pcfcross.inhom
Inhomogeneous Multitype Pair Correlation Function (Cross-Type)
Estimates the inhomogeneous cross-type pair correlation function for a multitype point pattern.
- Keywords
- spatial, nonparametric
Usage
pcfcross.inhom(X, i, j, lambdaI = NULL, lambdaJ = NULL, ...,
r = NULL, breaks = NULL,
kernel="epanechnikov", bw=NULL, stoyan=0.15,
correction = c("isotropic", "Ripley", "translate"),
sigma = NULL, varcov = NULL)
Arguments
- X
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).
- i
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 ofmarks(X)
.- j
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 ofmarks(X)
.- lambdaI
Optional. Values of the estimated intensity function of the points of type
i
. Either a vector giving the intensity values at the points of typei
, a pixel image (object of class"im"
) giving the intensity values at all locations, or afunction(x,y)
which can be evaluated to give the intensity value at any location.- lambdaJ
Optional. Values of the estimated intensity function of the points of type
j
. A numeric vector, pixel image orfunction(x,y)
.- r
Vector of values for the argument \(r\) at which \(g_{ij}(r)\) should be evaluated. There is a sensible default.
- breaks
This argument is for internal use only.
- kernel
Choice of smoothing kernel, passed to
density.default
.- bw
Bandwidth for smoothing kernel, passed to
density.default
.- …
Other arguments passed to the kernel density estimation function
density.default
.- stoyan
Bandwidth coefficient; see Details.
- correction
Choice of edge correction.
- sigma,varcov
Optional arguments passed to
density.ppp
to control the smoothing bandwidth, whenlambdaI
orlambdaJ
is estimated by kernel smoothing.
Details
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
.
If the arguments lambdaI
and lambdaJ
are missing or
null, they are estimated from X
by kernel smoothing using a
leave-one-out estimator.
Value
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
See Also
Examples
# NOT RUN {
data(amacrine)
plot(pcfcross.inhom(amacrine, "on", "off", stoyan=0.1),
legendpos="bottom")
# }