Additive model

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 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 include model selection, overfitting, and multicollinearity.

Description

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

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

or

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

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

See also

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

References

1. ↑ 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
2. ↑ Buja, A., Hastie, T., and Tibshirani, R. (1989). "Linear Smoothers and Additive Models", The Annals of Statistics 17(2):453–555. JSTOR 2241560

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