\HeaderA{te}{Define tensor product smooths in GAM formulae}{te}
\keyword{models}{te}
\keyword{smooth}{te}
\keyword{regression}{te}
\begin{Description}\relax
Function used in definition of tensor product smooth terms within
\code{gam} model formulae. The function does not evaluate a
smooth - it exists purely to help set up a model using tensor product 
based smooths.
\end{Description}
\begin{Usage}
\begin{verbatim}te(..., k=NA,bs="cr",m=0,d=NA,by=NA,fx=FALSE,mp=TRUE,np=TRUE,xt=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(s) of the bases used to represent the smooth term.
If not supplied then set to \code{5\textasciicircum{}d}. If supplied as a single number then this 
basis dimension is used for each basis. If supplied as an array then the elements are
the dimensions of the component (marginal) bases of the tensor
product. See \code{\LinkA{choose.k}{choose.k}} for further information.
\item[\code{bs}] array (or single character string) specifying the type for each 
marginal basis. \code{"cr"} for cubic regression spline; \code{"cs"} for cubic
regression spline with shrinkage; \code{"cc"} for periodic/cyclic 
cubic regression spline; \code{"tp"} for thin plate regression spline;
\code{"ts"} for t.p.r.s. with extra shrinkage. User defined bases can 
also be used here (see \code{\LinkA{smooth.construct}{smooth.construct}} for an example). If only one 
identifier is given then this is used for all bases.
\item[\code{m}] The order of the penalty for each t.p.r.s. term (e.g. 2 for
normal cubic spline penalty with 2nd derivatives). If a single number is given 
then it is used for all terms. \code{0} autoinitializes. \code{m} is ignored for the 
\code{"cr"} and \code{"cc"} bases.
\item[\code{d}] array of marginal basis dimensions. For example if you want a smooth for 3 covariates 
made up of a tensor product of a 2 dimensional t.p.r.s. basis and a 1-dimensional basis, then 
set \code{d=c(2,1)}.
\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.
\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{mp}] \code{TRUE} to use multiple penalties for the smooth. \code{FALSE} to use only 
a single penalty: single penalties are not recommended - they tend to allow only rather 
wiggly models.
\item[\code{np}] \code{TRUE} to use the `normal parameterization' for a tensor
product smooth. This parameterization represents any 1-d marginal smooths
using a parameterization where the parameters are function values at `knots'
spread evenly through the data. The parameterization makes the penalties
easily interpretable, however it can reduce numerical stability.
\item[\code{xt}] Either a single object, providing any extra information to be passed
to each marginal basis constructor, or a list of such objects, one for each
marginal basis. 
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
Smooths of several covariates can be constructed from tensor products of the bases
used to represent smooths of one (or sometimes more) of the covariates. To do this `marginal' bases
are produced with associated model matrices and penalty matrices, and these are then combined in the
manner described in \code{\LinkA{tensor.prod.model.matrix}{tensor.prod.model.matrix}} and \code{\LinkA{tensor.prod.penalties}{tensor.prod.penalties}}, to produce 
a single model matrix for the smooth, but multiple penalties (one for each marginal basis). The basis dimension 
of the whole smooth is the product of the basis dimensions of the marginal smooths.

An option for operating with a single penalty (The Kronecker product of the marginal penalties) is provided, but 
it is rarely of practical use: the penalty is typically so rank deficient that even the smoothest resulting model 
will have rather high estimated degrees of freedom. 

Tensor product smooths are especially useful for representing functions of covariates measured in different units, 
although they are typically not quite as nicely behaved as t.p.r.s. smooths for well scaled covariates.

Note also that GAMs constructed from lower rank tensor product smooths are
nested within GAMs constructed from higher rank tensor product smooths if the
same marginal bases are used in both cases (the marginal smooths themselves
are just special cases of tensor product smooths.)

The `normal parameterization' (\code{np=TRUE}) re-parameterizes the marginal
smooths of a tensor product smooth so that the parameters are function values
at a set of points spread evenly through the range of values of the covariate
of the smooth. This means that the penalty of the tensor product associated
with any particular covariate direction can be interpreted as the penalty of
the appropriate marginal smooth applied in that direction and averaged over
the smooth. Currently this is only done for marginals of a single
variable. This parameterization can reduce numerical stability for when used
with marginal smooths other than \code{"cc"}, \code{"cr"} and \code{"cs"}: if
this causes problems, set \code{np=FALSE}.

