Additive 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 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).

See also

• Generalized additive model
• Backfitting algorithm
• Alternating conditional expectation model
• Projection pursuit regression
• Generalized additive model for location, scale, and shape (GAMLSS)

References

• Buja, A., Hastie, T., and Tibshirani, R. (1989). "Linear Smoothers and Additive Models", The Annals of Statistics 17(2):453–555.
• Breiman, L. and Friedman, J.H. (1985). "Estimating Optimal Transformations for Multiple Regression and Correlation", Journal of the American Statistical Association 80:580–598.
• Friedman, J.H. and Stuetzle, W. (1981. "Projection Pursuit Regression", Journal of the American Statistical Association 76:817–823