md-nav__item">
  <a href="generated/statsmodels.gam.smooth_basis.BSplines.html" class="md-nav__link">
    
    <span class="md-ellipsis">
      <span title="/generated/statsmodels.gam.smooth_basis.bsplines.rst (reference label)">statsmodels.<wbr>gam.<wbr>smooth_<wbr>basis.<wbr>BSplines</span>
    </span>
  </a>
  
</li>
        
          <li class="md-nav__item">
  <a href="generated/statsmodels.gam.smooth_basis.CyclicCubicSplines.html" class="md-nav__link">
    
    <span class="md-ellipsis">
      <span title="/generated/statsmodels.gam.smooth_basis.cycliccubicsplines.rst (reference label)">statsmodels.<wbr>gam.<wbr>smooth_<wbr>basis.<wbr>Cyclic<wbr>Cubic<wbr>Splines</span>
    </span>
  </a>
  
</li>
        
      </ul>
    </nav>
  
</li>
        
          <li class="md-nav__item">
  <a href="#families-and-link-functions" class="md-nav__link">
    
    <span class="md-ellipsis">
      <span title="/gam.rst#families-and-link-functions (reference label)">Families and Link Functions</span>
    </span>
  </a>
  
</li>
        
      </ul>
    </nav>
  
</li>
      
    </ul>
  
</nav>
      
    </li>
  

    <li class="md-nav__item">
      <a href="rlm.html" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    <span title="/rlm.rst (reference label)">Robust Linear Models</span>
    
  </span>
  

      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="mixed_linear.html" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    <span title="/mixed_linear.rst (reference label)">Linear Mixed Effects Models</span>
    
  </span>
  

      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="discretemod.html" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    <span title="/discretemod.rst (reference label)">Regression with Discrete Dependent Variable</span>
    
  </span>
  

      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="mixed_glm.html" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    <span title="/mixed_glm.rst (reference label)">Generalized Linear Mixed Effects Models</span>
    
  </span>
  

      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="anova.html" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    <span title="/anova.rst (reference label)">ANOVA</span>
    
  </span>
  

      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="other_models.html" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    <span title="/other_models.rst (reference label)">Other Models othermod</span>
    
  </span>
  

      </a>
    </li>
  

          </ul>
        </nav>
      
    </li>
  
  
    <li class="md-nav__item">
      <a href="user-guide.html#time-series-analysis" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    <span title="/user-guide.rst#time-series-analysis (reference label)">Time Series Analysis</span>
    
  </span>
  

      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="user-guide.html#other-models" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    <span title="/user-guide.rst#other-models (reference label)">Other Models</span>
    
  </span>
  

      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="user-guide.html#statistics-and-tools" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    <span title="/user-guide.rst#statistics-and-tools (reference label)">Statistics and Tools</span>
    
  </span>
  

      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="user-guide.html#data-sets" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    <span title="/user-guide.rst#data-sets (reference label)">Data Sets</span>
    
  </span>
  

      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="user-guide.html#sandbox" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    <span title="/user-guide.rst#sandbox (reference label)">Sandbox</span>
    
  </span>
  

      </a>
    </li>
  

          </ul>
        </nav>
      
    </li>
  
  
    <li class="md-nav__item">
      <a href="examples/index.html" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    <span title="/examples/index.rst (reference label)">Examples</span>
    
  </span>
  

      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="api.html" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    <span title="/api.rst (reference label)">API Reference</span>
    
  </span>
  

      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="about.html" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    <span title="/about.rst (reference label)">About statsmodels</span>
    
  </span>
  

      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="dev/index.html" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    <span title="/dev/index.rst (reference label)">Developer Page</span>
    
  </span>
  

      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="release/index.html" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    <span title="/release/index.rst (reference label)">Release Notes</span>
    
  </span>
  

      </a>
    </li>
  

  </ul>
</nav>
                  </div>
                </div>
              </div>
            
            
              <div class="md-sidebar md-sidebar--secondary" data-md-component="sidebar" data-md-type="toc" >
                <div class="md-sidebar__scrollwrap">
                  <div class="md-sidebar__inner">
                    

<nav class="md-nav md-nav--secondary">
  
  
    <label class="md-nav__title" for="__toc">
      <span class="md-nav__icon md-icon"></span>
      Contents
    </label>
    <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
      
        <li class="md-nav__item">
  <a href="#examples" class="md-nav__link">
    
    <span class="md-ellipsis">
      <span title="/gam.rst#examples (reference label)">Examples</span>
    </span>
  </a>
  
    <nav class="md-nav" aria-label="Examples">
      <ul class="md-nav__list">
        
          <li class="md-nav__item">
  <a href="#references" class="md-nav__link">
    
    <span class="md-ellipsis">
      <span title="/gam.rst#references (reference label)">References</span>
    </span>
  </a>
  
</li>
        
      </ul>
    </nav>
  
</li>
      
        <li class="md-nav__item">
  <a href="#module-statsmodels.gam.generalized_additive_model" class="md-nav__link">
    
    <span class="md-ellipsis">
      <span title="statsmodels.gam.generalized_additive_model (Python module)">Module Reference</span>
    </span>
  </a>
  
    <nav class="md-nav" aria-label="Module Reference">
      <ul class="md-nav__list">
        