The function does not evaluate the variable arguments.
\end{Details}
\begin{Value}
A class \code{tensor.smooth.spec} object defining a tensor product smooth
to be turned into a basis and penalties by the \code{smooth.construct.tensor.smooth.spec} function. 

The returned object contains the following items:

\begin{ldescription}
\item[\code{margin}] A list of \code{smooth.spec} objects of the type returned by \code{\LinkA{s}{s}}, 
defining the basis from which the tensor product smooth is constructed.
\item[\code{term}] An array of text strings giving the names of the covariates that 
the term is a function of.
\item[\code{by}] is the name of any \code{by} variable as text (\code{"NA"} for none).
\item[\code{fx}] logical array with element for each penalty of the term
(tensor product smooths have multiple penalties). \code{TRUE} if the penalty is to 
be ignored, \code{FALSE}, otherwise. 
\item[\code{label}] A suitable text label for this smooth term.
\item[\code{dim}] The dimension of the smoother - i.e. the number of
covariates that it is a function of.
\item[\code{mp}] \code{TRUE} is multiple penalties are to be used (default).
\item[\code{np}] \code{TRUE} to re-parameterize 1-D marginal smooths in terms of function
values (defualt).
\end{ldescription}
\end{Value}
\begin{Author}\relax
Simon N. Wood \email{simon.wood@r-project.org}
\end{Author}
\begin{References}\relax
Wood, S.N. (2006a) Low rank scale invariant tensor product smooths for
generalized additive mixed models. Biometrics 62(4):1025-1036


\url{http://www.maths.bath.ac.uk/~sw283/}
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{s}{s}},\code{\LinkA{gam}{gam}},\code{\LinkA{gamm}{gamm}}
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}

# following shows how tensor pruduct deals nicely with 
# badly scaled covariates (range of x 5% of range of z )
test1<-function(x,z,sx=0.3,sz=0.4)  
{ x<-x*20
  (pi**sx*sz)*(1.2*exp(-(x-0.2)^2/sx^2-(z-0.3)^2/sz^2)+
  0.8*exp(-(x-0.7)^2/sx^2-(z-0.8)^2/sz^2))
}
n<-500
old.par<-par(mfrow=c(2,2))
x<-runif(n)/20;z<-runif(n);
xs<-seq(0,1,length=30)/20;zs<-seq(0,1,length=30)
pr<-data.frame(x=rep(xs,30),z=rep(zs,rep(30,30)))
truth<-matrix(test1(pr$x,pr$z),30,30)
f <- test1(x,z)
y <- f + rnorm(n)*0.2
b1<-gam(y~s(x,z))
persp(xs,zs,truth);title("truth")
vis.gam(b1);title("t.p.r.s")
b2<-gam(y~te(x,z))
vis.gam(b2);title("tensor product")
b3<-gam(y~te(x,z,bs=c("tp","tp")))
vis.gam(b3);title("tensor product")
par(old.par)

test2<-function(u,v,w,sv=0.3,sw=0.4)  
{ ((pi**sv*sw)*(1.2*exp(-(v-0.2)^2/sv^2-(w-0.3)^2/sw^2)+
  0.8*exp(-(v-0.7)^2/sv^2-(w-0.8)^2/sw^2)))*(u-0.5)^2*20
}
n <- 500
v <- runif(n);w<-runif(n);u<-runif(n)
f <- test2(u,v,w)
y <- f + rnorm(n)*0.2
# tensor product of a 2-d thin plate regression spline and 1-d cr spline
b <- gam(y~te(v,w,u,k=c(30,5),d=c(2,1),bs=c("tp","cr")))
op <- par(mfrow=c(2,2))
vis.gam(b,cond=list(u=0),color="heat",zlim=c(-0.2,3.5))
vis.gam(b,cond=list(u=.33),color="heat",zlim=c(-0.2,3.5))
vis.gam(b,cond=list(u=.67),color="heat",zlim=c(-0.2,3.5))
vis.gam(b,cond=list(u=1),color="heat",zlim=c(-0.2,3.5))
par(op)

\end{ExampleCode}
\end{Examples}

