Vibration of rotating structures: Difference between revisions
Jump to navigation
Jump to search
en>Mild Bill Hiccup m spelling |
en>Dolphin51 m Changed linking to avoid Centrifugal - a disambiguation page. |
||
| Line 1: | Line 1: | ||
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 [[Alternating conditional expectation model|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]]. | |||
==Description== | |||
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 | |||
: <math>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>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. {{jstor|2241560}}</ref> | |||
==See also== | |||
*[[Generalized additive model]] | |||
*[[Backfitting algorithm]] | |||
*[[Alternating conditional expectation model]] | |||
*[[Projection pursuit regression]] | |||
*[[Generalized additive model for location, scale, and shape]] (GAMLSS) | |||
*[[Median polish]] | |||
==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:Regression analysis]] | |||
[[Category:Nonparametric regression]] | |||
Latest revision as of 07:32, 29 January 2013
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.
Description
Given a data set of n statistical units, where represent predictors and is the outcome, the additive model takes the form
![{\displaystyle E[y_{i}|x_{i1},\ldots ,x_{ip}]=\beta _{0}+\sum _{j=1}^{p}f_{j}(x_{ij})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5706a34e005e06362a1877baf64b7282747b49db)
or
Where , and . The functions are unknown smooth functions fit from the data. Fitting the AM (i.e. the functions ) can be done using the backfitting algorithm proposed by Andreas Buja, Trevor Hastie and Robert Tibshirani (1989).[2]
![{\displaystyle E[\epsilon ]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7576dbb2c912e34b78ea232681fdca439bd90e8c)
![{\displaystyle E[f_{j}(X_{j})]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/def89ffacee203775b8655365ff77f93f3de8efe)
See also
- Generalized additive model
- Backfitting algorithm
- Alternating conditional expectation model
- Projection pursuit regression
- Generalized additive model for location, scale, and shape (GAMLSS)
- Median polish
References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
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. 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park.
- ↑ Friedman, J.H. and Stuetzle, W. (1981). "Projection Pursuit Regression", Journal of the American Statistical Association 76:817–823. 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park.
- ↑ Buja, A., Hastie, T., and Tibshirani, R. (1989). "Linear Smoothers and Additive Models", The Annals of Statistics 17(2):453–555. Template:Jstor