Local pair correlation function
Computes individual contributions to the pair correlation function from each data point.
localpcf(X, ..., delta=NULL, rmax=NULL, nr=512, stoyan=0.15, rvalue=NULL)
localpcfinhom(X, ..., delta=NULL, rmax=NULL, nr=512, stoyan=0.15,
lambda=NULL, sigma=NULL, varcov=NULL,
update=TRUE, leaveoneout=TRUE, rvalue=NULL)X |
A point pattern (object of class |
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 |
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 |
sigma,varcov,... |
These arguments are ignored by |
leaveoneout |
Logical value (passed to |
update |
Logical value indicating what to do when |
rvalue |
Optional. A single value of the distance argument r at which the local pair correlation should be computed. |
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 ith point
in X is computed by
g[i](r) = (a/(2 * pi * n) * sum[j] k(d[i,j] - r)
where the sum is over all points j != i,
a is the area of the observation window, n is the number
of points in X, and d[i,j] is the distance
between points i and j. Here k is the
Epanechnikov kernel,
k(t) = (3/(4*delta)) * max(0, 1 - 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/lambda
where 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.
Alternatively, if the argument rvalue is given, and it is a
single number, then the function will only be computed for this value
of r, and the results will be returned as a numeric vector,
with one entry of the vector for each point of the pattern X.
r |
the vector of values of the argument r at which the function K has been estimated |
theo |
the theoretical value K(r) = pi * r^2 or L(r)=r for a stationary Poisson process |
together with columns containing the values of the
local pair correlation function for each point in the pattern.
Column i corresponds to the ith point.
The last two columns contain the r and theo values.
Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk.
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)
# Extract the local pair correlation at distance 15 metres, for each point
g15 <- localpcf(X, rvalue=15, stoyan=0.5)
g15[1:10]
# Check that the value for point 7 agrees with the curve for point 7:
points(15, g15[7], col="red")
# Inhomogeneous
gi <- localpcfinhom(X, stoyan=0.5)
a <- plot(gi, main=c("inhomogeneous local pair correlation functions",
"Ponderosa pines"),
legend=FALSE, col=colo, lty=1)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.