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.
A point pattern (object of class "ppp").
Optional.
Values of the estimated intensity function.
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.
Arguments passed to density.default or
to densityBC controlling the kernel smoothing.
Vector of values for the argument \(r\) at which \(g(r)\) should be evaluated. There is a sensible default.
Logical value specifying whether to use adaptive kernel smoothing
(adaptive=TRUE) or fixed-bandwidth kernel smoothing
(adaptive=FALSE, the default).
Choice of smoothing kernel, passed to density.default.
Bandwidth for smoothing kernel. Either a single numeric value giving the standard deviation of the kernel, or a character string specifying a bandwidth selection rule, or a function that computes the selected bandwidth. See Details.
Kernel halfwidth \(h\) (incompatible with argument bw).
A numerical value.
The parameter h is defined as the
half-width of the support of the kernel, except for the Gaussian
kernel where h is the standard deviation.
Optional. List of additional arguments to be passed to bw
when bw is a function. Alternatively, bw may be a
function that should be applied to X to produce a list of
additional arguments.
Coefficient for Stoyan's bandwidth selection rule; see Details.
Numerical adjustment factor for the bandwidth.
The bandwidth actually used is adjust * bw.
This makes it easy to specify choices like ‘half the
selected bandwidth’.
Edge correction. A character vector specifying the choice (or choices) of edge correction. See Details.
Choice of divisor in the estimation formula:
either "r" (the default) or "d",
or the new alternatives "a" or "t". See Details.
String (partially matched) specifying a correction for the boundary effect
bias at \(r=0\). Possible values are
"none", "weighted", "convolution",
"reflection", "bdrykern" and "JonesFoster".
See Details, or help file for densityBC.
Logical. Whether to renormalise the estimate. See Details.
Integer (usually either 1 or 2). Normalisation power. See Details.
Logical. If lambda is a fitted model
(class "ppm", "kppm" or "dppm")
and 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.
Logical value (passed to density.ppp or
fitted.ppm) specifying whether to use a
leave-one-out rule when calculating the intensity.
Alternative to lambda.
Values of the estimated reciprocal \(1/\lambda\)
of the intensity function.
Either a vector giving the reciprocal intensity values
at the points of the pattern X,
a pixel image (object of class "im") giving the
reciprocal intensity values at all locations,
or a function(x,y) which can be evaluated to give the
reciprocal intensity value at any location.
Optional arguments passed to density.ppp
to control the smoothing bandwidth, when lambda is
estimated by kernel smoothing.
Numeric value. sigma will be multiplied by this value.
Optional. A pair correlation function that will be used as the
reference for the transformation to uniformity, when
divisor="t". Either a function in the R language
giving the pair correlation function, or a fitted model
(object of class "kppm", "dppm", "ppm"
or "slrm") or a theoretical point process model
(object of class "zclustermodel" or "detpointprocfamily")
for which the pair correlation function
can be computed.
Optional shrinkage coefficient. A single numeric value.
Logical value specifying whether to compute the kernel smoothing
using a Fast Fourier Transform algorithm (fast=TRUE)
or an exact analytic kernel sum (fast=FALSE).
Logical value indicating whether to compute an analytic approximation to the variance of the estimated pair correlation.
Optional. Calculations will be restricted to this subset of the window. See Details.
Logical.
If TRUE, the numerator and denominator of
each edge-corrected estimate will also be saved,
for use in analysing replicated point patterns.
Advanced use only. Precomputed data. See section on Advanced Use.
Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner rolfturner@posteo.net, Ege Rubak rubak@math.aau.dk, Martin Hazelton Martin.Hazelton@otago.ac.nz and Tilman Davies Tilman.Davies@otago.ac.nz.
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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