| Jmulti {spatstat} | R Documentation |
For a marked point pattern,
estimate the multitype J function
summarising dependence between the
points in subset I
and those in subset J.
Jmulti(X, I, J, eps=NULL, r=NULL, breaks=NULL, ..., disjoint=NULL)
X |
The observed point pattern, from which an estimate of the multitype distance distribution function JIJ(r) will be computed. It must be a marked point pattern. See under Details. |
I |
Subset of points of X from which distances are
measured.
|
J |
Subset of points in X to which distances are measured.
|
eps |
A positive number.
The pixel resolution of the discrete approximation to Euclidean
distance (see Jest). There is a sensible default.
|
r |
numeric vector. The values of the argument r at which the distribution function JIJ(r) should be evaluated. There is a sensible default. First-time users are strongly advised not to specify this argument. See below for important conditions on r. |
breaks |
An alternative to the argument r.
Not normally invoked by the user. See the Details section.
|
... |
Ignored. |
disjoint |
Optional flag indicating whether
the subsets I and J are disjoint.
If missing, this value will be computed by inspecting the
vectors I and J |
.
The function Jmulti
generalises Jest (for unmarked point
patterns) and Jdot and Jcross (for
multitype point patterns) to arbitrary marked point patterns.
Suppose X[I], X[J] are subsets, possibly overlapping, of a marked point process. Define
JIJ(r) = (1 - GIJ(r))/(1 - FJ(r))
where FJ(r) is the cumulative distribution function of the distance from a fixed location to the nearest point of X[J], and GJ(r) is the distribution function of the distance from a typical point of X[I] to the nearest distinct point of X[J].
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 logical vectors of length equal to
X$n, or integer vectors with entries in the range 1 to
X$n, etc.
It is assumed that X can be treated
as a realisation of a stationary (spatially homogeneous)
random spatial point process in the plane, observed through
a bounded window.
The window (which is specified in X as X$window)
may have arbitrary shape.
Biases due to edge effects are
treated in the same manner as in Jest.
The argument r is the vector of values for the
distance r at which JIJ(r) should be evaluated.
It is also used to determine the breakpoints
(in the sense of hist)
for the computation of histograms of distances. The reduced-sample and
Kaplan-Meier estimators are computed from histogram counts.
In the case of the Kaplan-Meier estimator this introduces a discretisation
error which is controlled by the fineness of the breakpoints.
First-time users would be strongly advised not to specify r.
However, if it is specified, r must satisfy r[1] = 0,
and max(r) must be larger than the radius of the largest disc
contained in the window. Furthermore, the successive entries of r
must be finely spaced.
An object of class "fv" (see fv.object).
Essentially a data frame containing six numeric columns
r |
the values of the argument r at which the function JIJ(r) has been estimated |
rs |
the ``reduced sample'' or ``border correction'' estimator of JIJ(r) |
km |
the spatial Kaplan-Meier estimator of JIJ(r) |
un |
the uncorrected estimate of JIJ(r),
formed by taking the ratio of uncorrected empirical estimators
of 1 - GIJ(r)
and 1 - FJ(r), see
Gdot and Fest.
|
theo |
the theoretical value of JIJ(r) for a marked Poisson process with the same estimated intensity, namely 1. |
Adrian Baddeley adrian@maths.uwa.edu.au http://www.maths.uwa.edu.au/~adrian/ and Rolf Turner rolf@math.unb.ca http://www.math.unb.ca/~rolf
Van Lieshout, M.N.M. and Baddeley, A.J. (1999) Indices of dependence between types in multivariate point patterns. Scandinavian Journal of Statistics 26, 511–532.
data(longleaf)
# Longleaf Pine data: marks represent diameter
Jm <- Jmulti(longleaf, longleaf$marks <= 15, longleaf$marks >= 25)
plot(Jm)