{{Short description|Statistical regression 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)<ref>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}}</ref> 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 <math>\{y_i,\, x_{i1}, \ldots, x_{ip}\}_{i=1}^n</math> of ''n'' statistical units, 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 functions fit from the data. Fitting the ''AM'' (i.e. the functions <math>f_j(x_{ij})</math>) can be done using the backfitting algorithm proposed by Andreas Buja, Trevor Hastie and Robert Tibshirani (1989).<ref>Buja, A., Hastie, T., and Tibshirani, R. (1989). "Linear Smoothers and Additive Models", ''The Annals of Statistics'' 17(2):453–555. {{JSTOR|2241560}}</ref>
==See also== *Generalized additive model *Backfitting algorithm *Projection pursuit regression *Generalized additive model for location, scale, and shape (GAMLSS) *Median polish *Projection pursuit
==References== {{reflist}}
==Further reading== *Breiman, L. and Friedman, J.H. (1985). "Estimating Optimal Transformations for Multiple Regression and Correlation", ''Journal of the American Statistical Association'' 80:580–598. {{doi|10.1080/01621459.1985.10478157}}
Category:Nonparametric regression Category:Regression models