localpcf(X, ..., delta=NULL, rmax=NULL, nr=512, stoyan=0.15)
"ppp"
).delta
."fv"
, see fv.object
,
which can be plotted directly using plot.fv
.
Essentially a data frame containing columnsi
corresponds to the i
th point.
The last two columns contain the r
and theo
values.localpcf
computes the contribution, from each individual
data point in a point pattern X
, to the
empirical pair correlation function of X
.
These contributions are sometimes known as LISA (local indicator
of spatial association) functions based on pair correlation.
Given a spatial point pattern X
, the local pcf
$g_i(r)$ associated with the $i$th point
in X
is computed by
$$g_i(r) = \frac a {2 \pi n} \sum_j k(d_{i,j} - r)$$
where the sum is over all points $j \neq i$,
$a$ is the area of the observation window, $n$ is the number
of points in X
, and $d_{ij}$ is the distance
between points i
and j
. Here k
is the
Epanechnikov kernel,
$$k(t) = \frac 3 { 4\delta} \max(0, 1 - \frac{t^2}{\delta^2}).$$
Edge correction is performed using the border method
(for the sake of computational efficiency):
the estimate $g_i(r)$ is set to NA
if
$r > b_i$, where $b_i$
is the distance from point $i$ to the boundary of the
observation window. The smoothing bandwidth $\delta$ may be specified.
If not, it is chosen by Stoyan's rule of thumb
$\delta = c/\hat\lambda$
where $\hat\lambda = n/a$ is the estimated intensity
and $c$ is a constant, usually taken to be 0.15.
The value of $c$ is controlled by the argument stoyan
.
localK
,
pcf
data(ponderosa)
X <- ponderosa
g <- localpcf(X)
plot(g, main="local pair correlation functions for ponderosa", legend=FALSE)
# plot only the local pair correlation function for point number 7
plot(g, est007 ~ r)
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