Additive model

Short description: Statistical regression model

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 [math]\displaystyle{ \{y_i,\, x_{i1}, \ldots, x_{ip}\}_{i=1}^n }[/math] of n statistical units, where [math]\displaystyle{ \{x_{i1}, \ldots, x_{ip}\}_{i=1}^n }[/math] represent predictors and [math]\displaystyle{ y_i }[/math] is the outcome, the additive model takes the form

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

or

[math]\displaystyle{ Y= \beta_0+\sum_{j=1}^p f_j(X_{j})+\varepsilon }[/math]

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

See also

• Generalized additive model
• Backfitting algorithm
• Projection pursuit regression
• Generalized additive model for location, scale, and shape (GAMLSS)
• Median polish
• Projection pursuit

Further reading

• Breiman, L. and Friedman, J.H. (1985). "Estimating Optimal Transformations for Multiple Regression and Correlation", Journal of the American Statistical Association 80:580–598. doi:10.1080/01621459.1985.10478157