markcrosscorr(X, r = NULL,
correction = c("isotropic", "Ripley", "translate"),
method = "density", ..., normalise = TRUE, Xname = NULL)
"ppp"
or something acceptable to
as.ppp
."isotropic"
, "Ripley"
, "translate"
,
"translation"
, "none"
or "best"
.
It specifies the edge correction(s) t"density"
,
"loess"
,
"sm"
and "smrep"
.normalise=FALSE
,
compute only the numerator of the expression for the
mark correlation.X
."fasp"
) containing
the mark cross-correlation functions for each possible pair
of columns of marks. Next, each pair of columns is considered, and the mark
cross-correlation is defined as
$$k_{mm}(r) = \frac{E_{0u}[M_i(0) M_j(u)]}{E[M_i,M_j]}$$
where $E_{0u}$ denotes the conditional expectation
given that there are points of the process at the locations
$0$ and $u$ separated by a distance $r$.
On the numerator,
$M_i(0)$ and $M_j(u)$
are the marks attached to locations $0$ and $u$ respectively
in the $i$th and $j$th columns of marks respectively.
On the denominator, $M_i$ and $M_j$ are
independent random values drawn from the
$i$th and $j$th columns of marks, respectively,
and $E$ is the usual expectation.
Note that $k_{mm}(r)$ is not a ``correlation''
in the usual statistical sense. It can take any
nonnegative real value. The value 1 suggests ``lack of correlation'':
if the marks attached to the points of X
are independent
and identically distributed, then
$k_{mm}(r) \equiv 1$.
The argument X
must be a point pattern (object of class
"ppp"
) or any data that are acceptable to as.ppp
.
It must be a marked point pattern.
The cross-correlations are estimated in the same manner as
for markcorr
.
markcorr
# The dataset 'betacells' has two columns of marks:
# 'type' (factor)
# 'area' (numeric)
if(interactive()) plot(betacells)
plot(markcrosscorr(betacells))
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