# Jmulti

##### Marked J Function

For a marked point pattern,
estimate the multitype $J$ function
summarising dependence between the
points in subset `I`

and those in subset $J$.

- Keywords
- spatial, nonparametric

##### Usage

`Jmulti(X, I, J, eps=NULL, r=NULL, breaks=NULL, ..., disjoint=NULL)`

##### Arguments

- 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. - 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 $J_{IJ}(r)$ should be evaluated. There is a sensible default. First-time users are strongly advised not to specify this argument. See below for important
- 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`

##### Details

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 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 $J_{IJ}(r)$ 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.

##### Value

- 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 $J_{IJ}(r)$ 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.

##### References

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.

##### See Also

##### Examples

```
data(longleaf)
# Longleaf Pine data: marks represent diameter
<testonly>longleaf <- longleaf[seq(1,longleaf$n, by=50), ]</testonly>
Jm <- Jmulti(longleaf, longleaf$marks <= 15, longleaf$marks >= 25)
plot(Jm)
```

*Documentation reproduced from package spatstat, version 1.15-2, License: GPL (>= 2)*