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Additive model

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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)[1] 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 i 1 , … , x i p } i = 1 n {\displaystyle \{y_{i},\,x_{i1},\ldots ,x_{ip}\}_{i=1}^{n}} {\displaystyle \{y_{i},\,x_{i1},\ldots ,x_{ip}\}_{i=1}^{n}} of n statistical units, where { x i 1 , … , x i p } i = 1 n {\displaystyle \{x_{i1},\ldots ,x_{ip}\}_{i=1}^{n}} {\displaystyle \{x_{i1},\ldots ,x_{ip}\}_{i=1}^{n}} represent predictors and y i {\displaystyle y_{i}} {\displaystyle y_{i}} is the outcome, the additive model takes the form

E [ y i | x i 1 , … , x i p ] = β 0 + ∑ j = 1 p f j ( x i j ) {\displaystyle \mathrm {E} [y_{i}|x_{i1},\ldots ,x_{ip}]=\beta _{0}+\sum _{j=1}^{p}f_{j}(x_{ij})} {\displaystyle \mathrm {E} [y_{i}|x_{i1},\ldots ,x_{ip}]=\beta _{0}+\sum _{j=1}^{p}f_{j}(x_{ij})}

or

Y = β 0 + ∑ j = 1 p f j ( X j ) + ε {\displaystyle Y=\beta _{0}+\sum _{j=1}^{p}f_{j}(X_{j})+\varepsilon } {\displaystyle Y=\beta _{0}+\sum _{j=1}^{p}f_{j}(X_{j})+\varepsilon }

Where E [ ϵ ] = 0 {\displaystyle \mathrm {E} [\epsilon ]=0} {\displaystyle \mathrm {E} [\epsilon ]=0}, V a r ( ϵ ) = σ 2 {\displaystyle \mathrm {Var} (\epsilon )=\sigma ^{2}} {\displaystyle \mathrm {Var} (\epsilon )=\sigma ^{2}} and E [ f j ( X j ) ] = 0 {\displaystyle \mathrm {E} [f_{j}(X_{j})]=0} {\displaystyle \mathrm {E} [f_{j}(X_{j})]=0}. The functions f j ( x i j ) {\displaystyle f_{j}(x_{ij})} {\displaystyle f_{j}(x_{ij})} are unknown smooth functions fit from the data. Fitting the AM (i.e. the functions f j ( x i j ) {\displaystyle f_{j}(x_{ij})} {\displaystyle f_{j}(x_{ij})}) can be done using the backfitting algorithm proposed by Andreas Buja, Trevor Hastie and Robert Tibshirani (1989).[2]

See also

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

Further reading