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
localLcrosstolocalKcross.- 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)
# }