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

- 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 Kaplan-Meier estimator of \(J_{IJ}(r)\)

- han
the Hanisch-style 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

`Gdot`

and`Fest`

.- theo
the theoretical value of \(J_{IJ}(r)\) for a marked Poisson process with the same estimated intensity, namely 1.

- 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

`X`

from which distances are measured. See Details.- J
Subset of points in

`X`

to which distances are measured. See Details.- 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 \(J_{IJ}(r)\) 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 \(r\).

- breaks
This argument is for internal use only.

- ...
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`

.- correction
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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