pcfcross {spatstat.explore} | R Documentation |
Multitype pair correlation function (cross-type)
Description
Calculates an estimate of the cross-type pair correlation function for a multitype point pattern.
Usage
pcfcross(X, i, j, ...,
r = NULL,
kernel = "epanechnikov", bw = NULL, stoyan = 0.15,
correction = c("isotropic", "Ripley", "translate"),
divisor = c("r", "d"),
ratio = FALSE)
Arguments
X |
The observed point pattern,
from which an estimate of the cross-type pair correlation function
|
i |
The type (mark value)
of the points in |
j |
The type (mark value)
of the points in |
... |
Ignored. |
r |
Vector of values for the argument |
kernel |
Choice of smoothing kernel,
passed to |
bw |
Bandwidth for smoothing kernel,
passed to |
stoyan |
Coefficient for default bandwidth rule; see Details. |
correction |
Choice of edge correction. |
divisor |
Choice of divisor in the estimation formula:
either |
ratio |
Logical.
If |
Details
The cross-type pair correlation function
is a generalisation of the pair correlation function pcf
to multitype point patterns.
For two locations and
separated by a distance
,
the probability
of finding a point of type
at location
and a point of type
at location
is
where is the intensity of the points
of type
.
For a completely random Poisson marked point process,
so
.
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
.
For a stationary multitype point process, the cross-type pair correlation
function between marks and
is formally defined as
where is the derivative of
the cross-type
function
.
of the point process. See
Kest
for information
about .
The command pcfcross
computes a kernel estimate of
the cross-type pair correlation function between marks and
.
-
If
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. -
If
divisor="d"
then a modified estimator is used: the contribution from an interpoint distanceto the estimate of
is divided by
instead of dividing by
. This usually improves the bias of the estimator when
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 ,
the argument
bw
is equivalent to .
If bw
is not specified, the default bandwidth
is determined by Stoyan's rule of thumb (Stoyan and Stoyan, 1994, page
285) applied to the points of type j
. That is,
,
where
is the (estimated) intensity of the
point process of type
j
,
and is a constant in the range from 0.1 to 0.2.
The argument
stoyan
determines the value of .
The companion function pcfdot
computes the
corresponding analogue of Kdot
.
Value
An object of class "fv"
, see fv.object
,
which can be plotted directly using plot.fv
.
Essentially a data frame containing columns
r |
the vector of values of the argument |
theo |
the theoretical value |
together with columns named
"border"
, "bord.modif"
,
"iso"
and/or "trans"
,
according to the selected edge corrections. These columns contain
estimates of the function
obtained by the edge corrections named.
Author(s)
Adrian Baddeley Adrian.Baddeley@curtin.edu.au and Rolf Turner rolfturner@posteo.net
See Also
Mark connection function markconnect
.
Multitype pair correlation pcfdot
, pcfmulti
.
Examples
p <- pcfcross(amacrine, "off", "on")
p <- pcfcross(amacrine, "off", "on", stoyan=0.1)
plot(p)