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)
# }