          <li class="md-nav__item">
  <a href="#model-class" class="md-nav__link">
    
    <span class="md-ellipsis">
      <span title="/gam.rst#model-class (reference label)">Model Class</span>
    </span>
  </a>
  
</li>
        
          <li class="md-nav__item">
  <a href="#results-classes" class="md-nav__link">
    
    <span class="md-ellipsis">
      <span title="/gam.rst#results-classes (reference label)">Results Classes</span>
    </span>
  </a>
  
</li>
        
          <li class="md-nav__item">
  <a href="#module-statsmodels.gam.smooth_basis" class="md-nav__link">
    
    <span class="md-ellipsis">
      <span title="statsmodels.gam.smooth_basis (Python module)">Smooth Basis Functions</span>
    </span>
  </a>
  
</li>
        
          <li class="md-nav__item">
  <a href="#families-and-link-functions" class="md-nav__link">
    
    <span class="md-ellipsis">
      <span title="/gam.rst#families-and-link-functions (reference label)">Families and Link Functions</span>
    </span>
  </a>
  
</li>
        
      </ul>
    </nav>
  
</li>
      
    </ul>
  
</nav>
                  </div>
                </div>
              </div>
            
          
            <div class="md-content" data-md-component="content">
              <article class="md-content__inner md-typeset" role="main">
                
                
<h1 id="generalized-additive-models-gam"><span id="gam"></span>Generalized Additive Models (GAM)<a class="headerlink" href="#generalized-additive-models-gam" title="Link to this heading">¶</a></h1>
<p>Generalized Additive Models allow for penalized estimation of smooth terms
in generalized linear models.</p>
<p>See <a class="reference internal" href="#module-reference">Module Reference</a> for commands and arguments.</p>
<h2 id="examples">Examples<a class="headerlink" href="#examples" title="Link to this heading">¶</a></h2>
<p>The following illustrates a Gaussian and a Poisson regression where
categorical variables are treated as linear terms and the effect of
two explanatory variables is captured by penalized B-splines.
The data is from the automobile dataset
<a class="reference external" href="https://archive.ics.uci.edu/ml/datasets/automobile">https://archive.ics.uci.edu/ml/datasets/automobile</a>
We can load a dataframe with selected columns from the unit test module.</p>
<div class="highlight-ipython notranslate"><div class="highlight"><pre><span></span><code><span class="gp">In [1]: </span><span class="kn">import</span><span class="w"> </span><span class="nn">statsmodels.api</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">sm</span>

<span class="gp">In [2]: </span><span class="kn">from</span><span class="w"> </span><span class="nn">statsmodels.gam.api</span><span class="w"> </span><span class="kn">import</span> <span class="n">GLMGam</span><span class="p">,</span> <span class="n">BSplines</span>

<span class="go"># import data</span>
<span class="gp">In [3]: </span><span class="kn">from</span><span class="w"> </span><span class="nn">statsmodels.gam.tests.test_penalized</span><span class="w"> </span><span class="kn">import</span> <span class="n">df_autos</span>

<span class="go"># create spline basis for weight and hp</span>
<span class="gp">In [4]: </span><span class="n">x_spline</span> <span class="o">=</span> <span class="n">df_autos</span><span class="p">[[</span><span class="s1">&#39;weight&#39;</span><span class="p">,</span> <span class="s1">&#39;hp&#39;</span><span class="p">]]</span>

<span class="gp">In [5]: </span><span class="n">bs</span> <span class="o">=</span> <span class="n">BSplines</span><span class="p">(</span><span class="n">x_spline</span><span class="p">,</span> <span class="n">df</span><span class="o">=</span><span class="p">[</span><span class="mi">12</span><span class="p">,</span> <span class="mi">10</span><span class="p">],</span> <span class="n">degree</span><span class="o">=</span><span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">])</span>

<span class="go"># penalization weight</span>
<span class="gp">In [6]: </span><span class="n">alpha</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">21833888.8</span><span class="p">,</span> <span class="mf">6460.38479</span><span class="p">])</span>

<span class="gp">In [7]: </span><span class="n">gam_bs</span> <span class="o">=</span> <span class="n">GLMGam</span><span class="o">.</span><span class="n">from_formula</span><span class="p">(</span><span class="s1">&#39;city_mpg ~ fuel + drive&#39;</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">df_autos</span><span class="p">,</span>
<span class="gp">   ...: </span>                             <span class="n">smoother</span><span class="o">=</span><span class="n">bs</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="n">alpha</span><span class="p">)</span>
<span class="gp">   ...: </span>

<span class="gp">In [8]: </span><span class="n">res_bs</span> <span class="o">=</span> <span class="n">gam_bs</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>

