# Additive model

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{{Short description|Statistical regression model}}
{{About|the statistical method|additive color models|Additive color}}
In [statistics](/source/statistics), an '''additive model''' ('''AM''') is a [nonparametric regression](/source/nonparametric_regression) method. It was suggested by [Jerome H. Friedman](/source/Jerome_H._Friedman) and Werner Stuetzle (1981)<ref>[Friedman, J.H.](/source/Friedman%2C_J.H.) and Stuetzle, W. (1981). "Projection Pursuit Regression", ''Journal of the American Statistical Association'' 76:817&ndash;823. {{doi|10.1080/01621459.1981.10477729}}</ref> and is an essential part of the [ACE](/source/Alternating_conditional_expectations) algorithm. The ''AM'' uses a one-dimensional [smoother](/source/Smoothing) to build a restricted class of nonparametric regression models. Because of this, it is less affected by the [curse of dimensionality](/source/curse_of_dimensionality) than a ''p''-dimensional smoother. Furthermore, the ''AM'' is more flexible than a [standard linear model](/source/linear_regression), 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](/source/model_selection), [overfitting](/source/overfitting), and [multicollinearity](/source/multicollinearity).

==Description==
Given a [data](/source/data) set <math>\{y_i,\, x_{i1}, \ldots, x_{ip}\}_{i=1}^n</math> of ''n'' [statistical unit](/source/statistical_unit)s, where <math>\{x_{i1}, \ldots, x_{ip}\}_{i=1}^n</math> represent predictors and <math>y_i</math> is the outcome, the ''additive model'' takes the form
: <math>\mathrm{E}[y_i|x_{i1}, \ldots, x_{ip}] = \beta_0+\sum_{j=1}^p f_j(x_{ij}) </math>
or
: <math>Y= \beta_0+\sum_{j=1}^p f_j(X_{j})+\varepsilon </math>
Where <math>\mathrm{E}[ \epsilon ] = 0</math>, <math>\mathrm{Var}(\epsilon) = \sigma^2</math> and <math>\mathrm{E}[ f_j(X_{j}) ] = 0</math>. The functions <math>f_j(x_{ij})</math> are unknown [smooth function](/source/smooth_function)s fit from the data. Fitting the ''AM'' (i.e. the functions <math>f_j(x_{ij})</math>) can be done using the [backfitting algorithm](/source/backfitting_algorithm) proposed by Andreas Buja, [Trevor Hastie](/source/Trevor_Hastie) and [Robert Tibshirani](/source/Robert_Tibshirani) (1989).<ref>Buja, A., Hastie, T., and Tibshirani, R. (1989). "Linear Smoothers and Additive Models", ''The Annals of Statistics'' 17(2):453&ndash;555. {{JSTOR|2241560}}</ref>

==See also==
*[Generalized additive model](/source/Generalized_additive_model)
*[Backfitting algorithm](/source/Backfitting_algorithm)
*[Projection pursuit regression](/source/Projection_pursuit_regression)
*[Generalized additive model for location, scale, and shape](/source/Generalized_additive_model_for_location%2C_scale%2C_and_shape) (GAMLSS)
*[Median polish](/source/Median_polish)
*[Projection pursuit](/source/Projection_pursuit)

==References==
{{reflist}}

==Further reading==
*Breiman, L. and [Friedman, J.H.](/source/Friedman%2C_J.H.) (1985). "Estimating Optimal Transformations for Multiple Regression and Correlation", ''[Journal of the American Statistical Association](/source/Journal_of_the_American_Statistical_Association)'' 80:580&ndash;598. {{doi|10.1080/01621459.1985.10478157}}

Category:Nonparametric regression
Category:Regression models

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