Additive Model (AM) Function: Difference between revisions
Jump to navigation
Jump to search
m (Text replacement - "\<P\>([\s]{1,7})([^\s])" to "<P> $2") |
m (Text replacement - ". "" to ". “") |
||
| Line 13: | Line 13: | ||
=== 2020 === | === 2020 === | ||
* (Wikipedia, 2020) ⇒ https://en.wikipedia.org/wiki/Additive_model Retrieved:2020-10-2. | * (Wikipedia, 2020) ⇒ https://en.wikipedia.org/wiki/Additive_model Retrieved:2020-10-2. | ||
** 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). | ** 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. </ref> and is an essential part of the [[Alternating conditional expectations|ACE]] algorithm. The ''AM'' uses a one-dimensional [[Smoothing|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 [[linear regression|standard linear model]], while being more interpretable than a general regression surface at the cost of approximation errors. Problems with ''AM'' include [[model selection]], [[overfitting]], and [[multicollinearity]]. | ||
=== 2020 === | === 2020 === | ||
* (Wikipedia, 2020) ⇒ https://en.wikipedia.org/wiki/Additive_model#Description Retrieved:2020-10-2. | * (Wikipedia, 2020) ⇒ https://en.wikipedia.org/wiki/Additive_model#Description Retrieved:2020-10-2. | ||
** Given a [[data]] set <math> \{y_i,\, x_{i1}, \ldots, x_{ip}\}_{i=1}^n </math> of ''n'' [[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 : <P> <math> E[y_i|x_{i1}, \ldots, x_{ip}] = \beta_0+\sum_{j=1}^p f_j(x_{ij}) </math> <P> or <P> <math> Y= \beta_0+\sum_{j=1}^p f_j(X_{j})+\varepsilon </math> Where <math> <P> E[ \epsilon ] = 0 </math> , <math> Var(\epsilon) = \sigma^2 </math> and <math> E[ f_j(X_{j}) ] = 0 </math> . The functions <math> f_j(x_{ij}) </math> are unknown [[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]] proposed by Andreas Buja, [[Trevor Hastie]] and [[Robert Tibshirani]] (1989). <ref> Buja, A., Hastie, T., and Tibshirani, R. (1989). | ** Given a [[data]] set <math> \{y_i,\, x_{i1}, \ldots, x_{ip}\}_{i=1}^n </math> of ''n'' [[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 : <P> <math> E[y_i|x_{i1}, \ldots, x_{ip}] = \beta_0+\sum_{j=1}^p f_j(x_{ij}) </math> <P> or <P> <math> Y= \beta_0+\sum_{j=1}^p f_j(X_{j})+\varepsilon </math> Where <math> <P> E[ \epsilon ] = 0 </math> , <math> Var(\epsilon) = \sigma^2 </math> and <math> E[ f_j(X_{j}) ] = 0 </math> . The functions <math> f_j(x_{ij}) </math> are unknown [[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]] 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. </ref> | ||
<references/> | <references/> | ||
Latest revision as of 04:35, 8 May 2024
An Additive Model (AM) Function is a model function of the form [math]\displaystyle{ Y= \beta_0+\sum_{j=1}^p f_j(X_{j})+\varepsilon }[/math].
- Context:
- It can be trained by an Additive Model Training System.
- It can (typically) be an Interpretable Model.
- …
- See: Additive Tree, Additive Modeling, Interaction Regression Model, Statistical Unit, Smooth Function, Backfitting Algorithm.
References
2020
- (Wikipedia, 2020) ⇒ https://en.wikipedia.org/wiki/Additive_model Retrieved:2020-10-2.
- 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 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 include model selection, overfitting, and multicollinearity.
2020
- (Wikipedia, 2020) ⇒ https://en.wikipedia.org/wiki/Additive_model#Description Retrieved:2020-10-2.
- Given a data set [math]\displaystyle{ \{y_i,\, x_{i1}, \ldots, x_{ip}\}_{i=1}^n }[/math] of n statistical units, where [math]\displaystyle{ \{x_{i1}, \ldots, x_{ip}\}_{i=1}^n }[/math] represent predictors and [math]\displaystyle{ y_i }[/math] is the outcome, the additive model takes the form :
[math]\displaystyle{ E[y_i|x_{i1}, \ldots, x_{ip}] = \beta_0+\sum_{j=1}^p f_j(x_{ij}) }[/math]
or
[math]\displaystyle{ Y= \beta_0+\sum_{j=1}^p f_j(X_{j})+\varepsilon }[/math] Where [math]\displaystyle{ \lt P\gt E[ \epsilon ] = 0 }[/math] , [math]\displaystyle{ Var(\epsilon) = \sigma^2 }[/math] and [math]\displaystyle{ E[ f_j(X_{j}) ] = 0 }[/math] . The functions [math]\displaystyle{ f_j(x_{ij}) }[/math] are unknown smooth functions fit from the data. Fitting the AM (i.e. the functions [math]\displaystyle{ f_j(x_{ij}) }[/math] ) can be done using the backfitting algorithm proposed by Andreas Buja, Trevor Hastie and Robert Tibshirani (1989). [2]
- Given a data set [math]\displaystyle{ \{y_i,\, x_{i1}, \ldots, x_{ip}\}_{i=1}^n }[/math] of n statistical units, where [math]\displaystyle{ \{x_{i1}, \ldots, x_{ip}\}_{i=1}^n }[/math] represent predictors and [math]\displaystyle{ y_i }[/math] is the outcome, the additive model takes the form :
- ↑ Friedman, J.H. and Stuetzle, W. (1981). “Projection Pursuit Regression", Journal of the American Statistical Association 76:817–823.
- ↑ Buja, A., Hastie, T., and Tibshirani, R. (1989). “Linear Smoothers and Additive Models", The Annals of Statistics 17(2):453–555.
2015
- http://slideplayer.com/slide/9920677/ Interaction regression models.
- QUOTE:
- QUOTE: