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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 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 mathematical modeling, overfitting is "the production of an analysis that corresponds too closely or exactly to a particular set of data, and may therefore fail to fit to additional data or predict future observations reliably". An overfitt ...
, and multicollinearity.


Description

Given a
data In the pursuit of knowledge, data (; ) is a collection of discrete values that convey information, describing quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpret ...
set \_^n of ''n''
statistical unit In statistics, a unit is one member of a set of entities being studied. It is the main source for the mathematical abstraction of a "random variable". Common examples of a unit would be a single person, animal, plant, manufactured item, or country ...
s, where \_^n represent predictors and y_i is the outcome, the ''additive model'' takes the form : \mathrm x_, \ldots, x_= \beta_0+\sum_^p f_j(x_) or : Y= \beta_0+\sum_^p f_j(X_)+\varepsilon Where \mathrm \epsilon = 0, \mathrm(\epsilon) = \sigma^2 and \mathrm f_j(X_) = 0. The functions f_j(x_) are unknown
smooth function In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
s fit from the data. Fitting the ''AM'' (i.e. the functions f_j(x_)) 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.


See also

* Generalized additive model * Backfitting algorithm *
Projection pursuit regression In statistics, projection pursuit regression (PPR) is a statistical model developed by Jerome H. Friedman and Werner Stuetzle which is an extension of additive models. This model adapts the additive models in that it first projects the data matri ...
* Generalized additive model for location, scale, and shape (GAMLSS) * Median polish *
Projection Pursuit Projection pursuit (PP) is a type of statistical technique which involves finding the most "interesting" possible projections in multidimensional data. Often, projections which deviate more from a normal distribution are considered to be more inte ...


References


Further reading

*Breiman, L. and Friedman, J.H. (1985). "Estimating Optimal Transformations for Multiple Regression and Correlation", ''
Journal of the American Statistical Association The ''Journal of the American Statistical Association (JASA)'' is the primary journal published by the American Statistical Association, the main professional body for statisticians in the United States. It is published four times a year in Ma ...
'' 80:580–598. {{doi, 10.1080/01621459.1985.10478157 Nonparametric regression Regression models