<span class="gp">In [9]: </span><span class="nb">print</span><span class="p">(</span><span class="n">res_bs</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
<span class="go">                 Generalized Linear Model Regression Results                  </span>
<span class="go">==============================================================================</span>
<span class="go">Dep. Variable:               city_mpg   No. Observations:                  203</span>
<span class="go">Model:                         GLMGam   Df Residuals:                   189.13</span>
<span class="go">Model Family:                Gaussian   Df Model:                        12.87</span>
<span class="go">Link Function:               Identity   Scale:                          4.8825</span>
<span class="go">Method:                         PIRLS   Log-Likelihood:                -441.81</span>
<span class="go">Date:                Wed, 02 Apr 2025   Deviance:                       923.45</span>
<span class="go">Time:                        08:41:23   Pearson chi2:                     923.</span>
<span class="go">No. Iterations:                     3   Pseudo R-squ. (CS):             0.9996</span>
<span class="go">Covariance Type:            nonrobust                                         </span>
<span class="go">================================================================================</span>
<span class="go">                   coef    std err          z      P&gt;|z|      [0.025      0.975]</span>
<span class="gt">--------------------------------------------------------------------------------</span>
<span class="n">Intercept</span>       <span class="mf">51.9923</span>      <span class="mf">1.997</span>     <span class="mf">26.034</span>      <span class="mf">0.000</span>      <span class="mf">48.078</span>      <span class="mf">55.906</span>
<span class="n">fuel</span><span class="p">[</span><span class="n">T</span><span class="o">.</span><span class="n">gas</span><span class="p">]</span>     <span class="o">-</span><span class="mf">5.8099</span>      <span class="mf">0.727</span>     <span class="o">-</span><span class="mf">7.989</span>      <span class="mf">0.000</span>      <span class="o">-</span><span class="mf">7.235</span>      <span class="o">-</span><span class="mf">4.385</span>
<span class="n">drive</span><span class="p">[</span><span class="n">T</span><span class="o">.</span><span class="n">fwd</span><span class="p">]</span>     <span class="mf">1.3910</span>      <span class="mf">0.819</span>      <span class="mf">1.699</span>      <span class="mf">0.089</span>      <span class="o">-</span><span class="mf">0.213</span>       <span class="mf">2.995</span>
<span class="n">drive</span><span class="p">[</span><span class="n">T</span><span class="o">.</span><span class="n">rwd</span><span class="p">]</span>     <span class="mf">1.0638</span>      <span class="mf">0.842</span>      <span class="mf">1.263</span>      <span class="mf">0.207</span>      <span class="o">-</span><span class="mf">0.587</span>       <span class="mf">2.715</span>
<span class="n">weight_s0</span>       <span class="o">-</span><span class="mf">3.5556</span>      <span class="mf">0.959</span>     <span class="o">-</span><span class="mf">3.707</span>      <span class="mf">0.000</span>      <span class="o">-</span><span class="mf">5.436</span>      <span class="o">-</span><span class="mf">1.676</span>
<span class="n">weight_s1</span>       <span class="o">-</span><span class="mf">9.0876</span>      <span class="mf">1.750</span>     <span class="o">-</span><span class="mf">5.193</span>      <span class="mf">0.000</span>     <span class="o">-</span><span class="mf">12.518</span>      <span class="o">-</span><span class="mf">5.658</span>
<span class="n">weight_s2</span>      <span class="o">-</span><span class="mf">13.0303</span>      <span class="mf">1.827</span>     <span class="o">-</span><span class="mf">7.132</span>      <span class="mf">0.000</span>     <span class="o">-</span><span class="mf">16.611</span>      <span class="o">-</span><span class="mf">9.450</span>
<span class="n">weight_s3</span>      <span class="o">-</span><span class="mf">14.2641</span>      <span class="mf">1.854</span>     <span class="o">-</span><span class="mf">7.695</span>      <span class="mf">0.000</span>     <span class="o">-</span><span class="mf">17.897</span>     <span class="o">-</span><span class="mf">10.631</span>
<span class="n">weight_s4</span>      <span class="o">-</span><span class="mf">15.1805</span>      <span class="mf">1.892</span>     <span class="o">-</span><span class="mf">8.024</span>      <span class="mf">0.000</span>     <span class="o">-</span><span class="mf">18.889</span>     <span class="o">-</span><span class="mf">11.472</span>
<span class="n">weight_s5</span>      <span class="o">-</span><span class="mf">15.9557</span>      <span class="mf">1.963</span>     <span class="o">-</span><span class="mf">8.128</span>      <span class="mf">0.000</span>     <span class="o">-</span><span class="mf">19.803</span>     <span class="o">-</span><span class="mf">12.108</span>
<span class="n">weight_s6</span>      <span class="o">-</span><span class="mf">16.6297</span>      <span class="mf">2.038</span>     <span class="o">-</span><span class="mf">8.161</span>      <span class="mf">0.000</span>     <span class="o">-</span><span class="mf">20.624</span>     <span class="o">-</span><span class="mf">12.636</span>
<span class="n">weight_s7</span>      <span class="o">-</span><span class="mf">16.9928</span>      <span class="mf">2.045</span>     <span class="o">-</span><span class="mf">8.308</span>      <span class="mf">0.000</span>     <span class="o">-</span><span class="mf">21.002</span>     <span class="o">-</span><span class="mf">12.984</span>
<span class="n">weight_s8</span>      <span class="o">-</span><span class="mf">19.3480</span>      <span class="mf">2.367</span>     <span class="o">-</span><span class="mf">8.174</span>      <span class="mf">0.000</span>     <span class="o">-</span><span class="mf">23.987</span>     <span class="o">-</span><span class="mf">14.709</span>
<span class="n">weight_s9</span>      <span class="o">-</span><span class="mf">20.7978</span>      <span class="mf">2.455</span>     <span class="o">-</span><span class="mf">8.472</span>      <span class="mf">0.000</span>     <span class="o">-</span><span class="mf">25.609</span>     <span class="o">-</span><span class="mf">15.986</span>
<span class="n">weight_s10</span>     <span class="o">-</span><span class="mf">20.8062</span>      <span class="mf">2.443</span>     <span class="o">-</span><span class="mf">8.517</span>      <span class="mf">0.000</span>     <span class="o">-</span><span class="mf">25.594</span>     <span class="o">-</span><span class="mf">16.018</span>
<span class="n">hp_s0</span>           <span class="o">-</span><span class="mf">1.4473</span>      <span class="mf">0.558</span>     <span class="o">-</span><span class="mf">2.592</span>      <span class="mf">0.010</span>      <span class="o">-</span><span class="mf">2.542</span>      <span class="o">-</span><span class="mf">0.353</span>
<span class="n">hp_s1</span>           <span class="o">-</span><span class="mf">3.4228</span>      <span class="mf">1.012</span>     <span class="o">-</span><span class="mf">3.381</span>      <span class="mf">0.001</span>      <span class="o">-</span><span class="mf">5.407</span>      <span class="o">-</span><span class="mf">1.438</span>
<span class="n">hp_s2</span>           <span class="o">-</span><span class="mf">5.9026</span>      <span class="mf">1.251</span>     <span class="o">-</span><span class="mf">4.717</span>      <span class="mf">0.000</span>      <span class="o">-</span><span class="mf">8.355</span>      <span class="o">-</span><span class="mf">3.450</span>
<span class="n">hp_s3</span>           <span class="o">-</span><span class="mf">7.2389</span>      <span class="mf">1.352</span>     <span class="o">-</span><span class="mf">5.354</span>      <span class="mf">0.000</span>      <span class="o">-</span><span class="mf">9.889</span>      <span class="o">-</span><span class="mf">4.589</span>
<span class="n">hp_s4</span>           <span class="o">-</span><span class="mf">9.1052</span>      <span class="mf">1.384</span>     <span class="o">-</span><span class="mf">6.581</span>      <span class="mf">0.000</span>     <span class="o">-</span><span class="mf">11.817</span>      <span class="o">-</span><span class="mf">6.393</span>
<span class="n">hp_s5</span>           <span class="o">-</span><span class="mf">9.9865</span>      <span class="mf">1.525</span>     <span class="o">-</span><span class="mf">6.547</span>      <span class="mf">0.000</span>     <span class="o">-</span><span class="mf">12.976</span>      <span class="o">-</span><span class="mf">6.997</span>
<span class="n">hp_s6</span>          <span class="o">-</span><span class="mf">13.3639</span>      <span class="mf">2.228</span>     <span class="o">-</span><span class="mf">5.998</span>      <span class="mf">0.000</span>     <span class="o">-</span><span class="mf">17.731</span>      <span class="o">-</span><span class="mf">8.997</span>
<span class="n">hp_s7</span>          <span class="o">-</span><span class="mf">13.8902</span>      <span class="mf">3.194</span>     <span class="o">-</span><span class="mf">4.349</span>      <span class="mf">0.000</span>     <span class="o">-</span><span class="mf">20.150</span>      <span class="o">-</span><span class="mf">7.630</span>
<span class="n">hp_s8</span>          <span class="o">-</span><span class="mf">11.9752</span>      <span class="mf">2.556</span>     <span class="o">-</span><span class="mf">4.685</span>      <span class="mf">0.000</span>     <span class="o">-</span><span class="mf">16.985</span>      <span class="o">-</span><span class="mf">6.965</span>
<span class="o">================================================================================</span>

