localKcross
Local Multitype K Function (Cross-Type)
for a multitype point pattern, computes the cross-type version of the local K function.
- Keywords
- spatial, nonparametric
Usage
localKcross(X, from, to, …, rmax = NULL,
correction = "Ripley", verbose = TRUE, rvalue=NULL)
localLcross(X, from, to, …, rmax = NULL, correction = "Ripley")
Arguments
- X
A multitype point pattern (object of class
"ppp"
with marks which are a factor).- …
Further arguments passed from
localLcross
tolocalKcross
.- rmax
Optional. Maximum desired value of the argument \(r\).
- from
Type of points from which distances should be measured. A single value; one of the possible levels of
marks(X)
, or an integer indicating which level.- to
Type of points to which distances should be measured. A single value; one of the possible levels of
marks(X)
, or an integer indicating which level.- 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
Given a multitype spatial point pattern X
,
the local cross-type \(K\) function localKcross
is the local version of the multitype \(K\) function
Kcross
.
Recall that Kcross(X, from, to)
is a sum of contributions
from all pairs of points in X
where
the first point belongs to from
and the second point belongs to type to
.
The local cross-type \(K\)
function is defined for each point X[i]
that belongs to
type from
, and it consists of all the contributions to
the cross-type \(K\) function that originate from point X[i]
:
$$
K_{i,from,to}(r) = \sqrt{\frac a {(n-1) \pi} \sum_j e_{ij}}
$$
where the sum is over all points \(j \neq i\)
belonging to type to
, 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 \(K_{i,from,to}(r)\)
can also be interpreted as one
of the summands that contributes to the global estimate of the
Kcross
function.
By default, the function \(K_{i,from,to}(r)\)
is computed for a range of \(r\) values
for each point \(i\) belonging to type from
.
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
belonging to type from
.
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
belonging to type from
.
The local cross-type \(L\) function localLcross
is computed by applying the transformation
\(L(r) = \sqrt{K(r)/(2\pi)}\).
Value
If rvalue
is given, the result is a numeric vector
of length equal to the number of points in the point pattern
that belong to type from
.
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
See Also
Kcross
,
Lcross
,
localK
,
localL
.
Inhomogeneous counterparts of localK
and localL
are computed by localKcross.inhom
and
localLinhom
.
Examples
# NOT RUN {
X <- amacrine
# compute all the local Lcross functions
L <- localLcross(X)
# plot all the local Lcross functions against r
plot(L, main="local Lcross functions for amacrine", legend=FALSE)
# plot only the local L function for point number 7
plot(L, iso007 ~ r)
# compute the values of L(r) for r = 0.1 metres
L12 <- localLcross(X, rvalue=0.1)
# }