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, ...,
r = NULL,
kernel = "epanechnikov", bw = NULL, stoyan = 0.15,
correction = c("isotropic", "Ripley", "translate"),
divisor = c("r", "d"))
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
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)
.- …
Ignored.
- r
Vector of values for the argument \(r\) at which \(g(r)\) should be evaluated. There is a sensible default.
- kernel
Choice of smoothing kernel, passed to
density.default
.- bw
Bandwidth for smoothing kernel, passed to
density.default
.- stoyan
Coefficient for default bandwidth rule; see Details.
- correction
Choice of edge correction.
- divisor
Choice of divisor in the estimation formula: either
"r"
(the default) or"d"
. See Details.
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
\(j\).
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 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) applied to the points of type j
. That is,
\(h = c/\sqrt{\lambda}\),
where \(\lambda\) is the (estimated) intensity of the
point process of type j
,
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 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
the vector of values of the argument \(r\) at which the function \(g_{i,j}\) has been estimated
the theoretical value \(g_{i,j}(r) = 1\) for independent marks.
See Also
Mark connection function markconnect
.
Examples
# NOT RUN {
data(amacrine)
p <- pcfcross(amacrine, "off", "on")
p <- pcfcross(amacrine, "off", "on", stoyan=0.1)
plot(p)
# }