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.