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, ..., correction = "Ripley", verbose = TRUE, rvalue=NULL)
localL(X, ..., correction = "Ripley", verbose = TRUE, rvalue=NULL)
Arguments
- X
- A point pattern (object of class
"ppp"
). - ...
- Ignored.
- correction
- String specifying the edge correction to be applied.
Options are
"none"
,"translate"
,"translation"
,"Ripley"
,"isotropic"
or"best"
. Only one correction may be - 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"
, seefv.object
, which can be plotted directly usingplot.fv
. Essentially a data frame containing columns r the vector of values of the argument $r$ at which the function $K$ has been estimated theo the theoretical value $K(r) = \pi r^2$ or $L(r)=r$ for a stationary Poisson process - together with columns containing the values of the
neighbourhood density function for each point in the pattern.
Column
i
corresponds to thei
th point. The last two columns contain ther
andtheo
values.
References
Getis, A. and Franklin, J. (1987) Second-order neighbourhood analysis of mapped point patterns. Ecology 68, 473--477.
See Also
Examples
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.ppp(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)