The command localL computes the neighbourhood density function,
a local version of the $L$-function (Besag's transformation of Ripley's
$K$-function) that was proposed by Getis and Franklin (1987).
The command localK computes the corresponding
local analogue of the K-function. Given a spatial point pattern X, the neighbourhood density function
$L_i(r)$ associated with the $i$th point
in X is computed by
$$L_i(r) = \sqrt{\frac a {(n-1) \pi} \sum_j e_{ij}}$$
where the sum is over all points $j \neq i$ 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 $L_i(r)$ can also be interpreted as one
of the summands that contributes to the global estimate of the L
function.
By default, the function $L_i(r)$ or
$K_i(r)$ is computed for a range of $r$ values
for each point $i$. 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.
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.
Inhomogeneous counterparts of localK and localL
are computed by localKinhom and localLinhom.