In statistics, an additive model (AM) is a nonparametric regression method. It was suggested by Jerome H. Friedman and Werner Stuetzle (1981)[1] and is an essential part of the ACE algorithm. The AM uses a one-dimensional smoother to build a restricted class of nonparametric regression models. Because of this, it is less affected by the curse of dimensionality than a p-dimensional smoother. Furthermore, the AM is more flexible than a standard linear model, while being more interpretable than a general regression surface at the cost of approximation errors. Problems with AM, like many other machine-learning methods, include model selection, overfitting, and multicollinearity.
Given a data set { y i , x i 1 , … , x i p } i = 1 n {\displaystyle \{y_{i},\,x_{i1},\ldots ,x_{ip}\}_{i=1}^{n}} of n statistical units, 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 additive model takes the form
or
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 smooth functions fit from the data. Fitting the AM (i.e. the functions f j ( x i j ) {\displaystyle f_{j}(x_{ij})} ) can be done using the backfitting algorithm proposed by Andreas Buja, Trevor Hastie and Robert Tibshirani (1989).[2]