This is a generalisation of the function Kdot
to include an adjustment for spatially inhomogeneous intensity,
in a manner similar to the function Kinhom. Briefly, given a multitype point process, consider the points without
their types, and suppose this unmarked point process
has intensity function
$\lambda(u)$ at spatial locations $u$.
Suppose we place a mass of $1/\lambda(\zeta)$
at each point $\zeta$ of the process. Then the expected total
mass per unit area is 1. The
inhomogeneous ``dot-type'' $K$ function
$K_{i\bullet}^{\mbox{inhom}}(r)$ equals the expected
total mass within a radius $r$ of a point of the process
of type $i$, discounting this point itself.
If the process of type $i$ points
were independent of the points of other types,
then $K_{i\bullet}^{\mbox{inhom}}(r)$
would equal $\pi r^2$.
Deviations between the empirical $K_{i\bullet}$ curve
and the theoretical curve $\pi r^2$
suggest dependence between the points of types $i$ and $j$ for
$j\neq i$.
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, and the mark vector
X$marks must be a factor.
The arguments i and j will be interpreted as
levels of the factor X$marks. (Warning: this means that
an integer value i=3 will be interpreted as the 3rd smallest level,
not the number 3).
The argument lambdaI supplies the values
of the intensity of the sub-process of points of type i.
It may be either
[object Object],[object Object]
The argument lambdadot should contain
estimated values of the intensity of the entire point process.
It may be either a pixel image, or a numeric vector of length equal
to the number of points in X.
For advanced use only, the optional argument lambdaIdot
is a matrix containing estimated
values of the products of these two intensities for each pair of
points, the first point of type i and the second of any type.
The argument r is the vector of values for the
distance $r$ at which $K_{i\bullet}(r)$ should be evaluated.
The values of $r$ must be increasing nonnegative numbers
and the maximum $r$ value must exceed the radius of the
largest disc contained in the window.
The argument correction chooses the edge correction
as explained e.g. in Kest.
The pair correlation function can also be applied to the
result of Kcross.inhom; see pcf.