Generalized additive model

In statistics, the generalized additive model (GAM) is a statistical model developed by Trevor Hastie and Rob Tibshirani for blending properties of generalized linear models with additive models.

The model specifies a distribution (such as a normal distribution, or a binomial distribution) and a link function g relating the expected value of the distribution to the predictors, and attempts to fit functions f_i(x_i) to satisfy:

$g(\operatorname {E} (Y))=\beta _{0}+f_{1}(x_{1})+f_{2}(x_{2})+\cdots +f_{m}(x_{m}).\,\!$

The functions f_i(x_i) may be fit using parametric or non-parametric means, thus providing the potential for better fits to data than other methods. The method hence is very general – a typical GAM might use a scatterplot smoothing function such as a locally weighted mean for f₁(x₁), and then use a factor model for f₂(x₂). By allowing nonparametric fits, well designed GAMs allow good fits to the training data with relaxed assumptions on the actual relationship, perhaps at the expense of interpretability of results.

Overfitting can be a problem with GAMs. The number of smoothing parameters can be specified, and this number should be reasonably small, certainly well under the degrees of freedom offered by the data. Cross-validation can be used to detect and/or reduce overfitting problems with GAMs (or other statistical methods). Other models such as GLMs may be preferable to GAMs unless GAMs improve predictive ability substantially for the application in question.

See also

• Additive model
• Generalized additive model for location, scale, and shape (GAMLSS)
• Backfitting algorithm

References

• Hastie, T. J. and Tibshirani, R. J. (1990). Generalized Additive Models. Chapman & Hall/CRC. ISBN 9780412343902.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Wood, S. N. (2006). Generalized Additive Models: An Introduction with R. Chapman & Hall/CRC. ISBN 9781584884743.