Estimates the inhomogeneous cross-type pair correlation function for a multitype point pattern.
pcfcross.inhom(X, i, j, lambdaI = NULL, lambdaJ = NULL, ...,
r = NULL, breaks = NULL,
kernel="epanechnikov", bw=NULL, adjust.bw = 1, stoyan=0.15,
correction = c("isotropic", "Ripley", "translate"),
sigma = NULL, adjust.sigma = 1, varcov = 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 inhomogeneous cross-type pair correlation function \(g_{ij}(r)\) has been estimated
vector of values equal to 1, the theoretical value of \(g_{ij}(r)\) for the Poisson process
vector of values of \(g_{ij}(r)\) estimated by translation correction
vector of values of \(g_{ij}(r)\) estimated by Ripley isotropic correction
as required.
The observed point pattern, from which an estimate of the inhomogeneous cross-type pair correlation function \(g_{ij}(r)\) will be computed. It must be a multitype point pattern (a marked point pattern whose marks are a factor).
The type (mark value)
of the points in X from which distances are measured.
A character string (or something that will be converted to a
character string).
Defaults to the first level of marks(X).
The type (mark value)
of the points in X to which distances are measured.
A character string (or something that will be
converted to a character string).
Defaults to the second level of marks(X).
Optional.
Values of the estimated intensity function of the points of type i.
Either a vector giving the intensity values
at the points of type i,
a pixel image (object of class "im") giving the
intensity values at all locations, or a function(x,y) which
can be evaluated to give the intensity value at any location.
Optional.
Values of the estimated intensity function of the points of type j.
A numeric vector, pixel image or function(x,y).
Vector of values for the argument \(r\) at which \(g_{ij}(r)\) should be evaluated. There is a sensible default.
This argument is for internal use only.
Choice of one-dimensional smoothing kernel,
passed to density.default.
Bandwidth for one-dimensional smoothing kernel,
passed to density.default.
Numeric value. bw will be multiplied by this value.
Other arguments passed to the one-dimensional kernel density estimation
function density.default.
Bandwidth coefficient; see Details.
Choice of edge correction.
Optional arguments passed to density.ppp
to control the smoothing bandwidth, when lambdaI or
lambdaJ is estimated by spatial kernel smoothing.
Numeric value. sigma will be multiplied by this value.
Adrian Baddeley Adrian.Baddeley@curtin.edu.au and Rolf Turner r.turner@auckland.ac.nz
The inhomogeneous cross-type pair correlation function \(g_{ij}(r)\) is a summary of the dependence between two types of points in a multitype 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, of types \(i\) and \(j\) respectively, at locations \(x\) and \(y\) separated by a distance \(r\) is equal to $$ p(r) = \lambda_i(x) lambda_j(y) g(r) \,{\rm d}x \, {\rm d}y $$ where \(\lambda_i\) is the intensity function of the process of points of type \(i\). For a multitype Poisson point process, this probability is \(p(r) = \lambda_i(x) \lambda_j(y)\) so \(g_{ij}(r) = 1\).
The command pcfcross.inhom estimates the inhomogeneous
pair correlation using a modified version of
the algorithm in pcf.ppp.
The arguments bw and adjust.bw control the
degree of one-dimensional smoothing of the estimate of pair correlation.
If the arguments lambdaI and/or lambdaJ are missing or
null, they will be estimated from X by spatial kernel smoothing
using a leave-one-out estimator, computed by density.ppp.
The arguments sigma, varcov
and adjust.sigma control the degree of spatial smoothing.
pcf.ppp,
pcfinhom,
pcfcross,
pcfdot.inhom
plot(pcfcross.inhom(amacrine, "on", "off", stoyan=0.1),
legendpos="bottom")
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