Additive model

<h2 id="description">Description</h2>
<p>Given a <a href="/facts/Data/AUBFN97o">data</a> set 
  
    
      
        {
        
          y
          
            i
          
        
        ,
        
        
          x
          
            i
            1
          
        
        ,
        …
        ,
        
          x
          
            i
            p
          
        
        
          }
          
            i
            =
            1
          
          
            n
          
        
      
    
    {\displaystyle \{y_{i},\,x_{i1},\ldots ,x_{ip}\}_{i=1}^{n}}
  
 of <i>n</i> <a href="/facts/Statistical_unit/oIW2uEBb">statistical units</a>, where 
  
    
      
        {
        
          x
          
            i
            1
          
        
        ,
        …
        ,
        
          x
          
            i
            p
          
        
        
          }
          
            i
            =
            1
          
          
            n
          
        
      
    
    {\displaystyle \{x_{i1},\ldots ,x_{ip}\}_{i=1}^{n}}
  
 represent predictors and 
  
    
      
        
          y
          
            i
          
        
      
    
    {\displaystyle y_{i}}
  
 is the outcome, the <i>additive model</i> takes the form
</p>

E
        
        [
        
          y
          
            i
          
        
        
          |
        
        
          x
          
            i
            1
          
        
        ,
        …
        ,
        
          x
          
            i
            p
          
        
        ]
        =
        
          β
          
            0
          
        
        +
        
          ∑
          
            j
            =
            1
          
          
            p
          
        
        
          f
          
            j
          
        
        (
        
          x
          
            i
            j
          
        
        )
      
    
    {\displaystyle \mathrm {E} [y_{i}|x_{i1},\ldots ,x_{ip}]=\beta _{0}+\sum _{j=1}^{p}f_{j}(x_{ij})}

Y
        =
        
          β
          
            0
          
        
        +
        
          ∑
          
            j
            =
            1
          
          
            p
          
        
        
          f
          
            j
          
        
        (
        
          X
          
            j
          
        
        )
        +
        ε
      
    
    {\displaystyle Y=\beta _{0}+\sum _{j=1}^{p}f_{j}(X_{j})+\varepsilon }

<p>Where 
  
    
      
        
          E
        
        [
        ϵ
        ]
        =
        0
      
    
    {\displaystyle \mathrm {E} [\epsilon ]=0}
  
, 
  
    
      
        
          V
          a
          r
        
        (
        ϵ
        )
        =
        
          σ
          
            2
          
        
      
    
    {\displaystyle \mathrm {Var} (\epsilon )=\sigma ^{2}}
  
 and 
  
    
      
        
          E
        
        [
        
          f
          
            j
          
        
        (
        
          X
          
            j
          
        
        )
        ]
        =
        0
      
    
    {\displaystyle \mathrm {E} [f_{j}(X_{j})]=0}
  
. The functions 
  
    
      
        
          f
          
            j
          
        
        (
        
          x
          
            i
            j
          
        
        )
      
    
    {\displaystyle f_{j}(x_{ij})}
  
 are unknown <a href="/facts/Smooth_function/mffL4ch2">smooth functions</a> fit from the data. Fitting the <i>AM</i> (i.e. the functions 
  
    
      
        
          f
          
            j
          
        
        (
        
          x
          
            i
            j
          
        
        )
      
    
    {\displaystyle f_{j}(x_{ij})}
  
) can be done using the <a href="/facts/Backfitting_algorithm/fpy26cXT">backfitting algorithm</a> proposed by Andreas Buja, <a href="/facts/Trevor_Hastie/9bSQQGo7">Trevor Hastie</a> and <a href="/facts/Robert_Tibshirani/BIzgnLuo">Robert Tibshirani</a> (1989).<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Generalized_additive_model/qgKRK09U">Generalized additive model</a></li>
<li><a href="/facts/Backfitting_algorithm/fpy26cXT">Backfitting algorithm</a></li>
<li><a href="/facts/Projection_pursuit_regression/SnnzJYLp">Projection pursuit regression</a></li>
<li><a href="/facts/Generalized_additive_model_for_location,_scale,_and_shape/1bHeLM8y">Generalized additive model for location, scale, and shape</a> (GAMLSS)</li>
<li><a href="/facts/Median_polish/UoEdpUMK">Median polish</a></li>
<li><a href="/facts/Projection_pursuit/EX9ETG8l">Projection pursuit</a></li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li>Breiman, L. and <a href="/facts/Friedman,_J.H./brTdEkVe">Friedman, J.H.</a> (1985). "Estimating Optimal Transformations for Multiple Regression and Correlation", <i><a href="/facts/Journal_of_the_American_Statistical_Association/H513IjxL">Journal of the American Statistical Association</a></i> 80:580–598. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1080%2F01621459.1985.10478157">10.1080/01621459.1985.10478157</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Friedman, J.H. and Stuetzle, W. (1981). "Projection Pursuit Regression", Journal of the American Statistical Association 76:817–823. doi:10.1080/01621459.1981.10477729 <a href="/wiki/Friedman,_J.H." target="_blank">/wiki/Friedman,_J.H.</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Buja, A., Hastie, T., and Tibshirani, R. (1989). "Linear Smoothers and Additive Models", The Annals of Statistics 17(2):453–555. JSTOR 2241560 <a href="/wiki/JSTOR_(identifier)" target="_blank">/wiki/JSTOR_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Additive model open-in-new

Additive model