Inhomogeneous Pair Correlation Function
Estimates the inhomogeneous pair correlation function of a point pattern using kernel methods.
pcfinhom(X, lambda = NULL, ..., r = NULL,
kernel = "epanechnikov", bw = NULL, stoyan = 0.15,
correction = c("translate", "Ripley"),
divisor = c("r", "d"),
renormalise = TRUE, normpower=1,
update = TRUE, leaveoneout = TRUE,
reciplambda = NULL,
sigma = NULL, varcov = NULL, close=NULL)X |
A point pattern (object of class |
lambda |
Optional.
Values of the estimated intensity function.
Either a vector giving the intensity values
at the points of the pattern |
r |
Vector of values for the argument r at which g(r) should be evaluated. There is a sensible default. |
kernel |
Choice of smoothing kernel, passed to |
bw |
Bandwidth for smoothing kernel,
passed to |
... |
Other arguments passed to the kernel density estimation
function |
stoyan |
Coefficient for Stoyan's bandwidth selection rule;
see |
correction |
Character string or character vector
specifying the choice of edge correction.
See |
divisor |
Choice of divisor in the estimation formula:
either |
renormalise |
Logical. Whether to renormalise the estimate. See Details. |
normpower |
Integer (usually either 1 or 2). Normalisation power. See Details. |
update |
Logical. If |
leaveoneout |
Logical value (passed to |
reciplambda |
Alternative to |
sigma,varcov |
Optional arguments passed to |
close |
Advanced use only. Precomputed data. See section on Advanced Use. |
The inhomogeneous pair correlation function ginhom(r) is a summary of the dependence between points in a spatial point process that does not have a uniform density of points.
The best intuitive interpretation is the following: the probability p(r) of finding two points at locations x and y separated by a distance r is equal to
p(r) = lambda(x) * lambda(y) * g(r) dx dy
where lambda is the intensity function of the point process. For a Poisson point process with intensity function lambda, this probability is p(r) = lambda(x) * lambda(y) so ginhom(r) = 1.
The inhomogeneous pair correlation function is related to the inhomogeneous K function through
ginhom(r) = Kinhom'(r)/ ( 2 * pi * r)
where Kinhom'(r)
is the derivative of Kinhom(r), the
inhomogeneous K function. See Kinhom for information
about Kinhom(r).
The command pcfinhom estimates the inhomogeneous
pair correlation using a modified version of
the algorithm in pcf.ppp.
If renormalise=TRUE (the default), then the estimates
are multiplied by c^normpower where
c = area(W)/sum[i] (1/lambda(x[i])).
This rescaling reduces the variability and bias of the estimate
in small samples and in cases of very strong inhomogeneity.
The default value of normpower is 1
but the most sensible value is 2, which would correspond to rescaling
the lambda values so that
sum[i] (1/lambda(x[i])) = area(W).
A function value table (object of class "fv").
Essentially a data frame containing the variables
r |
the vector of values of the argument r at which the inhomogeneous pair correlation function ginhom(r) has been estimated |
theo |
vector of values equal to 1, the theoretical value of ginhom(r) for the Poisson process |
trans |
vector of values of ginhom(r) estimated by translation correction |
iso |
vector of values of ginhom(r) estimated by Ripley isotropic correction |
as required.
To perform the same computation using several different bandwidths bw,
it is efficient to use the argument close.
This should be the result of closepairs(X, rmax)
for a suitably large value of rmax, namely
rmax >= max(r) + 3 * bw.
Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk.
data(residualspaper) X <- residualspaper$Fig4b plot(pcfinhom(X, stoyan=0.2, sigma=0.1)) fit <- ppm(X, ~polynom(x,y,2)) plot(pcfinhom(X, lambda=fit, normpower=2))
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