\HeaderA{gam.side}{Identifiability side conditions for a GAM}{gam.side}
\keyword{models}{gam.side}
\keyword{regression}{gam.side}
\begin{Description}\relax
GAM formulae with repeated variables only correspond to
identifiable models given some side conditions. This routine works 
out appropriate side conditions, based on zeroing redundant parameters.
It is called from \code{\LinkA{gam.setup}{gam.setup}} and is not intended to be called by users. 

The method identifies nested and repeated variables by their names, but
evaluates what constraints to impose, numerically. Constraints are always
applied to smooths of more variables in preference to smooths of fewer
variables. The numerical approach allows appropriate constraints to be
applied to models constructed using any smooths, including user defined smooths.
\end{Description}
\begin{Usage}
\begin{verbatim}
gam.side(sm,tol=.Machine$double.eps^.5)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{sm}] A list of smooth objects as returned by
\code{\LinkA{smooth.construct}{smooth.construct}}.
\item[\code{tol}] The tolernce to use when assessing linear dependence of smooths.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
Models such as  \code{y\textasciitilde{}s(x)+s(z)+s(x,z)} can be estimated by
\code{\LinkA{gam}{gam}}, but require identifiability constraints to be applied, to
make them identifiable. This routine does this, effectively setting redundant parameters
to zero. When the redundancy is between smooths of lower and higher numbers
of variables, the constraint is always applied to the smooth of the higher
number of variables. 

Dependent smooths are identified symbolically, but which constraints are 
needed to ensure identifiability of these smooths is determined numerically, using
\code{\LinkA{fixDependence}{fixDependence}}. This makes the routine rather general, and not
dependent on any particular basis.
\end{Details}
\begin{Value}
A list of smooths, with model matrices and penalty matrices adjusted
to automatically impose the required constraints. Any smooth that has been
modified will have an attribute \code{"del.index"}, listing the columns of its
model matrix that were deleted. This index is used in the creation of
prediction matrices for the term.
\end{Value}
\begin{Author}\relax
Simon N. Wood \email{simon.wood@r-project.org}
\end{Author}
\begin{Examples}
\begin{ExampleCode}
set.seed(0)
n<-400
sig2<-4
x0 <- runif(n, 0, 1)
x1 <- runif(n, 0, 1)
x2 <- runif(n, 0, 1)
pi <- asin(1) * 2
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, sqrt(abs(sig2)))
y <- f + e
b<-gam(y~s(x0)+s(x1)+s(x0,x1)+s(x2))
plot(b,pages=1)
test1<-function(x,z,sx=0.3,sz=0.4)  
{ (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);z<-runif(n);
y<-test1(x,z)+rnorm(n)*0.1
## a fully nested tensor product example
b<-gam(y~s(x,bs="cr",k=6)+s(z,bs="cr",k=6)+te(x,z,k=6))
plot(b)
par(old.par)
rm(list=c("f","x0","x1","x2","x","z","y","b","test1","n","sig2","pi","e"))
\end{ExampleCode}
\end{Examples}