<span class="c1"># plot smooth components</span>
<span class="gp">In [10]: </span><span class="n">res_bs</span><span class="o">.</span><span class="n">plot_partial</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">cpr</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="gh">Out[10]: </span><span class="go">&lt;Figure size 640x480 with 1 Axes&gt;</span>

<span class="gp">In [11]: </span><span class="n">res_bs</span><span class="o">.</span><span class="n">plot_partial</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">cpr</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="gh">Out[11]: </span><span class="go">&lt;Figure size 640x480 with 1 Axes&gt;</span>

<span class="gp">In [12]: </span><span class="n">alpha</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">8283989284.5829611</span><span class="p">,</span> <span class="mf">14628207.58927821</span><span class="p">])</span>

<span class="gp">In [13]: </span><span class="n">gam_bs</span> <span class="o">=</span> <span class="n">GLMGam</span><span class="o">.</span><span class="n">from_formula</span><span class="p">(</span><span class="s1">&#39;city_mpg ~ fuel + drive&#39;</span><span class="p">,</span> <span class="n">data</span><span class="o">=</span><span class="n">df_autos</span><span class="p">,</span>
<span class="gp">   ....: </span>                             <span class="n">smoother</span><span class="o">=</span><span class="n">bs</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="n">alpha</span><span class="p">,</span>
<span class="gp">   ....: </span>                             <span class="n">family</span><span class="o">=</span><span class="n">sm</span><span class="o">.</span><span class="n">families</span><span class="o">.</span><span class="n">Poisson</span><span class="p">())</span>
<span class="gp">   ....: </span>

<span class="gp">In [14]: </span><span class="n">res_bs</span> <span class="o">=</span> <span class="n">gam_bs</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>

