Additive model

The 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 the 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. Furhtermore, 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

$E[y_{i}|x_{i1},\ldots ,x_{ip}]=\sum _{j=1}^{p}f_{j}(x_{ij})$

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

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