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\name{ssanova9}
\alias{ssanova9}
\alias{para.arma}
\title{Fitting Smoothing Spline ANOVA Models with Correlated Data}
\description{
Fit smoothing spline ANOVA models with correlated Gaussian data.
The symbolic model specification via \code{formula} follows the same
rules as in \code{\link{lm}}.
}
\usage{
ssanova9(formula, type=NULL, data=list(), subset, offset,
na.action=na.omit, partial=NULL, method="v", alpha=1.4,
varht=1, id.basis=NULL, nbasis=NULL, seed=NULL, cov,
skip.iter=FALSE)
para.arma(fit)
}
\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{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 V, \code{method="m"} for M, and
\code{method="u"} for U; see the reference for definitions of U,
V, and M.}
\item{alpha}{Parameter modifying V or U; larger absolute values
yield smoother fits. 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{cov}{Input for covariance functions. See \code{\link{mkcov}}
for details.}
\item{skip.iter}{Flag indicating whether to use initial values of
theta and skip theta iteration. See notes on skipping theta
iteration.}
\item{fit}{\code{ssanova9} fit with ARMA error.}
}
\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{
The results may vary from run to run. For consistency, specify
\code{id.basis} or set \code{seed}.
}
\value{
\code{ssanova9} returns a list object of class
\code{c("ssanova9","ssanova")}.
The method \code{\link{summary.ssanova9}} 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.ssanova9}} 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.
\code{para.arma} returns the fitted ARMA coefficients for
\code{cov=list("arma",c(p,q))} in the call to \code{ssanova9}.
}
\author{Chong Gu, \email{chong@stat.purdue.edu}}
\references{
Han, C. and Gu, C. (2008), Optimal smoothing with correlated data,
\emph{Sankhya}, \bold{70-A}, 38--72.
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{
x <- runif(100); y <- 5 + 3*sin(2*pi*x) + rnorm(x)
## independent fit
fit <- ssanova9(y~x,cov=list("known",diag(1,100)))
## AR(1) fit
fit <- ssanova9(y~x,cov=list("arma",c(1,0)))
para.arma(fit)
## MA(1) fit
e <- rnorm(101); e <- e[-1]-.5*e[-101]
x <- runif(100); y <- 5 + 3*sin(2*pi*x) + e
fit <- ssanova9(y~x,cov=list("arma",c(0,1)))
para.arma(fit)
## Clean up
\dontrun{rm(x,y,e,fit)}
}
\keyword{smooth}
\keyword{models}
\keyword{regression}
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