Description Usage Arguments Details Value Author(s) References See Also Examples
For a marked point pattern, estimate the multitype J function summarising dependence between the points in subset I and those in subset J.
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X 
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. 
I 
Subset of points of 
J 
Subset of points in 
eps 
A positive number.
The pixel resolution of the discrete approximation to Euclidean
distance (see 
r 
numeric vector. The values of the argument r at which the distribution function J[IJ](r) should be evaluated. There is a sensible default. Firsttime users are strongly advised not to specify this argument. See below for important conditions on r. 
breaks 
This argument is for internal use only. 
... 
Ignored. 
disjoint 
Optional flag indicating whether
the subsets 
correction 
Optional. Character string specifying the edge correction(s)
to be used. Options are 
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) = (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 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 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 reducedsample and
KaplanMeier estimators are computed from histogram counts.
In the case of the KaplanMeier estimator this introduces a discretisation
error which is controlled by the fineness of the breakpoints.
Firsttime 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 J[IJ](r) has been estimated 
rs 
the “reduced sample” or “border correction” estimator of J[IJ](r) 
km 
the spatial KaplanMeier estimator of J[IJ](r) 
han 
the Hanischstyle estimator of J[IJ](r) 
un 
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

theo 
the theoretical value of J[IJ](r) for a marked Poisson process with the same estimated intensity, namely 1. 
.
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.
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