For a multitype point pattern, estimate the multitype \(J\) function summarising the interpoint dependence between points of type \(i\) and of type \(j\).

`Jcross(X, i, j, eps=NULL, r=NULL, breaks=NULL, ..., correction=NULL)`

An object of class `"fv"`

(see `fv.object`

).

Essentially a data frame containing six numeric columns

- J
the recommended estimator of \(J_{ij}(r)\), currently the Kaplan-Meier estimator.

- r
the values of the argument \(r\) at which the function \(J_{ij}(r)\) has been estimated

- km
the Kaplan-Meier estimator of \(J_{ij}(r)\)

- rs
the ``reduced sample'' or ``border correction'' estimator of \(J_{ij}(r)\)

- han
the Hanisch-style estimator of \(J_{ij}(r)\)

- un
the ``uncorrected'' estimator 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, namely 1.

The result also has two attributes `"G"`

and `"F"`

which are respectively the outputs of `Gcross`

and `Fest`

for the point pattern.

- X
The observed point pattern, from which an estimate of the multitype \(J\) function \(J_{ij}(r)\) will be computed. It must be a multitype point pattern (a marked point pattern whose marks are a factor). See under Details.

- i
The type (mark value) of the points in

`X`

from which distances are measured. A character string (or something that will be converted to a character string). Defaults to the first level of`marks(X)`

.- j
The type (mark value) of the points in

`X`

to which distances are measured. A character string (or something that will be converted to a character string). Defaults to the second level of`marks(X)`

.- eps
A positive number. The resolution of the discrete approximation to Euclidean distance (see below). There is a sensible default.

- r
Optional. Numeric vector. The values of the argument \(r\) at which the 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.

- 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.

The arguments `i`

and `j`

are always interpreted as
levels of the factor `X$marks`

. They are converted to character
strings if they are not already character strings.
The value `i=1`

does **not**
refer to the first level of the factor.

Adrian Baddeley Adrian.Baddeley@curtin.edu.au, Rolf Turner r.turner@auckland.ac.nz and Ege Rubak rubak@math.aau.dk.

This function `Jcross`

and its companions
`Jdot`

and `Jmulti`

are generalisations of the function `Jest`

to multitype point patterns.

A multitype point pattern is a spatial pattern of points classified into a finite number of possible ``colours'' or ``types''. In the spatstat package, a multitype pattern is represented as a single point pattern object in which the points carry marks, and the mark value attached to each point determines the type of that point.

The argument `X`

must be a point pattern (object of class
`"ppp"`

) or any data that are acceptable to `as.ppp`

.
It must be a marked point pattern, and the mark vector
`X$marks`

must be a factor.
The argument `i`

will be interpreted as a
level of the factor `X$marks`

. (Warning: this means that
an integer value `i=3`

will be interpreted as the number 3,
**not** the 3rd smallest level).

The ``type \(i\) to type \(j\)'' multitype \(J\) function of a stationary multitype point process \(X\) was introduced by Van lieshout and Baddeley (1999). It is defined by $$J_{ij}(r) = \frac{1 - G_{ij}(r)}{1 - F_{j}(r)}$$ where \(G_{ij}(r)\) is the distribution function of the distance from a type \(i\) point to the nearest point of type \(j\), and \(F_{j}(r)\) is the distribution function of the distance from a fixed point in space to the nearest point of type \(j\) in the pattern.

An estimate of \(J_{ij}(r)\) is a useful summary statistic in exploratory data analysis of a multitype point pattern. If the subprocess of type \(i\) points is independent of the subprocess of points of type \(j\), then \(J_{ij}(r) \equiv 1\). Hence deviations of the empirical estimate of \(J_{ij}\) from the value 1 may suggest dependence between types.

This algorithm estimates \(J_{ij}(r)\)
from the point pattern `X`

. It assumes 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`

,
using the Kaplan-Meier and border corrections.
The main work is done by `Gmulti`

and `Fest`

.

The argument `r`

is the vector of values for the
distance \(r\) at which \(J_{ij}(r)\) should be evaluated.
The values of \(r\) must be increasing nonnegative numbers
and the maximum \(r\) value must not exceed the radius of the
largest disc contained in the window.

Van Lieshout, M.N.M. and Baddeley, A.J. (1996)
A nonparametric measure of spatial interaction in point patterns.
*Statistica Neerlandica* **50**, 344--361.

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.

`Jdot`

,
`Jest`

,
`Jmulti`

```
# Lansing woods data: 6 types of trees
woods <- lansing
# \testonly{
woods <- woods[seq(1,npoints(woods), by=30)]
# }
Jhm <- Jcross(woods, "hickory", "maple")
# diagnostic plot for independence between hickories and maples
plot(Jhm)
# synthetic example with two types "a" and "b"
pp <- runifpoint(30) %mark% factor(sample(c("a","b"), 30, replace=TRUE))
J <- Jcross(pp)
```

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