pcfdot
Multitype pair correlation function (i-to-any)
Calculates an estimate of the multitype pair correlation function
(from points of type i to points of any type)
for a multitype point pattern.
- Keywords
- spatial, nonparametric
Usage
pcfdot(X, i, ...)Arguments
- X
- The observed point pattern, from which an estimate of the dot-type pair correlation function $g_{i\bullet}(r)$ 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. - ...
- Arguments passed to
pcf.ppp.
Details
This is a generalisation of the pair correlation function pcf
to multitype point patterns.
For two locations $x$ and $y$ separated by a nonzero
distance $r$,
the probability $p(r)$ of finding a point of type $i$ at location
$x$ and a point of any type at location $y$ is
$$p(r) = \lambda_i \lambda g_{i\bullet}(r) \,{\rm d}x \, {\rm d}y$$
where $\lambda$ is the intensity of all points,
and $\lambda_i$ is the intensity of the points
of type $i$.
For a completely random Poisson marked point process,
$p(r) = \lambda_i \lambda$
so $g_{i\bullet}(r) = 1$.
For a stationary multitype point process, the
type-i-to-any-type pair correlation
function between marks $i$ and $j$ is formally defined as
$$g_{i\bullet}(r) = \frac{K_{i\bullet}^\prime(r)}{2\pi r}$$
where $K_{i\bullet}^\prime$ is the derivative of
the type-i-to-any-type $K$ function
$K_{i\bullet}(r)$.
of the point process. See Kdot for information
about $K_{i\bullet}(r)$.
The command pcfdot computes a kernel estimate of
the multitype pair correlation function from points of type $i$
to points of any type.
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
multitype pair correlation.
See pcf.ppp for a list of arguments that control
the kernel estimation.
The companion function pcfcross computes the
corresponding analogue of Kcross.
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 $g_{i\bullet}$ has been estimated theo the theoretical value $g_{i\bullet}(r) = 1$ 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 pcfcross.
Pair correlation pcf,pcf.ppp.
Kdot
Examples
data(amacrine)
p <- pcfdot(amacrine, "on")
p <- pcfdot(amacrine, "on", stoyan=0.1)
plot(p)