Generalized Additive Model Selection

Description

This page is intended to provide some more information on how to select GAMs. Given a model structure specified by a gam model formula, gam() attempts to find the appropriate smoothness for each applicable model term using Generalized Cross Validation (GCV) or an Un-Biased Risk Estimator (UBRE), the latter being used in cases in which the scale parameter is assumed known. GCV and UBRE are covered in Craven and Wahba (1979) and Wahba (1990), see gam.method for more detail about the numerical optimization approaches available.

Automatic smoothness selection is unlikely to be successful with few data, particularly with multiple terms to be selected. In addition GCV and UBRE score can occasionally display local minima that can trap the minimisation algorithms. GCV/UBRE scores become constant with changing smoothing parameters at very low or very high smoothng parameters, and on occasion these `flat' regions can be separated from regions of lower score by a small `lip'. This seems to be the most common form of local minimum, but is usually avoidable by avoiding extreme smoothing parameters as starting values in optimization, and by avoiding big jumps in smoothing parameters while optimizing. Never the less, if you are suspicious of smoothing parameter estimates, try changing fit method (see gam.method) and see if the estimates change, or try changing some or all of the smoothing parameters `manually' (argument sp of gam).

In general the most logically consistent method to use for deciding which terms to include in the model is to compare GCV/UBRE scores for models with and without the term. More generally the score for the model with a smooth term can be compared to the score for the model with the smooth term replaced by appropriate parametric terms. Candidates for removal can be identified by reference to the approximate p-values provided by summary.gam. Candidates for replacement by parametric terms are smooth terms with estimated degrees of freedom close to their minimum possible.

One appealing approach to model selection is via shrinkage. Smooth classes cs.smooth and tprs.smooth (specified by "cs" and "ts" respectively) have smoothness penalties which include a small shrinkage component, so that for large enough smoothing parameters the smooth becomes identically zero. This allows automatic smoothing parameter selection methods to effectively remove the term from the model altogether. The shrinkage component of the penalty is set at a level that usually makes negligable contribution to the penalization of the model, only becoming effective when the term is effectively `completely smooth' according to the conventional penalty.

Author(s)

Simon N. Wood simon.wood@r-project.org

References

Craven and Wahba (1979) Smoothing Noisy Data with Spline Functions. Numer. Math. 31:377-403

Venables and Ripley (1999) Modern Applied Statistics with S-PLUS

Wahba (1990) Spline Models of Observational Data. SIAM.

Wood, S.N. (2000) Modelling and Smoothing Parameter Estimation with Multiple Quadratic Penalties. J.R.Statist.Soc.B 62(2):413-428

Wood, S.N. (2003) Thin plate regression splines. J.R.Statist.Soc.B 65(1):95-114

http://www.stats.gla.ac.uk/~simon/

Examples

## an example of GCV based model selection
library(mgcv)
set.seed(0) 
n<-400;sig<-2
x0 <- runif(n, 0, 1);x1 <- runif(n, 0, 1)
x2 <- runif(n, 0, 1);x3 <- runif(n, 0, 1)
x4 <- runif(n, 0, 1);x5 <- runif(n, 0, 1)
f <- 2 * sin(pi * x0)
f <- f + exp(2 * x1) - 3.75887
f <- f+0.2*x2^11*(10*(1-x2))^6+10*(10*x2)^3*(1-x2)^10-1.396
e <- rnorm(n, 0, sig)
y <- f + e
## Note the increased gamma parameter below to favour
## slightly smoother models (See Kim & Gu, 2004, JRSSB)...
b<-gam(y~s(x0,bs="ts")+s(x1,bs="ts")+s(x2,bs="ts")+
   s(x3,bs="ts")+s(x4,bs="ts")+s(x5,bs="ts"),gamma=1.4)
summary(b)
plot(b,pages=1)