I,
adjusted for spatially varying intensity.Kmulti.inhom(X, I, J, lambdaI=NULL, lambdaJ=NULL,
...,
r=NULL, breaks=NULL,
correction=c("border", "isotropic", "Ripley", "translate"),
lambdaIJ=NULL,
sigma=NULL, varcov=NULL)X
from which distances are measured. See Details.X to which
distances are measured. See Details.X[I].
Either a pixel image (object of class "im"),
a numeric vector containing the intensity values
at each of the points in X[I],
orX[J].
Either a pixel image (object of class "im"),
a numeric vector containing the intensity values
at each of the points in X[J],
orr.
Not normally invoked by the user. See the Details section."border", "bord.modif",
"isotropic", "Ripley", "translate",
"none" or "best".
It specifielambdaI and lambdaJ
for each pair of points, the first pointdensity.ppp
to control the smoothing bandwidth, when lambda is
estimated by kernel smoothing."fv" (see fv.object).Essentially a data frame containing numeric columns
"border", "bord.modif",
"iso" and/or "trans",
according to the selected edge corrections. These columns contain
estimates of the function $K_{IJ}(r)$
obtained by the edge corrections named.Kmulti.inhom
is the counterpart, for spatially-inhomogeneous marked point patterns,
of the multitype $K$ function Kmulti.Suppose $X$ is a marked point process, with marks of any kind. Suppose $X_I$, $X_J$ are two sub-processes, possibly overlapping. Typically $X_I$ would consist of those points of $X$ whose marks lie in a specified range of mark values, and similarly for $X_J$. Suppose that $\lambda_I(u)$, $\lambda_J(u)$ are the spatially-varying intensity functions of $X_I$ and $X_J$ respectively. Consider all the pairs of points $(u,v)$ in the point process $X$ such that the first point $u$ belongs to $X_I$, the second point $v$ belongs to $X_J$, and the distance between $u$ and $v$ is less than a specified distance $r$. Give this pair $(u,v)$ the numerical weight $1/(\lambda_I(u)\lambda_J(u))$. Calculate the sum of these weights over all pairs of points as described. This sum (after appropriate edge-correction and normalisation) is the estimated inhomogeneous multitype $K$ function.
The argument X must be a point pattern (object of class
"ppp") or any data that are acceptable to as.ppp.
The arguments I and J specify two subsets of the
point pattern. They may be any type of subset indices, for example,
logical vectors of length equal to npoints(X),
or integer vectors with entries in the range 1 to
npoints(X), or negative integer vectors.
Alternatively, I and J may be functions
that will be applied to the point pattern X to obtain
index vectors. If I is a function, then evaluating
I(X) should yield a valid subset index. This option
is useful when generating simulation envelopes using
envelope.
The argument lambdaI supplies the values
of the intensity of the sub-process identified by index I.
It may be either
[object Object],[object Object],[object Object],[object Object]
If lambdaI is omitted, then it will be estimated using
a `leave-one-out' kernel smoother, as described in Baddeley, Moller
and Waagepetersen (2000). The estimate of lambdaI for a given
point is computed by removing the point from the
point pattern, applying kernel smoothing to the remaining points using
density.ppp, and evaluating the smoothed intensity
at the point in question. The smoothing kernel bandwidth is controlled
by the arguments sigma and varcov, which are passed to
density.ppp along with any extra arguments.
Similarly lambdaJ supplies the values
of the intensity of the sub-process identified by index J.
The argument r is the vector of values for the
distance $r$ at which $K_{IJ}(r)$ should be evaluated.
It is also used to determine the breakpoints
(in the sense of hist)
for the computation of histograms of distances.
First-time users would be strongly advised not to specify r.
However, if it is specified, r must satisfy r[1] = 0,
and max(r) must be larger than the radius of the largest disc
contained in the window.
Biases due to edge effects are
treated in the same manner as in Kinhom.
The edge corrections implemented here are
[object Object],[object Object],[object Object]
The pair correlation function pcf can also be applied to the
result of Kmulti.inhom.
Kmulti,
Kdot.inhom,
Kcross.inhom,
pcf# Finnish Pines data: marked by diameter and height
plot(finpines, which.marks="height")
I <- (marks(finpines)$height <= 2)
J <- (marks(finpines)$height > 3)
K <- Kmulti.inhom(finpines, I, J)
plot(K)
# functions determining subsets
f1 <- function(X) { marks(X)$height <= 2 }
f2 <- function(X) { marks(X)$height > 3 }
K <- Kmulti.inhom(finpines, f1, f2)Run the code above in your browser using DataLab