<span class="gp">In [15]: </span><span class="nb">print</span><span class="p">(</span><span class="n">res_bs</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
<span class="go">                 Generalized Linear Model Regression Results                  </span>
<span class="go">==============================================================================</span>
<span class="go">Dep. Variable:               city_mpg   No. Observations:                  203</span>
<span class="go">Model:                         GLMGam   Df Residuals:                   194.75</span>
<span class="go">Model Family:                 Poisson   Df Model:                         7.25</span>
<span class="go">Link Function:                    Log   Scale:                          1.0000</span>
<span class="go">Method:                         PIRLS   Log-Likelihood:                -530.38</span>
<span class="go">Date:                Wed, 02 Apr 2025   Deviance:                       37.569</span>
<span class="go">Time:                        08:41:23   Pearson chi2:                     37.4</span>
<span class="go">No. Iterations:                     6   Pseudo R-squ. (CS):             0.7715</span>
<span class="go">Covariance Type:            nonrobust                                         </span>
<span class="go">================================================================================</span>
<span class="go">                   coef    std err          z      P&gt;|z|      [0.025      0.975]</span>
<span class="gt">--------------------------------------------------------------------------------</span>
<span class="n">Intercept</span>        <span class="mf">3.9960</span>      <span class="mf">0.130</span>     <span class="mf">30.844</span>      <span class="mf">0.000</span>       <span class="mf">3.742</span>       <span class="mf">4.250</span>
<span class="n">fuel</span><span class="p">[</span><span class="n">T</span><span class="o">.</span><span class="n">gas</span><span class="p">]</span>     <span class="o">-</span><span class="mf">0.2398</span>      <span class="mf">0.057</span>     <span class="o">-</span><span class="mf">4.222</span>      <span class="mf">0.000</span>      <span class="o">-</span><span class="mf">0.351</span>      <span class="o">-</span><span class="mf">0.128</span>
<span class="n">drive</span><span class="p">[</span><span class="n">T</span><span class="o">.</span><span class="n">fwd</span><span class="p">]</span>     <span class="mf">0.0386</span>      <span class="mf">0.075</span>      <span class="mf">0.513</span>      <span class="mf">0.608</span>      <span class="o">-</span><span class="mf">0.109</span>       <span class="mf">0.186</span>
<span class="n">drive</span><span class="p">[</span><span class="n">T</span><span class="o">.</span><span class="n">rwd</span><span class="p">]</span>     <span class="mf">0.0309</span>      <span class="mf">0.078</span>      <span class="mf">0.395</span>      <span class="mf">0.693</span>      <span class="o">-</span><span class="mf">0.122</span>       <span class="mf">0.184</span>
<span class="n">weight_s0</span>       <span class="o">-</span><span class="mf">0.0811</span>      <span class="mf">0.030</span>     <span class="o">-</span><span class="mf">2.689</span>      <span class="mf">0.007</span>      <span class="o">-</span><span class="mf">0.140</span>      <span class="o">-</span><span class="mf">0.022</span>
<span class="n">weight_s1</span>       <span class="o">-</span><span class="mf">0.1938</span>      <span class="mf">0.063</span>     <span class="o">-</span><span class="mf">3.067</span>      <span class="mf">0.002</span>      <span class="o">-</span><span class="mf">0.318</span>      <span class="o">-</span><span class="mf">0.070</span>
<span class="n">weight_s2</span>       <span class="o">-</span><span class="mf">0.3160</span>      <span class="mf">0.082</span>     <span class="o">-</span><span class="mf">3.864</span>      <span class="mf">0.000</span>      <span class="o">-</span><span class="mf">0.476</span>      <span class="o">-</span><span class="mf">0.156</span>
<span class="n">weight_s3</span>       <span class="o">-</span><span class="mf">0.3735</span>      <span class="mf">0.090</span>     <span class="o">-</span><span class="mf">4.160</span>      <span class="mf">0.000</span>      <span class="o">-</span><span class="mf">0.549</span>      <span class="o">-</span><span class="mf">0.198</span>
<span class="n">weight_s4</span>       <span class="o">-</span><span class="mf">0.4187</span>      <span class="mf">0.096</span>     <span class="o">-</span><span class="mf">4.360</span>      <span class="mf">0.000</span>      <span class="o">-</span><span class="mf">0.607</span>      <span class="o">-</span><span class="mf">0.230</span>
<span class="n">weight_s5</span>       <span class="o">-</span><span class="mf">0.4645</span>      <span class="mf">0.103</span>     <span class="o">-</span><span class="mf">4.495</span>      <span class="mf">0.000</span>      <span class="o">-</span><span class="mf">0.667</span>      <span class="o">-</span><span class="mf">0.262</span>
