Computes individual contributions to the pair correlation function from each data point.
localpcf(X, ..., delta=NULL, rmax=NULL, nr=512, stoyan=0.15)
localpcfinhom(X, ..., delta=NULL, rmax=NULL, nr=512, stoyan=0.15,
lambda=NULL, sigma=NULL, varcov=NULL,
update=TRUE, leaveoneout=TRUE)A point pattern (object of class "ppp").
Smoothing bandwidth for pair correlation. The halfwidth of the Epanechnikov kernel.
Optional. Maximum value of distance \(r\) for which pair correlation values \(g(r)\) should be computed.
Optional. Number of values of distance \(r\) for which pair correlation \(g(r)\) should be computed.
Optional. The value of the constant \(c\) in Stoyan's rule
of thumb for selecting the smoothing bandwidth delta.
Optional.
Values of the estimated intensity function, for the
inhomogeneous pair correlation.
Either a vector giving the intensity values
at the points of the pattern X,
a pixel image (object of class "im") giving the
intensity values at all locations, a fitted point process model
(object of class "ppm", "kppm" or "dppm")
or a function(x,y) which
can be evaluated to give the intensity value at any location.
These arguments are ignored by localpcf
but are passed by localpcfinhom (when lambda=NULL)
to the function density.ppp
to control the kernel smoothing estimation of lambda.
Logical value (passed to density.ppp or
fitted.ppm) specifying whether to use a
leave-one-out rule when calculating the intensity.
Logical value indicating what to do when lambda is a fitted model
(class "ppm", "kppm" or "dppm").
If update=TRUE (the default),
the model will first be refitted to the data X
(using update.ppm or update.kppm)
before the fitted intensity is computed.
If update=FALSE, the fitted intensity of the
model will be computed without re-fitting it to X.
An object of class "fv", see fv.object,
which can be plotted directly using plot.fv.
Essentially a data frame containing columns
the vector of values of the argument \(r\) at which the function \(K\) has been estimated
the theoretical value \(K(r) = \pi r^2\) or \(L(r)=r\) for a stationary Poisson process
localpcf computes the contribution, from each individual
data point in a point pattern X, to the
empirical pair correlation function of X.
These contributions are sometimes known as LISA (local indicator
of spatial association) functions based on pair correlation.
localpcfinhom computes the corresponding contribution
to the inhomogeneous empirical pair correlation function of X.
Given a spatial point pattern X, the local pcf
\(g_i(r)\) associated with the \(i\)th point
in X is computed by
$$
g_i(r) = \frac a {2 \pi n} \sum_j k(d_{i,j} - r)
$$
where the sum is over all points \(j \neq i\),
\(a\) is the area of the observation window, \(n\) is the number
of points in X, and \(d_{ij}\) is the distance
between points i and j. Here k is the
Epanechnikov kernel,
$$
k(t) = \frac 3 { 4\delta} \max(0, 1 - \frac{t^2}{\delta^2}).
$$
Edge correction is performed using the border method
(for the sake of computational efficiency):
the estimate \(g_i(r)\) is set to NA if
\(r > b_i\), where \(b_i\)
is the distance from point \(i\) to the boundary of the
observation window.
The smoothing bandwidth \(\delta\) may be specified.
If not, it is chosen by Stoyan's rule of thumb
\(\delta = c/\hat\lambda\)
where \(\hat\lambda = n/a\) is the estimated intensity
and \(c\) is a constant, usually taken to be 0.15.
The value of \(c\) is controlled by the argument stoyan.
For localpcfinhom, the optional argument lambda
specifies the values of the estimated intensity function.
If lambda is given, it should be either a
numeric vector giving the intensity values
at the points of the pattern X,
a pixel image (object of class "im") giving the
intensity values at all locations, a fitted point process model
(object of class "ppm", "kppm" or "dppm")
or a function(x,y) which
can be evaluated to give the intensity value at any location.
If lambda is not given, then it will be estimated
using a leave-one-out kernel density smoother as described
in pcfinhom.
# NOT RUN {
data(ponderosa)
X <- ponderosa
g <- localpcf(X, stoyan=0.5)
colo <- c(rep("grey", npoints(X)), "blue")
a <- plot(g, main=c("local pair correlation functions", "Ponderosa pines"),
legend=FALSE, col=colo, lty=1)
# plot only the local pair correlation function for point number 7
plot(g, est007 ~ r)
gi <- localpcfinhom(X, stoyan=0.5)
a <- plot(gi, main=c("inhomogeneous local pair correlation functions",
"Ponderosa pines"),
legend=FALSE, col=colo, lty=1)
# }
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