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.
localpcfinhom computes the corresponding contribution
to the inhomogeneous empirical pair correlation function of X.
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.
For localpcfinhom, the optional argument lambda
specifies the values of the estimated intensity function.
If lambda is given, it should be either a
numeric vector giving the intensity values
at the points of the pattern X,
a pixel image (object of class "im") giving the
intensity values at all locations, a fitted point process model
(object of class "ppm", "kppm" or "dppm")
or a function(x,y) which
can be evaluated to give the intensity value at any location.
If lambda is not given, then it will be estimated
using a leave-one-out kernel density smoother as described
in pcfinhom.
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.