<span class="n">weight_s6</span>       <span class="o">-</span><span class="mf">0.5092</span>      <span class="mf">0.112</span>     <span class="o">-</span><span class="mf">4.555</span>      <span class="mf">0.000</span>      <span class="o">-</span><span class="mf">0.728</span>      <span class="o">-</span><span class="mf">0.290</span>
<span class="n">weight_s7</span>       <span class="o">-</span><span class="mf">0.5469</span>      <span class="mf">0.119</span>     <span class="o">-</span><span class="mf">4.598</span>      <span class="mf">0.000</span>      <span class="o">-</span><span class="mf">0.780</span>      <span class="o">-</span><span class="mf">0.314</span>
<span class="n">weight_s8</span>       <span class="o">-</span><span class="mf">0.6211</span>      <span class="mf">0.137</span>     <span class="o">-</span><span class="mf">4.528</span>      <span class="mf">0.000</span>      <span class="o">-</span><span class="mf">0.890</span>      <span class="o">-</span><span class="mf">0.352</span>
<span class="n">weight_s9</span>       <span class="o">-</span><span class="mf">0.6866</span>      <span class="mf">0.153</span>     <span class="o">-</span><span class="mf">4.486</span>      <span class="mf">0.000</span>      <span class="o">-</span><span class="mf">0.987</span>      <span class="o">-</span><span class="mf">0.387</span>
<span class="n">weight_s10</span>      <span class="o">-</span><span class="mf">0.7370</span>      <span class="mf">0.174</span>     <span class="o">-</span><span class="mf">4.228</span>      <span class="mf">0.000</span>      <span class="o">-</span><span class="mf">1.079</span>      <span class="o">-</span><span class="mf">0.395</span>
<span class="n">hp_s0</span>           <span class="o">-</span><span class="mf">0.0247</span>      <span class="mf">0.010</span>     <span class="o">-</span><span class="mf">2.378</span>      <span class="mf">0.017</span>      <span class="o">-</span><span class="mf">0.045</span>      <span class="o">-</span><span class="mf">0.004</span>
<span class="n">hp_s1</span>           <span class="o">-</span><span class="mf">0.0557</span>      <span class="mf">0.022</span>     <span class="o">-</span><span class="mf">2.479</span>      <span class="mf">0.013</span>      <span class="o">-</span><span class="mf">0.100</span>      <span class="o">-</span><span class="mf">0.012</span>
<span class="n">hp_s2</span>           <span class="o">-</span><span class="mf">0.1046</span>      <span class="mf">0.038</span>     <span class="o">-</span><span class="mf">2.719</span>      <span class="mf">0.007</span>      <span class="o">-</span><span class="mf">0.180</span>      <span class="o">-</span><span class="mf">0.029</span>
<span class="n">hp_s3</span>           <span class="o">-</span><span class="mf">0.1438</span>      <span class="mf">0.050</span>     <span class="o">-</span><span class="mf">2.857</span>      <span class="mf">0.004</span>      <span class="o">-</span><span class="mf">0.242</span>      <span class="o">-</span><span class="mf">0.045</span>
<span class="n">hp_s4</span>           <span class="o">-</span><span class="mf">0.1919</span>      <span class="mf">0.063</span>     <span class="o">-</span><span class="mf">3.047</span>      <span class="mf">0.002</span>      <span class="o">-</span><span class="mf">0.315</span>      <span class="o">-</span><span class="mf">0.068</span>
<span class="n">hp_s5</span>           <span class="o">-</span><span class="mf">0.2567</span>      <span class="mf">0.079</span>     <span class="o">-</span><span class="mf">3.231</span>      <span class="mf">0.001</span>      <span class="o">-</span><span class="mf">0.412</span>      <span class="o">-</span><span class="mf">0.101</span>
<span class="n">hp_s6</span>           <span class="o">-</span><span class="mf">0.4152</span>      <span class="mf">0.120</span>     <span class="o">-</span><span class="mf">3.455</span>      <span class="mf">0.001</span>      <span class="o">-</span><span class="mf">0.651</span>      <span class="o">-</span><span class="mf">0.180</span>
<span class="n">hp_s7</span>           <span class="o">-</span><span class="mf">0.4889</span>      <span class="mf">0.152</span>     <span class="o">-</span><span class="mf">3.214</span>      <span class="mf">0.001</span>      <span class="o">-</span><span class="mf">0.787</span>      <span class="o">-</span><span class="mf">0.191</span>
<span class="n">hp_s8</span>           <span class="o">-</span><span class="mf">0.5470</span>      <span class="mf">0.195</span>     <span class="o">-</span><span class="mf">2.810</span>      <span class="mf">0.005</span>      <span class="o">-</span><span class="mf">0.928</span>      <span class="o">-</span><span class="mf">0.166</span>
<span class="o">================================================================================</span>

