Additive_model Knowpia

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, like many other machine-learning methods, include model selection, overfitting, and multicollinearity.

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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

\mathrm {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 $\mathrm {E} [\epsilon ]=0$ , $\mathrm {Var} (\epsilon )=\sigma ^{2}$ and $\mathrm {E} [f_{j}(X_{j})]=0$ . 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 edit

^ 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

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