Choosing the basis dimension, and checking the choice, when using
penalized regression smoothers.

Penalized regression smoothers gain computational efficiency by virtue of
being defined using a basis of relatively modest size, k. When setting
up models in the mgcv package, using s or te
terms in a model formula, k must be chosen: the defaults are essentially arbitrary.

In practice k-1 (or k) sets the upper limit on the degrees of freedom
associated with an s smooth (1 degree of freedom is usually lost to the identifiability
constraint on the smooth). For te smooths the upper limit of the
degrees of freedom is given by the product of the k values provided for
each marginal smooth less one, for the constraint. However the actual
effective degrees of freedom are controlled by the degree of penalization
selected during fitting, by GCV, AIC, REML or whatever is specified. The exception
to this is if a smooth is specified using the fx=TRUE option, in which
case it is unpenalized.

So, exact choice of k is not generally critical: it should be chosen to
be large enough that you are reasonably sure of having enough degrees of
freedom to represent the underlying ‘truth’ reasonably well, but small enough
to maintain reasonable computational efficiency. Clearly ‘large’ and ‘small’
are dependent on the particular problem being addressed.

As with all model assumptions, it is useful to be able to check the choice of
k informally. If the effective degrees of freedom for a model term are
estimated to be much less than k-1 then this is unlikely to be very
worthwhile, but as the EDF approach k-1, checking can be important. A
useful general purpose approach goes as follows: (i) fit your model and
extract the deviance residuals; (ii) for each smooth term in your model, fit
an equivalent, single, smooth to the residuals, using a substantially
increased k to see if there is pattern in the residuals that could
potentially be explained by increasing k. Examples are provided below.

The obvious, but more costly, alternative is simply to increase the suspect k
and refit the original model. If there are no statistically important changes as a result of
doing this, then k was large enough. (Change in the smoothness selection criterion,
and/or the effective degrees of freedom, when k is increased, provide the obvious
numerical measures for whether the fit has changed substantially.)

gam.check runs a simple simulation based check on the basis dimensions, which can
help to flag up terms for which k is too low. Grossly too small k
will also be visible from partial residuals available with plot.gam.

One scenario that can cause confusion is this: a model is fitted with
k=10 for a smooth term, and the EDF for the term is estimated as 7.6,
some way below the maximum of 9. The model is then refitted with k=20
and the EDF increases to 8.7 - what is happening - how come the EDF was not
8.7 the first time around? The explanation is that the function space with
k=20 contains a larger subspace of functions with EDF 8.7 than did the
function space with k=10: one of the functions in this larger subspace
fits the data a little better than did any function in the smaller
subspace. These subtleties seldom have much impact on the statistical
conclusions to be drawn from a model fit, however.

## Simulate some data ....
library(mgcv)
set.seed(1)
dat <- gamSim(1,n=400,scale=2)
## fit a GAM with quite low `k'
b<-gam(y~s(x0,k=6)+s(x1,k=6)+s(x2,k=6)+s(x3,k=6),data=dat)
plot(b,pages=1,residuals=TRUE) ## hint of a problem in s(x2)
## the following suggests a problem with s(x2)
gam.check(b)
## Another approach (see below for more obvious method)....
## check for residual pattern, removeable by increasing `k'
## typically `k', below, chould be substantially larger than
## the original, `k' but certainly less than n/2.
## Note use of cheap "cs" shrinkage smoothers, and gamma=1.4
## to reduce chance of overfitting...
rsd <- residuals(b)
gam(rsd~s(x0,k=40,bs="cs"),gamma=1.4,data=dat) ## fine
gam(rsd~s(x1,k=40,bs="cs"),gamma=1.4,data=dat) ## fine
gam(rsd~s(x2,k=40,bs="cs"),gamma=1.4,data=dat) ## `k' too low
gam(rsd~s(x3,k=40,bs="cs"),gamma=1.4,data=dat) ## fine
## refit...
b <- gam(y~s(x0,k=6)+s(x1,k=6)+s(x2,k=20)+s(x3,k=6),data=dat)
gam.check(b) ## better
## similar example with multi-dimensional smooth
b1 <- gam(y~s(x0)+s(x1,x2,k=15)+s(x3),data=dat)
rsd <- residuals(b1)
gam(rsd~s(x0,k=40,bs="cs"),gamma=1.4,data=dat) ## fine
gam(rsd~s(x1,x2,k=100,bs="ts"),gamma=1.4,data=dat) ## `k' too low
gam(rsd~s(x3,k=40,bs="cs"),gamma=1.4,data=dat) ## fine
gam.check(b1) ## shows same problem
## and a `te' example
b2 <- gam(y~s(x0)+te(x1,x2,k=4)+s(x3),data=dat)
rsd <- residuals(b2)
gam(rsd~s(x0,k=40,bs="cs"),gamma=1.4,data=dat) ## fine
gam(rsd~te(x1,x2,k=10,bs="cs"),gamma=1.4,data=dat) ## `k' too low
gam(rsd~s(x3,k=40,bs="cs"),gamma=1.4,data=dat) ## fine
gam.check(b2) ## shows same problem
## same approach works with other families in the original model
dat <- gamSim(1,n=400,scale=.25,dist="poisson")
bp<-gam(y~s(x0,k=5)+s(x1,k=5)+s(x2,k=5)+s(x3,k=5),
family=poisson,data=dat,method="ML")
gam.check(bp)
rsd <- residuals(bp)
gam(rsd~s(x0,k=40,bs="cs"),gamma=1.4,data=dat) ## fine
gam(rsd~s(x1,k=40,bs="cs"),gamma=1.4,data=dat) ## fine
gam(rsd~s(x2,k=40,bs="cs"),gamma=1.4,data=dat) ## `k' too low
gam(rsd~s(x3,k=40,bs="cs"),gamma=1.4,data=dat) ## fine
rm(dat)
## More obvious, but more expensive tactic... Just increase
## suspicious k until fit is stable.
set.seed(0)
dat <- gamSim(1,n=400,scale=2)
## fit a GAM with quite low `k'
b <- gam(y~s(x0,k=6)+s(x1,k=6)+s(x2,k=6)+s(x3,k=6),
data=dat,method="REML")
b
## edf for 3rd smooth is highest as proportion of k -- increase k
b <- gam(y~s(x0,k=6)+s(x1,k=6)+s(x2,k=12)+s(x3,k=6),
data=dat,method="REML")
b
## edf substantially up, -ve REML substantially down
b <- gam(y~s(x0,k=6)+s(x1,k=6)+s(x2,k=24)+s(x3,k=6),
data=dat,method="REML")
b
## slight edf increase and -ve REML change
b <- gam(y~s(x0,k=6)+s(x1,k=6)+s(x2,k=40)+s(x3,k=6),
data=dat,method="REML")
b
## defintely stabilized (but really k around 20 would have been fine)