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 i - 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
- Optional. An alternative to the argument
r
. Not normally invoked by the user. - 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 r the vector of values of the argument $r$ at which the inhomogeneous cross-type pair correlation function $g_{ij}(r)$ has been estimated theo vector of values equal to 1, the theoretical value of $g_{ij}(r)$ for the Poisson process trans vector of values of $g_{ij}(r)$ estimated by translation correction iso vector of values of $g_{ij}(r)$ estimated by Ripley isotropic correction - as required.
See Also
Examples
data(amacrine)
plot(pcfcross.inhom(amacrine, "on", "off", stoyan=0.1),
legendpos="bottom")