This computes a generalisation of the $K$ function
for inhomogeneous point patterns, proposed by
Baddeley, Moller and Waagepetersen (2000).
The ``ordinary'' $K$ function
(variously known as the reduced second order moment function
and Ripley's $K$ function), is
described under Kest. It is defined only
for stationary point processes.
The inhomogeneous $K$ function
$K_{\rm inhom}(r)$
is a direct generalisation to nonstationary point processes.
Suppose $x$ is a point process with non-constant intensity
$\lambda(u)$ at each location $u$.
Define $K_{\rm inhom}(r)$ to be the expected
value, given that $u$ is a point of $x$,
of the sum of all terms
$1/\lambda(u)\lambda(x_j)$
over all points $x_j$
in the process separated from $u$ by a distance less than $r$.
This reduces to the ordinary $K$ function if
$\lambda()$ is constant.
If $x$ is an inhomogeneous Poisson process with intensity
function $\lambda(u)$, then
$K_{\rm inhom}(r) = \pi r^2$. This allows us to inspect a point pattern for evidence of
interpoint interactions after allowing for spatial inhomogeneity
of the pattern. Values
$K_{\rm inhom}(r) > \pi r^2$
are suggestive of clustering.
The argument lambda should be a vector of length equal to the
number of points in the pattern X. It will be interpreted as
giving the (estimated) values of $\lambda(x_i)$ for
each point $x_i$ of the pattern $x$.
Edge corrections are used to correct bias in the estimation
of $K_{\rm inhom}$.
Each edge-corrected estimate of $K_{\rm inhom}(r)$ is
of the form
$$\widehat K_{\rm inhom}(r) = \sum_i \sum_j \frac{1{d_{ij} \le
r} e(x_i,x_j,r)}{\lambda(x_i)\lambda(x_j)}$$
where $d_{ij}$ is the distance between points
$x_i$ and $x_j$, and
$e(x_i,x_j,r)$ is
an edge correction factor. For the `border' correction,
$$e(x_i,x_j,r) =
\frac{1(b_i > r)}{\sum_j 1(b_j > r)/\lambda(x_j)}$$
where $b_i$ is the distance from $x_i$
to the boundary of the window. For the `modified border'
correction,
$$e(x_i,x_j,r) =
\frac{1(b_i > r)}{\mbox{area}(W \ominus r)}$$
where $W \ominus r$ is the eroded window obtained
by trimming a margin of width $r$ from the border of the original
window.
For the `translation' correction,
$$e(x_i,x_j,r) =
\frac 1 {\mbox{area}(W \cap (W + (x_j - x_i)))}$$
and for the `isotropic' correction,
$$e(x_i,x_j,r) =
\frac 1 {\mbox{area}(W) g(x_i,x_j)}$$
where $g(x_i,x_j)$ is the fraction of the
circumference of the circle with centre $x_i$ and radius
$||x_i - x_j||$ which lies inside the window.
The pair correlation function can also be applied to the
result of Kinhom; see pcf.