additive model

{{About|the statistical method|additive color models|Additive color}}

In statistics, an additive model (AM) is a nonparametric regression method. It was suggested by Jerome H. Friedman and Werner Stuetzle (1981)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}} 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 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.

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

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

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).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

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Further reading