\name{smooth.construct.ps.smooth.spec}
\alias{smooth.construct.ps.smooth.spec}
\alias{smooth.construct.cp.smooth.spec}
\alias{p.spline}
\alias{cyclic.p.spline}
%- Also NEED an `\alias' for EACH other topic documented here.
\title{P-splines in GAMs}
\description{\code{\link{gam}} can use univariate P-splines as proposed by Eilers and Marx (1996),
specified via terms like \code{s(x,bs="ps")}. These terms use B-spline bases
penalized by discrete penalties applied directly to
the basis coefficients. Cyclic P-splines are specified by model terms like \code{s(x,bs="cp",...)}.
These bases can be used in tensor product smooths (see \code{\link{te}}).
The advantage of P-splines is the flexible way that penalty and basis order can be mixed. This often provides a useful way of `taming' an otherwise poorly behave smooth. However, in regular use, splines with derivative based penalties (e.g. \code{"tp"} or \code{"cr"} bases) tend to result in slightly better MSE performance, presumably because the good approximation theoretic properties of splines are rather closely connected to the use of derivative penalties.
}
\usage{
\method{smooth.construct}{ps.smooth.spec}(object, data, knots)
\method{smooth.construct}{cp.smooth.spec}(object, data, knots)
}
\arguments{
\item{object}{a smooth specification object, usually generated by a term \code{s(x,bs="ps",...)} or
\code{s(x,bs="cp",...)}}
\item{data}{a list containing just the data (including any \code{by} variable) required by this term,
with names corresponding to \code{object$term} (and \code{object$by}). The \code{by} variable
is the last element.}
\item{knots}{a list containing any knots supplied for basis setup --- in same order and with same names as \code{data}.
Can be \code{NULL}. See details for further information.}
}
\value{ An object of class \code{"ps.smooth"} or \code{"cp.smooth"}. See \code{\link{smooth.construct}},
for the elements that this object will contain.
}
\details{A smooth term of the form \code{s(x,bs="ps",m=c(2,3))} specifies a 2nd order P-spline basis (cubic spline),
with a third order difference penalty (0th order is a ridge penalty) on the coefficients. If \code{m} is a single number then it is taken as the basis order and penalty order. The default is the `cubic spline like' \code{m=c(2,2)}.
The default basis dimension, \code{k}, is the larger of 10 and \code{m[1]+1} for a \code{"ps"} terms and the larger of 10 and \code{m[1]} for a \code{"cp"} term. \code{m[1]+1} and \code{m[1]} are the lower limits on basis dimension for the two types.
If knots are supplied, then the number of knots should be one more than the basis dimension
(i.e. \code{k+1}) for a \code{"cp"}smooth. For the \code{"ps"} basis the number of supplied knots should be \code{k + m[1] + 2}, and the range of the middle
\code{k-m[1]} knots should include all the covariate values. See example.
Alternatively, for both types of smooth, 2 knots can be supplied, denoting the
lower and upper limits between which the spline can be evaluated (Don't make this range too wide, however, or you
can end up with no information about some basis coefficients, because the corresponding basis functions have a
span that includes no data!). Note that P-splines don't make much sense with uneven knot spacing.
}
\references{
Eilers, P.H.C. and B.D. Marx (1996) Flexible Smoothing with B-splines and Penalties.
Statistical Science, 11(2):89-121
}
\author{ Simon N. Wood \email{simon.wood@r-project.org}}
\seealso{\code{\link{cSplineDes}}}
\examples{
## see ?gam
## cyclic example ...
set.seed(6)
x <- sort(runif(200)*10)
z <- runif(200)
f <- sin(x*2*pi/10)+.5
y <- rpois(exp(f),exp(f))
## finished simulating data, now fit model...
b <- gam(y ~ s(x,bs="cp") + s(z,bs="ps"),family=poisson)
## example with supplied knot ranges for x and z (can do just one)
b <- gam(y ~ s(x,bs="cp") + s(z,bs="ps"),family=poisson,
knots=list(x=c(0,10),z=c(0,1)))
## example with supplied knots...
bk <- gam(y ~ s(x,bs="cp",k=12) + s(z,bs="ps",k=13),family=poisson,
knots=list(x=seq(0,10,length=13),z=(-3):13/10))
## plot results...
par(mfrow=c(2,2))
plot(b,select=1,shade=TRUE);lines(x,f-mean(f),col=2)
plot(b,select=2,shade=TRUE);lines(z,0*z,col=2)
plot(bk,select=1,shade=TRUE);lines(x,f-mean(f),col=2)
plot(bk,select=2,shade=TRUE);lines(z,0*z,col=2)
}
\keyword{models} \keyword{regression}%-- one or more ..