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In [[statistics]], a '''generalized additive model (GAM)''' is a [[generalized linear model]] in which the linear predictor depends linearly on unknown smooth functions of some predictor variables, and interest focuses on inference about these smooth functions. | |||
GAMs were originally developed by [[Trevor Hastie]] and [[Robert Tibshirani]] <ref name=Hastie1990>{{cite book|author = Hastie, T. J. and Tibshirani, R. J.|title = Generalized Additive Models|publisher = Chapman & Hall/CRC|year = 1990|isbn=978-0-412-34390-2}}</ref> to blend properties of [[generalized linear model]]s with [[additive model]]s. | |||
The model relates a univariate response variable, ''Y'', to some predictor variables, ''x''<sub>''i''</sub>. An [[exponential family]] distribution is specified for Y (for example [[normal distribution|normal]], [[binomial distribution|binomial]] or [[Poisson distribution|Poisson]] distributions) along with a link function ''g'' (for example the identity or log functions) relating the expected value of ''Y'' to the predictor variables via a structure such as | |||
: <math>g(\operatorname{E}(Y))=\beta_0 + f_1(x_1) + f_2(x_2)+ \cdots + f_m(x_m).\,\!</math> | |||
The functions ''f''<sub>''i''</sub>(''x''<sub>''i''</sub>) may be functions with a specified parametric form (for example a polynomial, or a coefficient depending on the levels of a factor variable) or maybe specified non-parametrically, or semi-parametrically, simply as 'smooth functions', to be estimated by [[Nonparametric regression|non-parametric means]]. So a typical GAM might use a scatterplot smoothing function, such as a locally weighted mean, for ''f''<sub>1</sub>(''x''<sub>1</sub>), and then use a factor model for ''f''<sub>2</sub>(''x''<sub>2</sub>). This flexibility to allow non-parametric fits with relaxed assumptions on the actual relationship between response and predictor, provides the potential for better fits to data than purely parametric models, but arguably with some loss of interpretablity. | |||
== Estimation == | |||
The original GAM estimation method was the [[backfitting algorithm]],<ref name=Hastie1990 /> which provides a very general modular estimation method capable of using a wide variety of smoothing methods to estimate the ''f''<sub>''i''</sub>(''x''<sub>''i''</sub>). A disadvantage of backfitting is that it is difficult to integrate with well founded methods for choosing the degree of smoothness of the ''f''<sub>''i''</sub>(''x''<sub>''i''</sub>). As a result alternative methods have been developed in which smooth functions are represented semi-parametrically, using penalized regression splines,<ref name=Wood2006>{{cite book|author = Wood, S. N.|title = Generalized Additive Models: An Introduction with R|publisher = Chapman & Hall/CRC|year = 2006|isbn=978-1-58488-474-3}}</ref> in order to allow computationally efficient estimation of the degree of smoothness of the model components using generalized cross validation <ref name=Wood2000>Wood, S.N. (2000) Modelling and smoothing parameter estimation with multiple quadratic penalties. Journal of the Royal Statistical Society: Series B 62(2),413-428.</ref> or similar criteria. | |||
[[Overfitting]] can be a problem with GAMs. {{citation needed|date=October 2012}} The number of smoothing parameters can be specified, and this number should be reasonably small, certainly well under the [[degrees of freedom (statistics)|degrees of freedom]] offered by the data. [[Cross-validation (statistics)|Cross-validation]] can be used to detect and/or reduce overfitting problems with GAMs (or other statistical methods). {{citation needed|date=October 2012}} Other models such as [[Generalized linear model|GLMs]] may be preferable to GAMs unless GAMs improve predictive ability substantially (in validation sets) for the application in question. | |||
==See also== | |||
*[[Additive model]] | |||
*[[Backfitting algorithm]] | |||
*[[Generalized additive model for location, scale, and shape]] (GAMLSS) | |||
*[[Residual effective degrees of freedom]] | |||
==References== | |||
<references /> | |||
==External links== | |||
*[http://cran.r-project.org/web/packages/gam/index.html R Package for GAMs by backfitting] | |||
*[http://cran.r-project.org/web/packages/mgcv/index.html R Package for GAMs using penalized regression splines] | |||
[[Category:Generalized linear models]] | |||
[[Category:Nonparametric regression]] | |||
Revision as of 15:54, 6 March 2013
In statistics, a generalized additive model (GAM) is a generalized linear model in which the linear predictor depends linearly on unknown smooth functions of some predictor variables, and interest focuses on inference about these smooth functions. GAMs were originally developed by Trevor Hastie and Robert Tibshirani [1] to blend properties of generalized linear models with additive models.
The model relates a univariate response variable, Y, to some predictor variables, xi. An exponential family distribution is specified for Y (for example normal, binomial or Poisson distributions) along with a link function g (for example the identity or log functions) relating the expected value of Y to the predictor variables via a structure such as
The functions fi(xi) may be functions with a specified parametric form (for example a polynomial, or a coefficient depending on the levels of a factor variable) or maybe specified non-parametrically, or semi-parametrically, simply as 'smooth functions', to be estimated by non-parametric means. So a typical GAM might use a scatterplot smoothing function, such as a locally weighted mean, for f1(x1), and then use a factor model for f2(x2). This flexibility to allow non-parametric fits with relaxed assumptions on the actual relationship between response and predictor, provides the potential for better fits to data than purely parametric models, but arguably with some loss of interpretablity.
Estimation
The original GAM estimation method was the backfitting algorithm,[1] which provides a very general modular estimation method capable of using a wide variety of smoothing methods to estimate the fi(xi). A disadvantage of backfitting is that it is difficult to integrate with well founded methods for choosing the degree of smoothness of the fi(xi). As a result alternative methods have been developed in which smooth functions are represented semi-parametrically, using penalized regression splines,[2] in order to allow computationally efficient estimation of the degree of smoothness of the model components using generalized cross validation [3] or similar criteria.
Overfitting can be a problem with GAMs. Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. The number of smoothing parameters can be specified, and this number should be reasonably small, certainly well under the degrees of freedom offered by the data. Cross-validation can be used to detect and/or reduce overfitting problems with GAMs (or other statistical methods). Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. Other models such as GLMs may be preferable to GAMs unless GAMs improve predictive ability substantially (in validation sets) for the application in question.
See also
- Additive model
- Backfitting algorithm
- Generalized additive model for location, scale, and shape (GAMLSS)
- Residual effective degrees of freedom
References
- ↑ 1.0 1.1 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ Wood, S.N. (2000) Modelling and smoothing parameter estimation with multiple quadratic penalties. Journal of the Royal Statistical Society: Series B 62(2),413-428.