Calculates an estimate of the cross-type pair correlation function for a multitype point pattern.
pcfcross(X, i, j, ...,
r = NULL,
kernel = "epanechnikov", bw = NULL, stoyan = 0.15,
correction = c("isotropic", "Ripley", "translate"),
divisor = c("r", "d"),
ratio = FALSE)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.
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.
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).
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 of marks(X).
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 of marks(X).
Ignored.
Vector of values for the argument \(r\) at which \(g(r)\) should be evaluated. There is a sensible default.
Choice of smoothing kernel,
passed to density.default.
Bandwidth for smoothing kernel,
passed to density.default.
Coefficient for default bandwidth rule; see Details.
Choice of edge correction.
Choice of divisor in the estimation formula:
either "r" (the default) or "d". See Details.
Logical.
If TRUE, the numerator and denominator of
each edge-corrected estimate will also be saved,
for use in analysing replicated point patterns.
Adrian Baddeley Adrian.Baddeley@curtin.edu.au and Rolf Turner rolfturner@posteo.net
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.
Mark connection function markconnect.
Multitype pair correlation pcfdot, pcfmulti.
Pair correlation pcf,pcf.ppp.
Kcross
p <- pcfcross(amacrine, "off", "on")
p <- pcfcross(amacrine, "off", "on", stoyan=0.1)
plot(p)
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