localpcf
Local pair correlation function
Computes individual contributions to the pair correlation function from each data point.
- Keywords
- spatial, nonparametric
Usage
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)
Arguments
- X
A point pattern (object of class
"ppp"
).- delta
Smoothing bandwidth for pair correlation. The halfwidth of the Epanechnikov kernel.
- rmax
Optional. Maximum value of distance \(r\) for which pair correlation values \(g(r)\) should be computed.
- nr
Optional. Number of values of distance \(r\) for which pair correlation \(g(r)\) should be computed.
- stoyan
Optional. The value of the constant \(c\) in Stoyan's rule of thumb for selecting the smoothing bandwidth
delta
.- lambda
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 afunction(x,y)
which can be evaluated to give the intensity value at any location.- sigma,varcov,…
These arguments are ignored by
localpcf
but are passed bylocalpcfinhom
(whenlambda=NULL
) to the functiondensity.ppp
to control the kernel smoothing estimation oflambda
.- leaveoneout
Logical value (passed to
density.ppp
orfitted.ppm
) specifying whether to use a leave-one-out rule when calculating the intensity.- update
Logical value indicating what to do when
lambda
is a fitted model (class"ppm"
,"kppm"
or"dppm"
). Ifupdate=TRUE
(the default), the model will first be refitted to the dataX
(usingupdate.ppm
orupdate.kppm
) before the fitted intensity is computed. Ifupdate=FALSE
, the fitted intensity of the model will be computed without re-fitting it toX
.
Details
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
.
Value
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
See Also
Examples
# 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)
# }