<span class="c1"># Optimal penalization weights alpha can be obtained through generalized</span>
<span class="c1"># cross-validation or k-fold cross-validation.</span>
<span class="c1"># The alpha above are from the unit tests against the R mgcv package.</span>
<span class="gp">In [16]: </span><span class="n">gam_bs</span><span class="o">.</span><span class="n">select_penweight</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span>
<span class="gh">Out[16]: </span><span class="go">array([8.2840e+09, 1.4628e+07])</span>

<span class="gp">In [17]: </span><span class="n">gam_bs</span><span class="o">.</span><span class="n">select_penweight_kfold</span><span class="p">()[</span><span class="mi">0</span><span class="p">]</span>
<span class="gh">Out[17]: </span><span class="go">(np.float64(10000000.0), np.float64(79432.82347242805))</span>
</code></pre></div>
</div>
<h3 id="references">References<a class="headerlink" href="#references" title="Link to this heading">¶</a></h3>
<ul class="simple">
<li><p>Hastie, Trevor, and Robert Tibshirani. 1986. Generalized Additive Models. Statistical Science 1 (3): 297-310.</p></li>
<li><p>Wood, Simon N. 2006. Generalized Additive Models: An Introduction with R. Texts in Statistical Science. Boca Raton, FL: Chapman &amp; Hall/CRC.</p></li>
<li><p>Wood, Simon N. 2017. Generalized Additive Models: An Introduction with R. Second edition. Chapman &amp; Hall/CRC Texts in Statistical Science. Boca Raton: CRC Press/Taylor &amp; Francis Group.</p></li>
</ul>
<h2 id="module-statsmodels.gam.generalized_additive_model"><span id="module-reference"></span>Module Reference<a class="headerlink" href="#module-statsmodels.gam.generalized_additive_model" title="Link to this heading">¶</a></h2>
<h3 id="model-class">Model Class<a class="headerlink" href="#model-class" title="Link to this heading">¶</a></h3>
<table class="autosummary longtable docutils data align-default">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.gam.generalized_additive_model.GLMGam.html#statsmodels.gam.generalized_additive_model.GLMGam" title="statsmodels.gam.generalized_additive_model.GLMGam (Python class) — Generalized Additive Models (GAM)"><code class="xref py py-obj docutils literal notranslate"><span class="pre">GLMGam</span></code></a>(endog[, exog, smoother, alpha, ...])</p></td>
<td><p>Generalized Additive Models (GAM)</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/statsmodels.gam.generalized_additive_model.LogitGam.html#statsmodels.gam.generalized_additive_model.LogitGam" title="statsmodels.gam.generalized_additive_model.LogitGam (Python class) — Generalized Additive model for discrete Logit"><code class="xref py py-obj docutils literal notranslate"><span class="pre">LogitGam</span></code></a>(endog, smoother, alpha, *args, **kwargs)</p></td>
<td><p>Generalized Additive model for discrete Logit</p></td>
</tr>
</tbody>
</table>
<h3 id="results-classes">Results Classes<a class="headerlink" href="#results-classes" title="Link to this heading">¶</a></h3>
<table class="autosummary longtable docutils data align-default">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.gam.generalized_additive_model.GLMGamResults.html#statsmodels.gam.generalized_additive_model.GLMGamResults" title="statsmodels.gam.generalized_additive_model.GLMGamResults (Python class) — Results class for generalized additive models, GAM."><code class="xref py py-obj docutils literal notranslate"><span class="pre">GLMGamResults</span></code></a>(model, params, ...)</p></td>
<td><p>Results class for generalized additive models, GAM.</p></td>
</tr>
</tbody>
</table>
<h3 id="module-statsmodels.gam.smooth_basis"><span id="smooth-basis-functions"></span>Smooth Basis Functions<a class="headerlink" href="#module-statsmodels.gam.smooth_basis" title="Link to this heading">¶</a></h3>
<p>Currently there is verified support for two spline bases</p>
<table class="autosummary longtable docutils data align-default">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.gam.smooth_basis.BSplines.html#statsmodels.gam.smooth_basis.BSplines" title="statsmodels.gam.smooth_basis.BSplines (Python class) — additive smooth components using B-Splines"><code class="xref py py-obj docutils literal notranslate"><span class="pre">BSplines</span></code></a>(x, df, degree[, include_intercept, ...])</p></td>
<td><p>additive smooth components using B-Splines</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/statsmodels.gam.smooth_basis.CyclicCubicSplines.html#statsmodels.gam.smooth_basis.CyclicCubicSplines" title="statsmodels.gam.smooth_basis.CyclicCubicSplines (Python class) — additive smooth components using cyclic cubic regression splines"><code class="xref py py-obj docutils literal notranslate"><span class="pre">CyclicCubicSplines</span></code></a>(x, df[, constraints, ...])</p></td>
<td><p>additive smooth components using cyclic cubic regression splines</p></td>
</tr>
</tbody>
</table>
<p><cite>statsmodels.gam.smooth_basis</cite> includes additional splines and a (global)
polynomial smoother basis but those have not been verified yet.</p>
<h3 id="families-and-link-functions">Families and Link Functions<a class="headerlink" href="#families-and-link-functions" title="Link to this heading">¶</a></h3>
<p>The distribution families in <cite>GLMGam</cite> are the same as for GLM and so are
the corresponding link functions.
Current unit tests only cover Gaussian and Poisson, and GLMGam might not
work for all options that are available in GLM.</p>


  <aside class="md-source-file">
    
      
  <span class="md-source-file__fact">
    <span class="md-icon" title="Last update">
      <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M21 13.1c-.1 0-.3.1-.4.2l-1 1 2.1 2.1 1-1c.2-.2.2-.6 0-.8l-1.3-1.3c-.1-.1-.2-.2-.4-.2m-1.9 1.8-6.1 6V23h2.1l6.1-6.1zM12.5 7v5.2l4 2.4-1 1L11 13V7zM11 21.9c-5.1-.5-9-4.8-9-9.9C2 6.5 6.5 2 12 2c5.3 0 9.6 4.1 10 9.3-.3-.1-.6-.2-1-.2s-.7.1-1 .2C19.6 7.2 16.2 4 12 4c-4.4 0-8 3.6-8 8 0 4.1 3.1 7.5 7.1 7.9l-.1.2z"/></svg>
    </span>
    Apr 02, 2025
  </span>

    
  </aside>


              </article>
            </div>
          
          
        </div>
        
      </main>
      
        <footer class="md-footer">
  
  
  <div class="md-footer-meta md-typeset">
    
    <div class="md-footer-meta__inner md-grid">
      <div class="md-copyright">
  
    <div class="md-footer-copyright__highlight">
        &#169; Copyright 2009-2025, Josef Perktold, Skipper Seabold, Jonathan Taylor, statsmodels-developers.
        
