\name{hopskel}
\alias{hopskel}
\alias{hopskel.test}
\title{Hopkins-Skellam Test}
\description{
Perform the Hopkins-Skellam test of Complete Spatial Randomness,
or simply calculate the test statistic.
}
\usage{
hopskel(X)
hopskel.test(X, \dots,
alternative=c("two.sided", "less", "greater",
"clustered", "regular"),
method=c("asymptotic", "MonteCarlo"),
nsim=999)
}
\arguments{
\item{X}{
Point pattern (object of class \code{"ppp"}).
}
\item{alternative}{
String indicating the type of alternative for the
hypothesis test. Partially matched.
}
\item{method}{
Method of performing the test. Partially matched.
}
\item{nsim}{
Number of Monte Carlo simulations to perform, if a Monte Carlo
p-value is required.
}
\item{\dots}{Ignored.}
}
\details{
Hopkins and Skellam (1954) proposed a test of Complete Spatial
Randomness based on comparing nearest-neighbour distances with
point-event distances.
If the point pattern \code{X} contains \code{n}
points, we first compute the nearest-neighbour distances
\eqn{P_1, \ldots, P_n}{P[1], ..., P[n]}
so that \eqn{P_i}{P[i]} is the distance from the \eqn{i}th data
point to the nearest other data point. Then we
generate another completely random pattern \code{U} with
the same number \code{n} of points, and compute for each point of \code{U}
the distance to the nearest point of \code{X}, giving
distances \eqn{I_1, \ldots, I_n}{I[1], ..., I[n]}.
The test statistic is
\deqn{
A = \frac{\sum_i P_i^2}{\sum_i I_i^2}
}{
A = (sum[i] P[i]^2) / (sum[i] I[i]^2)
}
The null distribution of \eqn{A} is roughly
an \eqn{F} distribution with shape parameters \eqn{(2n,2n)}.
(This is equivalent to using the test statistic \eqn{H=A/(1+A)}
and referring \eqn{H} to the Beta distribution with parameters
\eqn{(n,n)}).
The function \code{hopskel} calculates the Hopkins-Skellam test statistic
\eqn{A}, and returns its numeric value. This can be used as a simple
summary of spatial pattern: the value \eqn{H=1} is consistent
with Complete Spatial Randomness, while values \eqn{H < 1} are
consistent with spatial clustering, and values \eqn{H > 1} are consistent
with spatial regularity.
The function \code{hopskel.test} performs the test.
If \code{method="asymptotic"} (the default), the test statistic \eqn{H}
is referred to the \eqn{F} distribution. If \code{method="MonteCarlo"},
a Monte Carlo test is performed using \code{nsim} simulated point
patterns.
}
\value{
The value of \code{hopskel} is a single number.
The value of \code{hopskel.test} is an object of class \code{"htest"}
representing the outcome of the test. It can be printed.
}
\references{
Hopkins, B. and Skellam, J.G. (1954)
A new method of determining the type of distribution
of plant individuals. \emph{Annals of Botany} \bold{18},
213--227.
}
\seealso{
\code{\link{clarkevans}},
\code{\link{clarkevans.test}},
\code{\link{nndist}},
\code{\link{nncross}}
}
\examples{
hopskel(redwood)
hopskel.test(redwood, alternative="clustered")
}
\author{
\spatstatAuthors.
}
\keyword{spatial}
\keyword{nonparametric}
\keyword{htest}