In statistics, an additive model (AM) is a nonparametric regression method. It was suggested by Jerome H. Friedman and Werner Stuetzle (1981) 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 e.g. 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.

## Description

Given a data set $\displaystyle{ \{y_i,\, x_{i1}, \ldots, x_{ip}\}_{i=1}^n }$ of n statistical units, where $\displaystyle{ \{x_{i1}, \ldots, x_{ip}\}_{i=1}^n }$ represent predictors and $\displaystyle{ y_i }$ is the outcome, the additive model takes the form

$\displaystyle{ \mathrm{E}[y_i|x_{i1}, \ldots, x_{ip}] = \beta_0+\sum_{j=1}^p f_j(x_{ij}) }$

or

$\displaystyle{ Y= \beta_0+\sum_{j=1}^p f_j(X_{j})+\varepsilon }$

Where $\displaystyle{ \mathrm{E}[ \epsilon ] = 0 }$, $\displaystyle{ \mathrm{Var}(\epsilon) = \sigma^2 }$ and $\displaystyle{ \mathrm{E}[ f_j(X_{j}) ] = 0 }$. The functions $\displaystyle{ f_j(x_{ij}) }$ are unknown smooth functions fit from the data. Fitting the AM (i.e. the functions $\displaystyle{ f_j(x_{ij}) }$) can be done using the backfitting algorithm proposed by Andreas Buja, Trevor Hastie and Robert Tibshirani (1989).