    </div>
  
    Created using
    <a href="https://www.sphinx-doc.org/" target="_blank" rel="noopener">Sphinx</a>
    7.3.7.
     and
    <a href="https://github.com/jbms/sphinx-immaterial/" target="_blank" rel="noopener">Sphinx-Immaterial</a>
  
</div>
      
        <div class="md-social">
  
    
    <a href="https://github.com/statsmodels/statsmodels/" target="_blank" rel="noopener" title="Source on github.com" class="md-social__link">
      <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 496 512"><!--! Font Awesome Free 6.7.2 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2024 Fonticons, Inc.--><path d="M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6m-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3m44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9M244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8M97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1m-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7m32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1m-11.4-14.7c-1.6 1-1.6 3.6 0 5.9s4.3 3.3 5.6 2.3c1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2"/></svg>
    </a>
  
    
    <a href="https://pypi.org/project/statsmodels/" target="_blank" rel="noopener" title="pypi.org" class="md-social__link">
      <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512"><!--! Font Awesome Free 6.7.2 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2024 Fonticons, Inc.--><path d="M439.8 200.5c-7.7-30.9-22.3-54.2-53.4-54.2h-40.1v47.4c0 36.8-31.2 67.8-66.8 67.8H172.7c-29.2 0-53.4 25-53.4 54.3v101.8c0 29 25.2 46 53.4 54.3 33.8 9.9 66.3 11.7 106.8 0 26.9-7.8 53.4-23.5 53.4-54.3v-40.7H226.2v-13.6h160.2c31.1 0 42.6-21.7 53.4-54.2 11.2-33.5 10.7-65.7 0-108.6M286.2 404c11.1 0 20.1 9.1 20.1 20.3 0 11.3-9 20.4-20.1 20.4-11 0-20.1-9.2-20.1-20.4.1-11.3 9.1-20.3 20.1-20.3M167.8 248.1h106.8c29.7 0 53.4-24.5 53.4-54.3V91.9c0-29-24.4-50.7-53.4-55.6-35.8-5.9-74.7-5.6-106.8.1-45.2 8-53.4 24.7-53.4 55.6v40.7h106.9v13.6h-147c-31.1 0-58.3 18.7-66.8 54.2-9.8 40.7-10.2 66.1 0 108.6 7.6 31.6 25.7 54.2 56.8 54.2H101v-48.8c0-35.3 30.5-66.4 66.8-66.4m-6.7-142.6c-11.1 0-20.1-9.1-20.1-20.3.1-11.3 9-20.4 20.1-20.4 11 0 20.1 9.2 20.1 20.4s-9 20.3-20.1 20.3"/></svg>
    </a>
  
    
    <a href="https://doi.org/10.5281/zenodo.593847" target="_blank" rel="noopener" title="doi.org" class="md-social__link">
      <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512"><!--! Font Awesome Free 6.7.2 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2024 Fonticons, Inc.--><path d="M0 216C0 149.7 53.7 96 120 96h8c17.7 0 32 14.3 32 32s-14.3 32-32 32h-8c-30.9 0-56 25.1-56 56v8h64c35.3 0 64 28.7 64 64v64c0 35.3-28.7 64-64 64H64c-35.3 0-64-28.7-64-64V216m256 0c0-66.3 53.7-120 120-120h8c17.7 0 32 14.3 32 32s-14.3 32-32 32h-8c-30.9 0-56 25.1-56 56v8h64c35.3 0 64 28.7 64 64v64c0 35.3-28.7 64-64 64h-64c-35.3 0-64-28.7-64-64V216"/></svg>
    </a>
  
</div>
      
    </div>
    
  </div>
</footer>
      
    </div>
    <div class="md-dialog" data-md-component="dialog">
      <div class="md-dialog__inner md-typeset"></div>
    </div>
    
    
    <script id="__config" type="application/json">{"base": ".", "features": [], "translations": {"clipboard.copied": "Copied to clipboard", "clipboard.copy": "Copy to clipboard", "search.result.more.one": "1 more on this page", "search.result.more.other": "# more on this page", "search.result.none": "No matching documents", "search.result.one": "1 matching document", "search.result.other": "# matching documents", "search.result.placeholder": "Type to start searching", "search.result.term.missing": "Missing", "select.version": "Select version"}, "version": {"provider": "mike", "staticVersions": null, "versionPath": "../versions-v3.json"}}</script>
    
      
        <script src="_static/sphinx_immaterial_theme.32136f45f91ae6956.min.js?v=a7a9472a"></script>
        <script crossorigin="anonymous" integrity="sha256-Ae2Vz/4ePdIu6ZyI/5ZGsYnb+m0JlOmKPjt6XZ9JJkA=" src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.4/require.min.js"></script>
        <script>window.MathJax = {"tex": {"inlineMath": [["$", "$"], ["\\(", "\\)"]], "processEscapes": true}, "options": {"ignoreHtmlClass": "tex2jax_ignore|mathjax_ignore|document", "processHtmlClass": "tex2jax_process|mathjax_process|math|output_area"}}</script>
        <script id="MathJax-script" src="_static/mathjax/tex-mml-chtml.js?v=cadf963e"></script>
    
  </body>
</html>