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\name{ssanova}
\alias{ssanova}
\title{Fitting Smoothing Spline ANOVA Models}
\description{
Fit smoothing spline ANOVA models in Gaussian regression. The
symbolic model specification via \code{formula} follows the same
rules as in \code{\link{lm}}.
}
\usage{
ssanova(formula, type=NULL, data=list(), weights, subset, offset,
na.action=na.omit, partial=NULL, method="v", alpha=1.4,
varht=1, id.basis=NULL, nbasis=NULL, seed=NULL, random=NULL,
skip.iter=FALSE)
}
\arguments{
\item{formula}{Symbolic description of the model to be fit.}
\item{type}{List specifying the type of spline for each variable.
See \code{\link{mkterm}} for details.}
\item{data}{Optional data frame containing the variables in the
model.}
\item{weights}{Optional vector of weights to be used in the
fitting process.}
\item{subset}{Optional vector specifying a subset of observations
to be used in the fitting process.}
\item{offset}{Optional offset term with known parameter 1.}
\item{na.action}{Function which indicates what should happen when
the data contain NAs.}
\item{partial}{Optional symbolic description of parametric terms in
partial spline models.}
\item{method}{Method for smoothing parameter selection. Supported
are \code{method="v"} for GCV, \code{method="m"} for GML (REML),
and \code{method="u"} for Mallows' CL.}
\item{alpha}{Parameter modifying GCV or Mallows' CL; larger absolute
values yield smoother fits; negative value invokes a stable and
more accurate GCV/CL evaluation algorithm but may take two to
five times as long. Ignored when \code{method="m"} are
specified.}
\item{varht}{External variance estimate needed for
\code{method="u"}. Ignored when \code{method="v"} or
\code{method="m"} are specified.}
\item{id.basis}{Index designating selected "knots".}
\item{nbasis}{Number of "knots" to be selected. Ignored when
\code{id.basis} is supplied.}
\item{seed}{Seed to be used for the random generation of "knots".
Ignored when \code{id.basis} is supplied.}
\item{random}{Input for parametric random effects in nonparametric
mixed-effect models. See \code{\link{mkran}} for details.}
\item{skip.iter}{Flag indicating whether to use initial values of
theta and skip theta iteration. See notes on skipping theta
iteration.}
}
\details{
The model specification via \code{formula} is intuitive. For
example, \code{y~x1*x2} yields a model of the form
\deqn{
y = C + f_{1}(x1) + f_{2}(x2) + f_{12}(x1,x2) + e
}
with the terms denoted by \code{"1"}, \code{"x1"}, \code{"x2"}, and
\code{"x1:x2"}.
The model terms are sums of unpenalized and penalized
terms. Attached to every penalized term there is a smoothing
parameter, and the model complexity is largely determined by the
number of smoothing parameters.
A subset of the observations are selected as "knots." Unless
specified via \code{id.basis} or \code{nbasis}, the number of
"knots" \eqn{q} is determined by \eqn{max(30,10n^{2/9})}, which is
appropriate for the default cubic splines for numerical vectors.
Using \eqn{q} "knots," \code{ssanova} calculates an approximate
solution to the penalized least squares problem using algorithms of
the order \eqn{O(nq^{2})}, which for \eqn{q<<n} scale better than
the \eqn{O(n^{3})} algorithms of \code{\link{ssanova0}}. For the
exact solution, one may set \eqn{q=n} in \code{ssanova}, but
\code{\link{ssanova0}} would be much faster.
}
\section{Skipping Theta Iteration}{
For the selection of multiple smoothing parameters,
\code{\link{nlm}} is used to minimize the selection criterion such
as the GCV score. When the number of smoothing parameters is large,
the process can be time-consuming due to the great amount of
function evaluations involved.
The starting values for the \code{nlm} iteration are obtained using
Algorith 3.2 in Gu and Wahba (1991). These starting values usually
yield good estimates themselves, leaving the subsequent quasi-Newton
iteration to pick up the "last 10\%" performance with extra effort
many times of the initial one. Thus, it is often a good idea to
skip the iteration by specifying \code{skip.iter=TRUE}, especially
in high-dimensions and/or with multi-way interactions.
\code{skip.iter=TRUE} could be made the default in future releases.
}
\note{
To use GCV and Mallows' CL unmodified, set \code{alpha=1}.
For simpler models and moderate sample sizes, the exact solution of
\code{\link{ssanova0}} can be faster.
The results may vary from run to run. For consistency, specify
\code{id.basis} or set \code{seed}.
In \emph{gss} versions earlier than 1.0, \code{ssanova} was under
the name \code{ssanova1}.
}
\value{
\code{ssanova} returns a list object of class \code{"ssanova"}.
The method \code{\link{summary.ssanova}} can be used to obtain
summaries of the fits. The method \code{\link{predict.ssanova}} can
be used to evaluate the fits at arbitrary points along with standard
errors. The method \code{\link{project.ssanova}} can be used to
calculate the Kullback-Leibler projection for model selection. The
methods \code{\link{residuals.ssanova}} and
\code{\link{fitted.ssanova}} extract the respective traits
from the fits.
}
\author{Chong Gu, \email{chong@stat.purdue.edu}}
\references{
Wahba, G. (1990), \emph{Spline Models for Observational Data}.
Philadelphia: SIAM.
Gu, C. and Wahba, G. (1991), Minimizing GCV/GML scores with multiple
smoothing parameters via the Newton method. \emph{SIAM Journal on
Scientific and Statistical Computing}, \bold{12}, 383--398.
Kim, Y.-J. and Gu, C. (2004), Smoothing spline Gaussian regression:
more scalable computation via efficient approximation.
\emph{Journal of the Royal Statistical Society, Ser. B}, \bold{66},
337--356.
Gu, C. (2013), \emph{Smoothing Spline ANOVA Models (2nd Ed)}. New
York: Springer-Verlag.
Chong Gu (2014), Smoothing Spline ANOVA Models: R Package gss.
\emph{Journal of Statistical Software}, 58(5), 1-25. URL
http://www.jstatsoft.org/v58/i05/.
}
\examples{
## Fit a cubic spline
x <- runif(100); y <- 5 + 3*sin(2*pi*x) + rnorm(x)
cubic.fit <- ssanova(y~x)
## Obtain estimates and standard errors on a grid
new <- data.frame(x=seq(min(x),max(x),len=50))
est <- predict(cubic.fit,new,se=TRUE)
## Plot the fit and the Bayesian confidence intervals
plot(x,y,col=1); lines(new$x,est$fit,col=2)
lines(new$x,est$fit+1.96*est$se,col=3)
lines(new$x,est$fit-1.96*est$se,col=3)
## Clean up
\dontrun{rm(x,y,cubic.fit,new,est)
dev.off()}
## Fit a tensor product cubic spline
data(nox)
nox.fit <- ssanova(log10(nox)~comp*equi,data=nox)
## Fit a spline with cubic and nominal marginals
nox$comp<-as.factor(nox$comp)
nox.fit.n <- ssanova(log10(nox)~comp*equi,data=nox)
## Fit a spline with cubic and ordinal marginals
nox$comp<-as.ordered(nox$comp)
nox.fit.o <- ssanova(log10(nox)~comp*equi,data=nox)
## Clean up
\dontrun{rm(nox,nox.fit,nox.fit.n,nox.fit.o)}
}
\keyword{smooth}
\keyword{models}
\keyword{regression}
|
