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,
correction=NULL)An object of class "fv" (see fv.object).
Essentially a data frame containing six numeric columns
the values of the argument \(r\) at which the function \(J_{IJ}(r)\) has been estimated
the ``reduced sample'' or ``border correction'' estimator of \(J_{IJ}(r)\)
the spatial Kaplan-Meier estimator of \(J_{IJ}(r)\)
the Hanisch-style estimator of \(J_{IJ}(r)\)
the uncorrected estimate of \(J_{IJ}(r)\),
formed by taking the ratio of uncorrected empirical estimators
of \(1 - G_{IJ}(r)\)
and \(1 - F_{J}(r)\), see
Gdot and Fest.
the theoretical value of \(J_{IJ}(r)\) for a marked Poisson process with the same estimated intensity, namely 1.
The observed point pattern, from which an estimate of the multitype distance distribution function \(J_{IJ}(r)\) will be computed. It must be a marked point pattern. See under Details.
Subset of points of X from which distances are
measured. See Details.
Subset of points in X to which distances are measured.
See Details.
A positive number.
The pixel resolution of the discrete approximation to Euclidean
distance (see Jest). There is a sensible default.
numeric vector. The values of the argument \(r\) at which the distribution function \(J_{IJ}(r)\) 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 \(r\).
This argument is for internal use only.
Ignored.
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.
Optional. Character string specifying the edge correction(s)
to be used. Options are "none", "rs", "km",
"Hanisch" and "best".
Alternatively correction="all" selects all options.
Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk.
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 $$J_{IJ}(r) = \frac{1 - G_{IJ}(r)}{1 - F_J(r)}$$ where \(F_J(r)\) is the cumulative distribution function of the distance from a fixed location to the nearest point of \(X_J\), and \(G_{IJ}(r)\) 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 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.
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 Window(X))
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 \(J_{IJ}(r)\) should be evaluated.
It is also used to determine the breakpoints
(in the sense of hist)
for the computation of histograms of distances. 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.
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.
Jcross,
Jdot,
Jest
trees <- longleaf
# Longleaf Pine data: marks represent diameter
# \testonly{
trees <- trees[seq(1,npoints(trees), by=50)]
# }
Jm <- Jmulti(trees, marks(trees) <= 15, marks(trees) >= 25)
plot(Jm)
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