\HeaderA{corExp}{Exponential Correlation Structure}{corExp}
\keyword{models}{corExp}
\begin{Description}\relax
This function is a constructor for the \code{corExp} class,
representing an exponential spatial correlation structure. Letting
\eqn{d}{} denote the range and \eqn{n}{} denote the nugget
effect, the correlation between two observations a distance
\eqn{r}{} apart is \eqn{\exp(-r/d)}{} when no nugget effect
is present and \eqn{(1-n) \exp(-r/d)}{} when a nugget
effect is assumed. Objects created using this constructor must later be
initialized using the appropriate \code{Initialize} method.
\end{Description}
\begin{Usage}
\begin{verbatim}
corExp(value, form, nugget, metric, fixed)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{value}] an optional vector with the parameter values in
constrained form. If \code{nugget} is \code{FALSE}, \code{value} can
have only one element, corresponding to the "range" of the
exponential correlation structure, which must be greater than
zero. If \code{nugget} is \code{TRUE}, meaning that a nugget effect
is present, \code{value} can contain one or two elements, the first
being the "range" and the second the "nugget effect" (one minus the
correlation between two observations taken arbitrarily close
together); the first must be greater than zero and the second must be
between zero and one. Defaults to \code{numeric(0)}, which results in
a range of 90\% of the minimum distance and a nugget effect of 0.1
being assigned to the parameters when \code{object} is initialized.
\item[\code{form}] a one sided formula of the form \code{\textasciitilde{} S1+...+Sp}, or
\code{\textasciitilde{} S1+...+Sp | g}, specifying spatial covariates \code{S1}
through \code{Sp} and,  optionally, a grouping factor \code{g}. 
When a grouping factor is present in \code{form}, the correlation
structure is assumed to apply only to observations within the same
grouping level; observations with different grouping levels are
assumed to be uncorrelated. Defaults to \code{\textasciitilde{} 1}, which corresponds
to using the order of the observations in the data as a covariate,
and no groups.
\item[\code{nugget}] an optional logical value indicating whether a nugget
effect is present. Defaults to \code{FALSE}.
\item[\code{metric}] an optional character string specifying the distance
metric to be used. The currently available options are
\code{"euclidean"} for the root sum-of-squares of distances;
\code{"maximum"} for the maximum difference; and \code{"manhattan"}
for the sum of the absolute differences. Partial matching of
arguments is used, so only the first three characters need to be
provided. Defaults to \code{"euclidean"}.
\item[\code{fixed}] an optional logical value indicating whether the
coefficients should be allowed to vary in the optimization, or kept
fixed at their initial value. Defaults to \code{FALSE}, in which case
the coefficients are allowed to vary.
\end{ldescription}
\end{Arguments}
\begin{Value}
an object of class \code{corExp}, also inheriting from class
\code{corSpatial}, representing an exponential spatial correlation
structure.
\end{Value}
\begin{Author}\relax
Jose Pinheiro \email{Jose.Pinheiro@pharma.novartis.com} and Douglas Bates \email{bates@stat.wisc.edu}
\end{Author}
\begin{References}\relax
Cressie, N.A.C. (1993), "Statistics for Spatial Data", J. Wiley \& Sons.
Venables, W.N. and Ripley, B.D. (1997) "Modern Applied Statistics with
S-plus", 2nd Edition, Springer-Verlag.

Littel, Milliken, Stroup, and Wolfinger (1996) "SAS Systems for Mixed
Models", SAS Institute.

Pinheiro, J.C., and Bates, D.M. (2000) "Mixed-Effects Models
in S and S-PLUS", Springer, esp. p. 238.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{corClasses}{corClasses}},
\code{\LinkA{Initialize.corStruct}{Initialize.corStruct}},
\code{\LinkA{summary.corStruct}{summary.corStruct}},
\code{\LinkA{dist}{dist}}
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
sp1 <- corExp(form = ~ x + y + z)

# Pinheiro and Bates, p. 238
spatDat <- data.frame(x = (0:4)/4, y = (0:4)/4)

cs1Exp <- corExp(1, form = ~ x + y)
cs1Exp <- Initialize(cs1Exp, spatDat)
corMatrix(cs1Exp)

cs2Exp <- corExp(1, form = ~ x + y, metric = "man")
cs2Exp <- Initialize(cs2Exp, spatDat)
corMatrix(cs2Exp)

cs3Exp <- corExp(c(1, 0.2), form = ~ x + y,
                 nugget = TRUE)
cs3Exp <- Initialize(cs3Exp, spatDat)
corMatrix(cs3Exp)

# example lme(..., corExp ...)
# Pinheiro and Bates, pp. 222-247
# p. 222
options(contrasts = c("contr.treatment", "contr.poly"))
fm1BW.lme <- lme(weight ~ Time * Diet, BodyWeight,
                   random = ~ Time)
# p. 223
fm2BW.lme <- update(fm1BW.lme, weights = varPower())
# p. 246
fm3BW.lme <- update(fm2BW.lme,
           correlation = corExp(form = ~ Time))
# p. 247
fm4BW.lme <-
      update(fm3BW.lme, correlation = corExp(form =  ~ Time,
                        nugget = TRUE))
anova(fm3BW.lme, fm4BW.lme)

\end{ExampleCode}
\end{Examples}

