pcf.ppp.Rd
\name{pcf.ppp}
\alias{pcf.ppp}
\title{Pair Correlation Function of Point Pattern}
\description{
Estimates the pair correlation function of
a point pattern using kernel methods.
}
\usage{
\method{pcf}{ppp}(X, \dots, r = NULL, kernel="epanechnikov", bw=NULL,
stoyan=0.15,
correction=c("translate", "Ripley"),
divisor = c("r", "d"),
var.approx = FALSE,
domain=NULL,
ratio=FALSE, close=NULL)
}
\arguments{
\item{X}{
A point pattern (object of class \code{"ppp"}).
}
\item{r}{
Vector of values for the argument \eqn{r} at which \eqn{g(r)}
should be evaluated. There is a sensible default.
}
\item{kernel}{
Choice of smoothing kernel,
passed to \code{density}.
}
\item{bw}{
Bandwidth for smoothing kernel, passed to \code{density}.
}
\item{\dots}{
Other arguments passed to the kernel density estimation
function \code{density}.
}
\item{stoyan}{
Bandwidth coefficient; see Details.
}
\item{correction}{
Choice of edge correction.
}
\item{divisor}{
Choice of divisor in the estimation formula:
either \code{"r"} (the default) or \code{"d"}. See Details.
}
\item{var.approx}{
Logical value indicating whether to compute an analytic
approximation to the variance of the estimated pair correlation.
}
\item{domain}{
Optional. Calculations will be restricted to this subset
of the window. See Details.
}
\item{ratio}{
Logical.
If \code{TRUE}, the numerator and denominator of
each edge-corrected estimate will also be saved,
for use in analysing replicated point patterns.
}
\item{close}{
Advanced use only. Precomputed data. See section on Advanced Use.
}
}
\value{
A function value table
(object of class \code{"fv"}).
Essentially a data frame containing the variables
\item{r}{the vector of values of the argument \eqn{r}
at which the pair correlation function \eqn{g(r)} has been estimated
}
\item{theo}{vector of values equal to 1,
the theoretical value of \eqn{g(r)} for the Poisson process
}
\item{trans}{vector of values of \eqn{g(r)}
estimated by translation correction
}
\item{iso}{vector of values of \eqn{g(r)}
estimated by Ripley isotropic correction
}
\item{v}{vector of approximate values of the variance of
the estimate of \eqn{g(r)}
}
as required.
If \code{ratio=TRUE} then the return value also has two
attributes called \code{"numerator"} and \code{"denominator"}
which are \code{"fv"} objects
containing the numerators and denominators of each
estimate of \eqn{g(r)}.
The return value also has an attribute \code{"bw"} giving the
smoothing bandwidth that was used.
}
\details{
The pair correlation function \eqn{g(r)}
is a summary of the dependence between points in a spatial point
process. The best intuitive interpretation is the following: the probability
\eqn{p(r)} of finding two points at locations \eqn{x} and \eqn{y}
separated by a distance \eqn{r} is equal to
\deqn{
p(r) = \lambda^2 g(r) \,{\rm d}x \, {\rm d}y
}{
p(r) = lambda^2 * g(r) dx dy
}
where \eqn{\lambda}{lambda} is the intensity of the point process.
For a completely random (uniform Poisson) process,
\eqn{p(r) = \lambda^2 \,{\rm d}x \, {\rm d}y}{p(r) = lambda^2 dx dy}
so \eqn{g(r) = 1}.
Formally, the pair correlation function of a stationary point process
is defined by
\deqn{
g(r) = \frac{K'(r)}{2\pi r}
}{
g(r) = K'(r)/ ( 2 * pi * r)
}
where \eqn{K'(r)} is the derivative of \eqn{K(r)}, the
reduced second moment function (aka ``Ripley's \eqn{K} function'')
of the point process. See \code{\link{Kest}} for information
about \eqn{K(r)}.
For a stationary Poisson process, the
pair correlation function is identically equal to 1. Values
\eqn{g(r) < 1} suggest inhibition between points;
values greater than 1 suggest clustering.
This routine computes an estimate of \eqn{g(r)}
by kernel smoothing.
\itemize{
\item
If \code{divisor="r"} (the default), then the standard
kernel estimator (Stoyan and Stoyan, 1994, pages 284--285)
is used. By default, the recommendations of Stoyan and Stoyan (1994)
are followed exactly.
\item
If \code{divisor="d"} then a modified estimator is used:
the contribution from
an interpoint distance \eqn{d_{ij}}{d[ij]} to the
estimate of \eqn{g(r)} is divided by \eqn{d_{ij}}{d[ij]}
instead of dividing by \eqn{r}. This usually improves the
bias of the estimator when \eqn{r} is close to zero.
