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}})

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]

References

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

Additive model

Description

See also

References

Further reading