\HeaderA{s}{Defining smooths in GAM formulae}{s}
\keyword{models}{s}
\keyword{smooth}{s}
\keyword{regression}{s}
\begin{Description}\relax
Function used in definition of smooth terms within
\code{gam} model formulae. The function does not evaluate a (spline)
smooth - it exists purely to help set up a model using spline based smooths.
\end{Description}
\begin{Usage}
\begin{verbatim}s(..., k=-1,fx=FALSE,bs="tp",m=0,by=NA)\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{...}] a list of variables that are the covariates that this
smooth is a function of.
\item[\code{k}] the dimension of the basis used to represent the smooth term.
The default depends on the number of variables that the smooth is a
function of. \code{k} should not be less than the dimension of the
null space of the penalty for the term (see
\code{\LinkA{null.space.dimension}{null.space.dimension}}), but will be reset if
it is.
\item[\code{fx}] indicates whether the term is a fixed d.f. regression
spline (\code{TRUE}) or a penalized regression spline (\code{FALSE}).
\item[\code{bs}] this can be \code{"cr"} for a cubic regression spline, \code{"cs"}
for a cubic regression spline with shrinkage, \code{"cc"} for
a cyclic (periodic) spline,
\code{"tp"} for a thin plate regression spline, \code{"ts"} for a thin
plate regression spline with shrinkage or a user defined 
character string for other user defined smooth classes. Of the built in alternatives, only thin plate
regression splines can be used for multidimensional smooths. Note that the
\code{"cr"} and \code{"cc"} bases are faster to set up than the \code{"tp"} basis, particularly
on large data sets (although the \code{knots} argument to \code{gam} can be
used to get round this).
\item[\code{m}] The order of the penalty for this t.p.r.s. term (e.g. 2 for
normal cubic spline penalty with 2nd derivatives). O signals
autoinitialization, which sets the order to the lowest value
satisfying 2m>d+1, where d is the number of covariates: this choise
ensures visual smoothness. In addition, m must satisfy the technical
restriction 2m>d, otherwise it will be autoinitialized.
\item[\code{by}] specifies a covariate by which the whole smooth term is to
be multiplied. This is particularly useful for creating models in
which a smooth interacts with a factor: in this case the \code{by}
variable would usually be the dummy variable coding one level of the
factor. See the examples below. This is the means by which `variable
coefficient models' can be specified.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
The function does not evaluate the variable arguments. To use this function to specify use of
your own smooths, note the relationships between the inputs and the output object and see the example
in \code{\LinkA{smooth.construct}{smooth.construct}}.
\end{Details}
\begin{Value}
A class \code{xx.smooth.spec} object, where \code{xx} is a basis identifying code given by
the \code{bs} argument of \code{s}. These \code{smooth.spec} objects define smooths and are turned into
bases and penalties by \code{smooth.construct} method functions. 

The returned object contains the following items:

\begin{ldescription}
\item[\code{term}] An array of text strings giving the names of the covariates that 
the term is a function of.
\item[\code{bs.dim}] The dimension of the basis used to represent the smooth.
\item[\code{fixed}] TRUE if the term is to be treated as a pure regression
spline (with fixed degrees of freedom); FALSE if it is to be treated
as a penalized regression spline
\item[\code{dim}] The dimension of the smoother - i.e. the number of
covariates that it is a function of.
\item[\code{p.order}] The order of the t.p.r.s. penalty, or 0 for
auto-selection of the penalty order.
\item[\code{by}] is the name of any \code{by} variable as text (\code{"NA"} for none).
\item[\code{full.call}] Text for pasting into a string to be converted to a
gam formula, which has the values of function options given explicitly -
this is useful for constructing a fully expanded gam formula which can
be used without needing access to any variables that may have been used
to define \code{k}, \code{fx}, \code{bs} or \code{m} in the original call. i.e. this is text which
when parsed and evaluated generates a call to \code{s()} with all the
options spelled out explicitly.
\item[\code{label}] A suitable text label for this smooth term.
\end{ldescription}
\end{Value}
\begin{Author}\relax
Simon N. Wood \email{simon.wood@r-project.org}
\end{Author}
\begin{References}\relax
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

\url{http://www.stats.gla.ac.uk/~simon/}
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{te}{te}}, \code{\LinkA{gam}{gam}}, \code{\LinkA{gamm}{gamm}}
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
# example utilising `by' variables
library(mgcv)
set.seed(0)
n<-200;sig2<-4
x1 <- runif(n, 0, 1);x2 <- runif(n, 0, 1);x3 <- runif(n, 0, 1)
fac<-c(rep(1,n/2),rep(2,n/2)) # create factor
fac.1<-rep(0,n)+(fac==1);fac.2<-1-fac.1 # and dummy variables
fac<-as.factor(fac)
f1 <-  exp(2 * x1) - 3.75887
f2 <-  0.2 * x1^11 * (10 * (1 - x1))^6 + 10 * (10 * x1)^3 * (1 - x1)^10
f<-f1*fac.1+f2*fac.2+x2
e <- rnorm(n, 0, sqrt(abs(sig2)))
y <- f + e
# NOTE: smooths will be centered, so need to include fac in model....
b<-gam(y~fac+s(x1,by=fac.1)+s(x1,by=fac.2)+x2) 
plot(b,pages=1)
\end{ExampleCode}
\end{Examples}