}
There is also a choice of spatial edge corrections
(which are needed to avoid bias due to edge effects
associated with the boundary of the spatial window):
\itemize{
\item
If \code{correction="translate"} or \code{correction="translation"}
then the translation correction
is used. For \code{divisor="r"} the translation-corrected estimate
is given in equation (15.15), page 284 of Stoyan and Stoyan (1994).
\item
If \code{correction="Ripley"} then Ripley's isotropic edge correction
is used. For \code{divisor="r"} the isotropic-corrected estimate
is given in equation (15.18), page 285 of Stoyan and Stoyan (1994).
\item
If \code{correction=c("translate", "Ripley")} then both estimates
will be computed.
}
Alternatively \code{correction="all"} selects all options.
The choice of smoothing kernel is controlled by the
argument \code{kernel} which is passed to \code{\link{density}}.
The default is the Epanechnikov kernel, recommended by
Stoyan and Stoyan (1994, page 285).
The bandwidth of the smoothing kernel can be controlled by the
argument \code{bw}. Its precise interpretation
is explained in the documentation for \code{\link{density.default}}.
For the Epanechnikov kernel, the argument \code{bw} is
equivalent to \eqn{h/\sqrt{5}}{h/sqrt(5)}.
Stoyan and Stoyan (1994, page 285) recommend using the Epanechnikov
kernel with support \eqn{[-h,h]} chosen by the rule of thumn
\eqn{h = c/\sqrt{\lambda}}{h = c/sqrt(lambda)},
where \eqn{\lambda}{lambda} is the (estimated) intensity of the
point process, and \eqn{c} is a constant in the range from 0.1 to 0.2.
See equation (15.16).
If \code{bw} is missing, then this rule of thumb will be applied.
The argument \code{stoyan} determines the value of \eqn{c}.
The smoothing bandwidth that was used in the calculation is returned
as an attribute of the final result.
The argument \code{r} is the vector of values for the
distance \eqn{r} at which \eqn{g(r)} should be evaluated.
There is a sensible default.
If it is specified, \code{r} must be a vector of increasing numbers
starting from \code{r[1] = 0},
and \code{max(r)} must not exceed half the diameter of
the window.
If the argument \code{domain} is given, estimation will be restricted
to this region. That is, the estimate of
\eqn{g(r)} will be based on pairs of points in which the first point lies
inside \code{domain} and the second point is unrestricted.
The argument \code{domain}
should be a window (object of class \code{"owin"}) or something acceptable to
\code{\link{as.owin}}. It must be a subset of the
window of the point pattern \code{X}.
To compute a confidence band for the true value of the
pair correlation function, use \code{\link{lohboot}}.
If \code{var.approx = TRUE}, the variance of the
estimate of the pair correlation will also be calculated using
an analytic approximation (Illian et al, 2008, page 234)
which is valid for stationary point processes which are not
too clustered. This calculation is not yet implemented when
the argument \code{domain} is given.
}
\section{Advanced Use}{
To perform the same computation using several different bandwidths \code{bw},
it is efficient to use the argument \code{close}.
This should be the result of \code{\link{closepairs}(X, rmax)}
for a suitably large value of \code{rmax}, namely
\code{rmax >= max(r) + 3 * bw}.
}
\references{
Illian, J., Penttinen, A., Stoyan, H. and Stoyan, D. (2008)
\emph{Statistical Analysis and Modelling of Spatial Point Patterns.}
Wiley.
Stoyan, D. and Stoyan, H. (1994)
\emph{Fractals, random shapes and point fields:
methods of geometrical statistics.}
John Wiley and Sons.
}
\seealso{
\code{\link{Kest}},
\code{\link{pcf}},
\code{\link{density}},
\code{\link{lohboot}}.
}
\examples{
X <- simdat
\testonly{
X <- X[seq(1,npoints(X), by=4)]
}
p <- pcf(X)
plot(p, main="pair correlation function for X")
# indicates inhibition at distances r < 0.3
pd <- pcf(X, divisor="d")
# compare estimates
plot(p, cbind(iso, theo) ~ r, col=c("blue", "red"),
ylim.covers=0, main="", lwd=c(2,1), lty=c(1,3), legend=FALSE)
plot(pd, iso ~ r, col="green", lwd=2, add=TRUE)
legend("center", col=c("blue", "green"), lty=1, lwd=2,
legend=c("divisor=r","divisor=d"))
# calculate approximate variance and show POINTWISE confidence bands
pv <- pcf(X, var.approx=TRUE)
plot(pv, cbind(iso, iso+2*sqrt(v), iso-2*sqrt(v)) ~ r)
}
\author{
\spatstatAuthors
and Martin Hazelton.
}
\keyword{spatial}
\keyword{nonparametric}
