gam model formulae. The function does not evaluate a
smooth - it exists purely to help set up a model using tensor product
based smooths.te(..., k=NA,bs="cr",m=0,d=NA,by=NA,fx=FALSE,mp=TRUE,np=TRUE,xt=NA)5^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 "cr" for cubic regression spline; "cs" for cubic
regression spline with shrinkage; "cc" for periodic/cyclic
cubic regression spline; <0 autoinitializes. m is ignored for the
"crd=c(2,1).by
variable would usually be the dummy variable TRUE) or a penalized regression spline (FALSE).TRUE to use multiple penalties for the smooth. FALSE to use only
a single penalty: single penalties are not recommended - they tend to allow only rather
wiggly models.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 tensor.smooth.spec object defining a tensor product smooth
to be turned into a basis and penalties by the smooth.construct.tensor.smooth.spec function. The returned object contains the following items:
smooth.spec objects of the type returned by s,
defining the basis from which the tensor product smooth is constructed.by variable as text ("NA" for none).TRUE if the penalty is to
be ignored, FALSE, otherwise.TRUE is multiple penalties are to be used (default).TRUE to re-parameterize 1-D marginal smooths in terms of function
values (defualt).tensor.prod.model.matrix and 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' (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 "cc", "cr" and "cs": if
this causes problems, set np=FALSE.
The function does not evaluate the variable arguments.
s,gam,gamm# 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))
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