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Additive model

Statistical regression model

Statistical regression model

Note

the statistical method

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 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).

References

References

  1. [[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
  2. 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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