Estimates the inhomogeneous pair correlation function of a point pattern using kernel methods.
pcfinhom(X, lambda = NULL, ...,
r = NULL, adaptive = FALSE,
kernel = "epanechnikov", bw = NULL, h = NULL,
bw.args = list(), stoyan = 0.15, adjust = 1,
correction = c("translate", "Ripley"),
divisor = c("r", "d", "a", "t"),
zerocor=c("weighted", "reflection", "convolution",
"bdrykern", "JonesFoster", "none"),
renormalise = TRUE, normpower = 1,
update = TRUE, leaveoneout = TRUE,
reciplambda = NULL, sigma = NULL, adjust.sigma = 1, varcov = NULL,
gref = NULL, tau = 0, fast = TRUE, var.approx = FALSE,
domain = NULL, ratio = FALSE, close = NULL)A function value table
(object of class "fv").
Essentially a data frame containing the variables
the vector of values of the argument \(r\) at which the pair correlation function \(g(r)\) has been estimated
vector of values equal to 1, the theoretical value of \(g(r)\) for the Poisson process
vector of values of \(g(r)\) estimated by translation correction
vector of values of \(g(r)\) estimated by Ripley isotropic correction
vector of approximate values of the variance of the estimate of \(g(r)\)
as required.
If ratio=TRUE then the return value also has two
attributes called "numerator" and "denominator"
which are "fv" objects
containing the numerators and denominators of each
estimate of \(g(r)\).
The return value also has an attribute "bw" giving the
smoothing bandwidth that was used, and an attribute "info"
containing details of the algorithm parameters.
The inhomogeneous pair correlation function \(g_{\rm inhom}(r)\) 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 $$ p(r) = \lambda(x) lambda(y) g(r) \,{\rm d}x \, {\rm d}y $$ 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 \(g_{\rm inhom}(r) = 1\).
The inhomogeneous pair correlation function
is related to the inhomogeneous \(K\) function through
$$
g_{\rm inhom}(r) = \frac{K'_{\rm inhom}(r)}{2\pi r}
$$
where \(K'_{\rm inhom}(r)\)
is the derivative of \(K_{\rm inhom}(r)\), the
inhomogeneous \(K\) function. See Kinhom for information
about \(K_{\rm inhom}(r)\).
The command pcfinhom estimates the inhomogeneous
pair correlation using a modified version of
the algorithm in pcf. In this modified version,
the contribution from each pair of points \(X[i], X[j]\) is
weighted by
\(1/(\lambda(X[i]) \lambda(X[j]))\).
The arguments divisor, correction and zerocor
are interpreted as described in the help file for pcf.
If renormalise=TRUE (the default), then the estimates
are multiplied by \(c^{\mbox{normpower}}\) where
\(
c = \mbox{area}(W)/\sum (1/\lambda(x_i)).
\)
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
\(
\sum (1/\lambda(x_i)) = \mbox{area}(W).
\)
Baddeley, A., Davies, T.M. and Hazelton, M.L. (2025) An improved estimator of the pair correlation function of a spatial point process. Biometrika, to appear.
pcf,
bw.bdh,
bw.pcfinhom
g <- pcfinhom(japanesepines, divisor="a")
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