L(t)=sqrt(K(t)/pi), where
K(t) is Ripleys K function
for a spatial point pattern and
L(t) is linear for a completely random
point process.
Lhat(object, maxdist=<<see below>>, ndist=100, boundary=bbox(object),
plot.it=T)
"spp" representing a spatial point pattern,
or a data frame or matrix with first two columns containing
locations of a point pattern.
Lhat should
be estimated.
Defaults to half the length of a diagonal of the sample's bounding box.
Lhat.
Default is
100.
The distances for which
Lhat will be estimated are calculated as
seq(0,maxdist,ndist), both
maxdist and
ndist will change if
not reasonable for the given
object.
boundary should be
given as a list with named components
"x" and
"y" denoting the
corners of the rectangular region. For example, for the unit square
the boundary could be given as
bbox(x=c(0,1),y=c(0,1)), the bounding
box of two diagonally opposed points.
Defaults to a rectangle covering the range of points.
K-estimates be plotted?
Default is
TRUE.
dist,
contains the distances at which
Lhat was computed,
and the second column, called
Lhat, contains the values of
L(dist).
L(t)=t. See DETAILS.
plot.it=TRUE, a plot of the value of
L(t) against distance
will be produced on the current graphics device.
Khat computes Ripleys (1976) estimate of K(t) for a spatial point pattern:
K(t) = (1/intensity) * E[number of events < or = distance
t of an arbitrary event].
The theoretical K-function for a Poisson completely spatially random process is
K(t) = pi t^2, so L(t) = sqrt(K(t)/pi) is equal to
t, the distances.
The default plots
L(t) versus
t which should approximate a straight line
for a homogeneous process with no spatial dependence.
See function
Khat for estimation of
K(t).
Ripley, Brian D. (1976). The second-order analysis of stationary point processes. Journal of Applied Probability 13,255-266.
lansing.spp <- as.spp(lansing) lansing.khat <- Lhat(lansing.spp) Lhat(wheat) abline(0,1)