Given a spatial point pattern with several columns of marks, this function computes the mark correlation function between each pair of columns of marks.
markcrosscorr(X, r = NULL,
correction = c("isotropic", "Ripley", "translate"),
method = "density", ...,
rmax = NULL,
weights=NULL, normalise = TRUE, Xname = NULL)A function array (object of class "fasp") containing
the mark cross-correlation functions for each possible pair
of columns of marks.
Each function in the array also
has an attribute "smooth.args" containing
the smoothing parameters that were used to compute the estimate.
First, all columns of marks are converted to numerical values. A factor with \(m\) possible levels is converted to \(m\) columns of dummy (indicator) values.
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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