pcfcross
Multitype pair correlation function (cross-type)
Calculates an estimate of the cross-type pair correlation function for a multitype point pattern.
- Keywords
- spatial, nonparametric
Usage
pcfcross(X, i, j, ...)Arguments
- X
- The observed point pattern, from which an estimate of the 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
- Number or character string identifying the type (mark value)
of the points in
Xfrom which distances are measured. - j
- Number or character string identifying the type (mark value)
of the points in
Xto which distances are measured. - ...
- Arguments passed to
pcf.ppp.
Details
The cross-type pair correlation function
is a generalisation of the pair correlation function pcf
to multitype point patterns.
For two locations $x$ and $y$ separated by a distance $r$,
the probability $p(r)$ of finding a point of type $i$ at location
$x$ and a point of type $j$ at location $y$ is
$$p(r) = \lambda_i \lambda_j g_{i,j}(r) \,{\rm d}x \, {\rm d}y$$
where $\lambda_i$ is the intensity of the points
of type $i$.
For a completely random Poisson marked point process,
$p(r) = \lambda_i \lambda_j$
so $g_{i,j}(r) = 1$.
Indeed for any marked point pattern in which the points of type i
are independent of the points of type j,
the theoretical value of the cross-type pair correlation is
$g_{i,j}(r) = 1$.
For a stationary multitype point process, the cross-type pair correlation
function between marks $i$ and $j$ is formally defined as
$$g_{i,j}(r) = \frac{K_{i,j}^\prime(r)}{2\pi r}$$
where $K_{i,j}^\prime$ is the derivative of
the cross-type $K$ function $K_{i,j}(r)$.
of the point process. See Kest for information
about $K(r)$.
The command pcfcross computes a kernel estimate of
the cross-type pair correlation function between marks $i$ and
$j$. It uses pcf.ppp to compute kernel estimates
of the pair correlation functions for several unmarked point patterns,
and uses the bilinear properties of second moments to obtain the
cross-type pair correlation.
See pcf.ppp for a list of arguments that control
the kernel estimation.
The companion function pcfdot computes the
corresponding analogue of Kdot.
Value
- An object of class
"fv", seefv.object, which can be plotted directly usingplot.fv.Essentially a data frame containing columns
r the vector of values of the argument $r$ at which the function $g_{i,j}$ has been estimated theo the theoretical value $g_{i,j}(r) = 1$ for independent marks. - together with columns named
"border","bord.modif","iso"and/or"trans", according to the selected edge corrections. These columns contain estimates of the function $g_{i,j}$ obtained by the edge corrections named.
See Also
Mark connection function markconnect.
Multitype pair correlation pcfdot.
Pair correlation pcf,pcf.ppp.
Kcross
Examples
data(amacrine)
p <- pcfcross(amacrine, "off", "on")
p <- pcfcross(amacrine, "off", "on", stoyan=0.1)
plot(p)