i
to points of any type)
for a multitype point pattern.pcfdot(X, i, ..., r = NULL,
kernel = "epanechnikov", bw = NULL, stoyan = 0.15,
correction = c("isotropic", "Ripley", "translate"),
divisor = c("r", "d"))
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)
.density.default
.density.default
."r"
(the default) or "d"
. See Details."fv"
, see fv.object
,
which can be plotted directly using plot.fv
.Essentially a data frame containing columns
"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.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.
divisor="r"
(the default), then the multitype
counterpart of the standard
kernel estimator (Stoyan and Stoyan, 1994, pages 284--285)
is used. By default, the recommendations of Stoyan and Stoyan (1994)
are followed exactly.divisor="d"
then a modified estimator is used:
the contribution from
an interpoint distance$d_{ij}$to the
estimate of$g(r)$is divided by$d_{ij}$instead of dividing by$r$. This usually improves the
bias of the estimator when$r$is close to zero. There is also a choice of spatial edge corrections
(which are needed to avoid bias due to edge effects
associated with the boundary of the spatial window):
correction="translate"
is the Ohser-Stoyan translation
correction, and correction="isotropic"
or "Ripley"
is Ripley's isotropic correction.
The choice of smoothing kernel is controlled by the
argument kernel
which is passed to density
.
The default is the Epanechnikov kernel.
The bandwidth of the smoothing kernel can be controlled by the
argument bw
. Its precise interpretation
is explained in the documentation for density.default
.
For the Epanechnikov kernel with support $[-h,h]$,
the argument bw
is equivalent to $h/\sqrt{5}$.
If bw
is not specified, the default bandwidth
is determined by Stoyan's rule of thumb (Stoyan and Stoyan, 1994, page
285). That is,
$h = c/\sqrt{\lambda}$,
where $\lambda$ is the (estimated) intensity of the
unmarked point process,
and $c$ is a constant in the range from 0.1 to 0.2.
The argument stoyan
determines the value of $c$.
The companion function pcfcross
computes the
corresponding analogue of Kcross
.
markconnect
. Multitype pair correlation pcfcross
, pcfmulti
.
Pair correlation pcf
,pcf.ppp
.
Kdot
data(amacrine)
p <- pcfdot(amacrine, "on")
p <- pcfdot(amacrine, "on", stoyan=0.1)
plot(p)
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