Given a multitype spatial point pattern X,
the local cross-type \(K\) function localKcross
is the local version of the multitype \(K\) function
Kcross.
Recall that Kcross(X, from, to) is a sum of contributions
from all pairs of points in X where
the first point belongs to from
and the second point belongs to type to.
The local cross-type \(K\)
function is defined for each point X[i] that belongs to
type from, and it consists of all the contributions to
the cross-type \(K\) function that originate from point X[i]:
$$
K_{i,from,to}(r) = \sqrt{\frac a {(n-1) \pi} \sum_j e_{ij}}
$$
where the sum is over all points \(j \neq i\)
belonging to type to, that lie
within a distance \(r\) of the \(i\)th point,
\(a\) is the area of the observation window, \(n\) is the number
of points in X, and \(e_{ij}\) is an edge correction
term (as described in Kest).
The value of \(K_{i,from,to}(r)\)
can also be interpreted as one
of the summands that contributes to the global estimate of the
Kcross function.
By default, the function \(K_{i,from,to}(r)\)
is computed for a range of \(r\) values
for each point \(i\) belonging to type from.
The results are stored as a function value
table (object of class "fv") with a column of the table
containing the function estimates for each point of the pattern
X belonging to type from.
Alternatively, if the argument rvalue is given, and it is a
single number, then the function will only be computed for this value
of \(r\), and the results will be returned as a numeric vector,
with one entry of the vector for each point of the pattern X
belonging to type from.
The local cross-type \(L\) function localLcross
is computed by applying the transformation
\(L(r) = \sqrt{K(r)/(2\pi)}\).