Additive model

In statistics , an additive model ( AM ) is a nonparametric regression model . It was suggested by Jerome H. Friedman and Werner Stuetzle (1981). The additive model uses a one-dimensional smoother to form a constrained class of nonparametric regression models. Therefore, the model is less affected by the curse of dimensionality than, for example, a p -dimensional smoother. The AM is more flexible than ordinary linear regression models . Problems that can arise with the additive model are overfitting and multicollinearity .

The standard model of additive regression

Given are the observations of a continuous response and explanatory variables , the effects of which on the response can be modeled by a linear predictor . In addition, the observations of a continuous explanatory variable are available, the effects of which are analyzed and modeled non-parametrically . If there are no interaction effects between the explanatory variables, the standard model of additive regression is given by: ${\displaystyle (y_{i},x_{i1},x_{i2},\ldots ,x_{ik})\quad ,i=1,\ldots ,n}$ ${\displaystyle y}$ ${\displaystyle x_{1},x_{2},\ldots ,x_{k}}$ ${\displaystyle (z_{i1},z_{i2},\ldots ,z_{iq})\quad ,i=1,\ldots ,n}$

${\displaystyle y_{i}=f_{1}(z_{i1})+\ldots +f_{q}(z_{iq})+\beta _{0}+x_{i1}\beta _{1}+x_{i2}\beta _{2}+\dotsc +x_{ik}\beta _{k}+\varepsilon _{i}=f_{1}(z_{i1})+\ldots +f_{q}(z_{iq})+\eta _{i}^{lin.}+\varepsilon _{i}=\eta _{i}^{add.}+\varepsilon _{i}}$,

where for the linear predictors and applies. ${\displaystyle \eta _{i}^{lin.}=\beta _{0}+x_{i1}\beta _{1}+x_{i2}\beta _{2}+\dotsc +x_{ik}\beta _{k}}$ ${\displaystyle \eta _{i}^{add.}=f_{1}(z_{i1})+\ldots +f_{q}(z_{iq})+\eta _{i}^{lin.}}$

The functions represent the non-linear smoothing effects of the explanatory variables and are modeled and estimated using non-parametric methods. The same assumptions are made with regard to the disturbance variables as in the classic linear model  (see Linear Single Regression # Assumptions about the disturbance variables ). The identification problem arises with additive models . ${\displaystyle f_{1}(z_{1}),\ldots ,f_{q}(z_{q})}$

See also

• Generalized additive models for position, scale and shape parameters
• Scatter diagram smoother

literature

• Ludwig Fahrmeir , Thomas Kneib , Stefan Lang: Regression: Models, Methods and Applications. 2nd Edition. Springer Verlag, 2009, ISBN 978-3-642-01836-7 .

Individual evidence

1. ↑ Friedman, JH and Stuetzle, W. (1981). Projection Pursuit Regression , Journal of the American Statistical Association 76: 817-823. doi: 10.1080 / 01621459.1981.10477729
2. ^ Ludwig Fahrmeir, Thomas Kneib , Stefan Lang, Brian Marx: Regression: models, methods and applications. Springer Science & Business Media, 2013, ISBN 978-3-642-34332-2 , p. 536.
3. ^ Ludwig Fahrmeir, Thomas Kneib, Stefan Lang, Brian Marx: Regression: models, methods and applications. Springer Science & Business Media, 2013, ISBN 978-3-642-34332-2 , p. 536.