localK
Neighbourhood density function
Computes the neighbourhood density function, a local version of the \(K\)-function or \(L\)-function, defined by Getis and Franklin (1987).
- Keywords
- spatial, nonparametric
Usage
localK(X, ..., rmax = NULL, correction = "Ripley", verbose = TRUE, rvalue=NULL)
localL(X, ..., rmax = NULL, correction = "Ripley", verbose = TRUE, rvalue=NULL)Arguments
- X
A point pattern (object of class
"ppp").- …
Ignored.
- rmax
Optional. Maximum desired value of the argument \(r\).
- correction
String specifying the edge correction to be applied. Options are
"none","translate","translation","Ripley","isotropic"or"best". Only one correction may be specified.- verbose
Logical flag indicating whether to print progress reports during the calculation.
- rvalue
Optional. A single value of the distance argument \(r\) at which the function L or K should be computed.
Details
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.
Value
If rvalue is given, the result is a numeric vector
of length equal to the number of points in the point pattern.
If rvalue is absent, the result is
an object of class "fv", see fv.object,
which can be plotted directly using plot.fv.
Essentially a data frame containing columns
the vector of values of the argument \(r\) at which the function \(K\) has been estimated
the theoretical value \(K(r) = \pi r^2\) or \(L(r)=r\) for a stationary Poisson process
References
Getis, A. and Franklin, J. (1987) Second-order neighbourhood analysis of mapped point patterns. Ecology 68, 473--477.
See Also
Examples
# NOT RUN {
data(ponderosa)
X <- ponderosa
# compute all the local L functions
L <- localL(X)
# plot all the local L functions against r
plot(L, main="local L functions for ponderosa", legend=FALSE)
# plot only the local L function for point number 7
plot(L, iso007 ~ r)
# compute the values of L(r) for r = 12 metres
L12 <- localL(X, rvalue=12)
# Spatially interpolate the values of L12
# Compare Figure 5(b) of Getis and Franklin (1987)
X12 <- X %mark% L12
Z <- Smooth(X12, sigma=5, dimyx=128)
plot(Z, col=topo.colors(128), main="smoothed neighbourhood density")
contour(Z, add=TRUE)
points(X, pch=16, cex=0.5)
# }